From 7deba5a0fc1fc69d9479c8d65b83ef4027d6c1aa Mon Sep 17 00:00:00 2001 From: leanprover-community-bot Date: Tue, 4 Apr 2023 16:42:48 +0000 Subject: [PATCH] bump to nightly-2023-04-04-16 mathlib commit https://github.com/leanprover-community/mathlib/commit/3cacc945118c8c637d89950af01da78307f59325 --- .../Calculus/LagrangeMultipliers.lean | 5 +- Mathbin/Analysis/InnerProductSpace/PiL2.lean | 9 +- .../InnerProductSpace/Projection.lean | 8 +- Mathbin/Analysis/NormedSpace/Dual.lean | 2 +- .../Analysis/NormedSpace/FiniteDimension.lean | 2 +- Mathbin/Analysis/NormedSpace/Multilinear.lean | 31 +- .../Analysis/NormedSpace/OperatorNorm.lean | 7 + Mathbin/LinearAlgebra/Isomorphisms.lean | 88 + Mathbin/Topology/Algebra/Module/Basic.lean | 1714 ++++++++++++++++- .../Algebra/Module/FiniteDimension.lean | 2 +- Mathbin/Topology/Algebra/Module/WeakDual.lean | 2 +- .../Topology/VectorBundle/Constructions.lean | 2 +- lake-manifest.json | 8 +- lakefile.lean | 4 +- 14 files changed, 1816 insertions(+), 68 deletions(-) diff --git a/Mathbin/Analysis/Calculus/LagrangeMultipliers.lean b/Mathbin/Analysis/Calculus/LagrangeMultipliers.lean index 8a0bbad7d7..45e07e159c 100644 --- a/Mathbin/Analysis/Calculus/LagrangeMultipliers.lean +++ b/Mathbin/Analysis/Calculus/LagrangeMultipliers.lean @@ -78,9 +78,8 @@ theorem IsLocalExtrOn.exists_linear_map_of_hasStrictFderivAt -- squeezed `simp [mul_comm]` to speed up elaboration simp only [mul_comm, Algebra.id.smul_eq_mul, LinearEquiv.trans_apply, LinearEquiv.prod_apply, LinearEquiv.refl_apply, LinearMap.ringLmapEquivSelf_symm_apply, LinearMap.coprodEquiv_apply, - ContinuousLinearMap.toLinearMap_eq_coe, ContinuousLinearMap.coe_prod, - LinearMap.coprod_comp_prod, LinearMap.add_apply, LinearMap.coe_comp, - ContinuousLinearMap.coe_coe, LinearMap.coe_smulRight, LinearMap.one_apply] + [anonymous], ContinuousLinearMap.coe_prod, LinearMap.coprod_comp_prod, LinearMap.add_apply, + LinearMap.coe_comp, ContinuousLinearMap.coe_coe, LinearMap.coe_smulRight, LinearMap.one_apply] #align is_local_extr_on.exists_linear_map_of_has_strict_fderiv_at IsLocalExtrOn.exists_linear_map_of_hasStrictFderivAt /-- Lagrange multipliers theorem: if `φ : E → ℝ` has a local extremum on the set `{x | f x = f x₀}` diff --git a/Mathbin/Analysis/InnerProductSpace/PiL2.lean b/Mathbin/Analysis/InnerProductSpace/PiL2.lean index 0617b3e5c3..fd9bac43f8 100644 --- a/Mathbin/Analysis/InnerProductSpace/PiL2.lean +++ b/Mathbin/Analysis/InnerProductSpace/PiL2.lean @@ -887,9 +887,9 @@ noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[ rw [← sq_eq_sq (norm_nonneg _) (norm_nonneg _), norm_sq_eq_add_norm_sq_projection x S] simp only [sq, Mx_decomp] rw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth] - simp only [LinearIsometry.norm_map, p1, p2, ContinuousLinearMap.toLinearMap_eq_coe, - add_left_inj, mul_eq_mul_left_iff, norm_eq_zero, true_or_iff, eq_self_iff_true, - ContinuousLinearMap.coe_coe, Submodule.coe_norm, Submodule.coe_eq_zero] + simp only [LinearIsometry.norm_map, p1, p2, [anonymous], add_left_inj, mul_eq_mul_left_iff, + norm_eq_zero, true_or_iff, eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm, + Submodule.coe_eq_zero] exact { toLinearMap := M norm_map' := M_norm_map } @@ -898,8 +898,7 @@ noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[ theorem LinearIsometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S) : L.extend s = L s := by haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S - simp only [LinearIsometry.extend, ContinuousLinearMap.toLinearMap_eq_coe, ← - LinearIsometry.coe_toLinearMap] + simp only [LinearIsometry.extend, [anonymous], ← LinearIsometry.coe_toLinearMap] simp only [add_right_eq_self, LinearIsometry.coe_toLinearMap, LinearIsometryEquiv.coe_toLinearIsometry, LinearIsometry.coe_comp, Function.comp_apply, orthogonalProjection_mem_subspace_eq_self, LinearMap.coe_comp, ContinuousLinearMap.coe_coe, diff --git a/Mathbin/Analysis/InnerProductSpace/Projection.lean b/Mathbin/Analysis/InnerProductSpace/Projection.lean index 7fce0105e5..8e49a5d966 100644 --- a/Mathbin/Analysis/InnerProductSpace/Projection.lean +++ b/Mathbin/Analysis/InnerProductSpace/Projection.lean @@ -705,8 +705,8 @@ def reflection : E ≃ₗᵢ[𝕜] E := convert norm_sub_eq_norm_add this using 2 · rw [LinearEquiv.coe_mk, reflectionLinearEquiv, LinearEquiv.toFun_eq_coe, LinearEquiv.coe_ofInvolutive, LinearMap.sub_apply, LinearMap.id_apply, bit0, - LinearMap.add_apply, LinearMap.comp_apply, Submodule.subtype_apply, - ContinuousLinearMap.toLinearMap_eq_coe, ContinuousLinearMap.coe_coe] + LinearMap.add_apply, LinearMap.comp_apply, Submodule.subtype_apply, [anonymous], + ContinuousLinearMap.coe_coe] dsimp [w, v] abel · simp only [add_sub_cancel'_right, eq_self_iff_true] } @@ -1220,8 +1220,8 @@ theorem LinearIsometryEquiv.reflections_generate_dim_aux [FiniteDimensional ℝ symm ext x have := LinearMap.congr_fun (linear_map.ker_eq_top.mp this) x - simpa only [sub_eq_zero, ContinuousLinearMap.toLinearMap_eq_coe, ContinuousLinearMap.coe_sub, - LinearMap.sub_apply, LinearMap.zero_apply] using this + simpa only [sub_eq_zero, [anonymous], ContinuousLinearMap.coe_sub, LinearMap.sub_apply, + LinearMap.zero_apply] using this · -- Inductive step. Let `W` be the fixed subspace of `φ`. We suppose its complement to have -- dimension at most n + 1. let W := ker (ContinuousLinearMap.id ℝ F - φ) diff --git a/Mathbin/Analysis/NormedSpace/Dual.lean b/Mathbin/Analysis/NormedSpace/Dual.lean index eaed7e6332..c49f1f79f1 100644 --- a/Mathbin/Analysis/NormedSpace/Dual.lean +++ b/Mathbin/Analysis/NormedSpace/Dual.lean @@ -102,7 +102,7 @@ theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ /-- The dual pairing as a bilinear form. -/ def dualPairing : Dual 𝕜 E →ₗ[𝕜] E →ₗ[𝕜] 𝕜 := - ContinuousLinearMap.coeLm 𝕜 + ContinuousLinearMap.coeLM 𝕜 #align normed_space.dual_pairing NormedSpace.dualPairing @[simp] diff --git a/Mathbin/Analysis/NormedSpace/FiniteDimension.lean b/Mathbin/Analysis/NormedSpace/FiniteDimension.lean index 34777f49f0..e096dc73d2 100644 --- a/Mathbin/Analysis/NormedSpace/FiniteDimension.lean +++ b/Mathbin/Analysis/NormedSpace/FiniteDimension.lean @@ -178,7 +178,7 @@ theorem ContinuousLinearMap.continuous_det : Continuous fun f : E →L[𝕜] E = refine' Continuous.matrix_det _ exact ((LinearMap.toMatrix b b).toLinearMap.comp - (ContinuousLinearMap.coeLm 𝕜)).continuous_of_finiteDimensional + (ContinuousLinearMap.coeLM 𝕜)).continuous_of_finiteDimensional · unfold LinearMap.det simpa only [h, MonoidHom.one_apply, dif_neg, not_false_iff] using continuous_const #align continuous_linear_map.continuous_det ContinuousLinearMap.continuous_det diff --git a/Mathbin/Analysis/NormedSpace/Multilinear.lean b/Mathbin/Analysis/NormedSpace/Multilinear.lean index 6e560a7876..85ac3853b3 100644 --- a/Mathbin/Analysis/NormedSpace/Multilinear.lean +++ b/Mathbin/Analysis/NormedSpace/Multilinear.lean @@ -1027,17 +1027,17 @@ def ContinuousLinearEquiv.compContinuousMultilinearMapL (g : G ≃L[𝕜] G') : left_inv := by intro f ext1 m - simp only [comp_continuous_multilinear_mapL, ContinuousLinearEquiv.coe_def_rev, - to_linear_map_eq_coe, LinearMap.toFun_eq_coe, coe_coe, LinearMap.mkContinuous₂_apply, - LinearMap.mk₂_apply, comp_continuous_multilinear_map_coe, ContinuousLinearEquiv.coe_coe, - Function.comp_apply, ContinuousLinearEquiv.symm_apply_apply] + simp only [comp_continuous_multilinear_mapL, [anonymous], to_linear_map_eq_coe, + LinearMap.toFun_eq_coe, coe_coe, LinearMap.mkContinuous₂_apply, LinearMap.mk₂_apply, + comp_continuous_multilinear_map_coe, ContinuousLinearEquiv.coe_coe, Function.comp_apply, + ContinuousLinearEquiv.symm_apply_apply] right_inv := by intro f ext1 m - simp only [comp_continuous_multilinear_mapL, ContinuousLinearEquiv.coe_def_rev, - to_linear_map_eq_coe, LinearMap.mkContinuous₂_apply, LinearMap.mk₂_apply, - LinearMap.toFun_eq_coe, coe_coe, comp_continuous_multilinear_map_coe, - ContinuousLinearEquiv.coe_coe, Function.comp_apply, ContinuousLinearEquiv.apply_symm_apply] + simp only [comp_continuous_multilinear_mapL, [anonymous], to_linear_map_eq_coe, + LinearMap.mkContinuous₂_apply, LinearMap.mk₂_apply, LinearMap.toFun_eq_coe, coe_coe, + comp_continuous_multilinear_map_coe, ContinuousLinearEquiv.coe_coe, Function.comp_apply, + ContinuousLinearEquiv.apply_symm_apply] continuous_toFun := (compContinuousMultilinearMapL 𝕜 _ _ _ g.toContinuousLinearMap).Continuous continuous_invFun := (compContinuousMultilinearMapL 𝕜 _ _ _ g.symm.toContinuousLinearMap).Continuous } @@ -1201,8 +1201,8 @@ theorem norm_comp_continuous_linearIsometry_le (g : ContinuousMultilinearMap by apply op_norm_le_bound _ (norm_nonneg _) fun m => _ apply (g.le_op_norm _).trans _ - simp only [ContinuousLinearMap.toLinearMap_eq_coe, ContinuousLinearMap.coe_coe, - LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.norm_map] + simp only [[anonymous], ContinuousLinearMap.coe_coe, LinearIsometry.coe_toContinuousLinearMap, + LinearIsometry.norm_map] #align continuous_multilinear_map.norm_comp_continuous_linear_isometry_le ContinuousMultilinearMap.norm_comp_continuous_linearIsometry_le theorem norm_comp_continuous_linearIsometryEquiv (g : ContinuousMultilinearMap 𝕜 E₁ G) @@ -1269,15 +1269,14 @@ def compContinuousLinearMapEquivL (f : ∀ i, E i ≃L[𝕜] E₁ i) : left_inv := by intro g ext1 m - simp only [ContinuousLinearMap.toLinearMap_eq_coe, LinearMap.toFun_eq_coe, - ContinuousLinearMap.coe_coe, comp_continuous_linear_mapL_apply, - comp_continuous_linear_map_apply, ContinuousLinearEquiv.coe_coe, - ContinuousLinearEquiv.apply_symm_apply] + simp only [[anonymous], LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe, + comp_continuous_linear_mapL_apply, comp_continuous_linear_map_apply, + ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply] right_inv := by intro g ext1 m - simp only [ContinuousLinearMap.toLinearMap_eq_coe, comp_continuous_linear_mapL_apply, - LinearMap.toFun_eq_coe, ContinuousLinearMap.coe_coe, comp_continuous_linear_map_apply, + simp only [[anonymous], comp_continuous_linear_mapL_apply, LinearMap.toFun_eq_coe, + ContinuousLinearMap.coe_coe, comp_continuous_linear_map_apply, ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_apply_apply] } #align continuous_multilinear_map.comp_continuous_linear_map_equivL ContinuousMultilinearMap.compContinuousLinearMapEquivL diff --git a/Mathbin/Analysis/NormedSpace/OperatorNorm.lean b/Mathbin/Analysis/NormedSpace/OperatorNorm.lean index 644df81f28..2f7742c4ac 100644 --- a/Mathbin/Analysis/NormedSpace/OperatorNorm.lean +++ b/Mathbin/Analysis/NormedSpace/OperatorNorm.lean @@ -1084,6 +1084,13 @@ section NonUnital variable (𝕜) (𝕜' : Type _) [NonUnitalSeminormedRing 𝕜'] [NormedSpace 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' 𝕜'] [SMulCommClass 𝕜 𝕜' 𝕜'] +/- warning: continuous_linear_map.mul clashes with continuous_linear_map.has_mul -> ContinuousLinearMap.mul +warning: continuous_linear_map.mul -> ContinuousLinearMap.mul is a dubious translation: +lean 3 declaration is + forall (𝕜 : Type.{u1}) [_inst_7 : NontriviallyNormedField.{u1} 𝕜] (𝕜' : Type.{u2}) [_inst_20 : NonUnitalSeminormedRing.{u2} 𝕜'] [_inst_21 : NormedSpace.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)] [_inst_22 : IsScalarTower.{u1, u2, u2} 𝕜 𝕜' 𝕜' (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 𝕜' (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 𝕜' (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 𝕜' (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7)))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 𝕜' (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21))))) (Mul.toSMul.{u2} 𝕜' (Distrib.toHasMul.{u2} 𝕜' (NonUnitalNonAssocSemiring.toDistrib.{u2} 𝕜' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜' (NonUnitalRing.toNonUnitalNonAssocRing.{u2} 𝕜' (NonUnitalSeminormedRing.toNonUnitalRing.{u2} 𝕜' _inst_20)))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 𝕜' (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 𝕜' (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 𝕜' (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7)))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 𝕜' (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21)))))] [_inst_23 : SMulCommClass.{u1, u2, u2} 𝕜 𝕜' 𝕜' (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 𝕜' (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 𝕜' (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 𝕜' (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7)))))) (AddZeroClass.toHasZero.{u2} 𝕜' (AddMonoid.toAddZeroClass.{u2} 𝕜' (AddCommMonoid.toAddMonoid.{u2} 𝕜' (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 𝕜' (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21))))) (Mul.toSMul.{u2} 𝕜' (Distrib.toHasMul.{u2} 𝕜' (NonUnitalNonAssocSemiring.toDistrib.{u2} 𝕜' (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} 𝕜' (NonUnitalRing.toNonUnitalNonAssocRing.{u2} 𝕜' (NonUnitalSeminormedRing.toNonUnitalRing.{u2} 𝕜' _inst_20))))))], ContinuousLinearMap.{u1, u1, u2, u2} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (ContinuousLinearMap.{u1, u1, u2, u2} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21)) (ContinuousLinearMap.topologicalSpace.{u1, u1, u2, u2} 𝕜 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))) 𝕜' 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20)) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (ContinuousLinearMap.Mul._proof_1.{u2} 𝕜' _inst_20)) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u2} 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (ContinuousLinearMap.Mul._proof_2.{u2} 𝕜' _inst_20)) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (ContinuousLinearMap.module.{u1, u1, u1, u2, u2} 𝕜 𝕜 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) 𝕜' (UniformSpace.toTopologicalSpace.{u2} 𝕜' (PseudoMetricSpace.toUniformSpace.{u2} 𝕜' (NonUnitalSeminormedRing.toPseudoMetricSpace.{u2} 𝕜' _inst_20))) (AddCommGroup.toAddCommMonoid.{u2} 𝕜' (SeminormedAddCommGroup.toAddCommGroup.{u2} 𝕜' (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20))) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (NormedSpace.toModule.{u1, u2} 𝕜 𝕜' (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u2} 𝕜' _inst_20) _inst_21) (ContinuousLinearMap.Mul._proof_3.{u1, u2} 𝕜 _inst_7 𝕜' _inst_20 _inst_21) (ContinuousLinearMap.Mul._proof_4.{u1, u2} 𝕜 _inst_7 𝕜' _inst_20 _inst_21) (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (NontriviallyNormedField.toNormedField.{u1} 𝕜 _inst_7))))))) (ContinuousLinearMap.Mul._proof_5.{u2} 𝕜' _inst_20)) +but is expected to have type + forall {𝕜 : Type.{u1}} [_inst_7 : Semiring.{u1} 𝕜] {𝕜' : Type.{u2}} [_inst_20 : TopologicalSpace.{u2} 𝕜'] [_inst_21 : AddCommMonoid.{u2} 𝕜'] [_inst_22 : Module.{u1, u2} 𝕜 𝕜' _inst_7 _inst_21], Mul.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} 𝕜 𝕜 _inst_7 _inst_7 (RingHom.id.{u1} 𝕜 (Semiring.toNonAssocSemiring.{u1} 𝕜 _inst_7)) 𝕜' _inst_20 _inst_21 𝕜' _inst_20 _inst_21 _inst_22 _inst_22) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.mul ContinuousLinearMap.mulₓ'. -/ /-- Multiplication in a non-unital normed algebra as a continuous bilinear map. -/ def mul : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' := (LinearMap.mul 𝕜 𝕜').mkContinuous₂ 1 fun x y => by simpa using norm_mul_le x y diff --git a/Mathbin/LinearAlgebra/Isomorphisms.lean b/Mathbin/LinearAlgebra/Isomorphisms.lean index 6cd69df4da..83b31e12ff 100644 --- a/Mathbin/LinearAlgebra/Isomorphisms.lean +++ b/Mathbin/LinearAlgebra/Isomorphisms.lean @@ -39,6 +39,12 @@ open Submodule section IsomorphismLaws +/- warning: linear_map.quot_ker_equiv_range -> LinearMap.quotKerEquivRange is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6), LinearEquiv.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.quotKerEquivRange._proof_1.{u1} R _inst_1) (LinearMap.quotKerEquivRange._proof_2.{u1} R _inst_1) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6) M₂ (Submodule.setLike.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6)) (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (LinearMap.quotKerEquivRange._proof_3.{u1} R _inst_1) f)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) (Submodule.addCommMonoid.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6 (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (LinearMap.quotKerEquivRange._proof_3.{u1} R _inst_1) f)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6 (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (LinearMap.quotKerEquivRange._proof_3.{u1} R _inst_1) f)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6), LinearEquiv.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Subtype.{succ u3} M₂ (fun (x : M₂) => Membership.mem.{u3, u3} M₂ (Submodule.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6) (SetLike.instMembership.{u3, u3} (Submodule.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6) M₂ (Submodule.setLike.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6)) x (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) f))) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) (Submodule.addCommMonoid.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6 (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) f)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_6 (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) f)) +Case conversion may be inaccurate. Consider using '#align linear_map.quot_ker_equiv_range LinearMap.quotKerEquivRangeₓ'. -/ /-- The first isomorphism law for modules. The quotient of `M` by the kernel of `f` is linearly equivalent to the range of `f`. -/ noncomputable def quotKerEquivRange : (M ⧸ f.ker) ≃ₗ[R] f.range := @@ -47,24 +53,43 @@ noncomputable def quotKerEquivRange : (M ⧸ f.ker) ≃ₗ[R] f.range := (LinearEquiv.ofEq _ _ <| Submodule.range_liftQ _ _ _) #align linear_map.quot_ker_equiv_range LinearMap.quotKerEquivRange +/- warning: linear_map.quot_ker_equiv_of_surjective -> LinearMap.quotKerEquivOfSurjective is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R 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Consider using '#align linear_map.quot_ker_equiv_of_surjective LinearMap.quotKerEquivOfSurjectiveₓ'. -/ /-- The first isomorphism theorem for surjective linear maps. -/ noncomputable def quotKerEquivOfSurjective (f : M →ₗ[R] M₂) (hf : Function.Surjective f) : (M ⧸ f.ker) ≃ₗ[R] M₂ := f.quotKerEquivRange.trans (LinearEquiv.ofTop f.range (LinearMap.range_eq_top.2 hf)) #align linear_map.quot_ker_equiv_of_surjective LinearMap.quotKerEquivOfSurjective +/- warning: linear_map.quot_ker_equiv_range_apply_mk -> LinearMap.quotKerEquivRange_apply_mk is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R 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M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) 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(Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) 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(Ring.toSemiring.{u2} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)))) (Module.toDistribMulAction.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, 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Consider using '#align linear_map.quot_ker_equiv_range_apply_mk LinearMap.quotKerEquivRange_apply_mkₓ'. -/ @[simp] theorem quotKerEquivRange_apply_mk (x : M) : (f.quotKerEquivRange (Submodule.Quotient.mk x) : M₂) = f x := rfl #align linear_map.quot_ker_equiv_range_apply_mk LinearMap.quotKerEquivRange_apply_mk +/- warning: linear_map.quot_ker_equiv_range_symm_apply_image -> LinearMap.quotKerEquivRange_symm_apply_image is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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(AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) (fun (_x : LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) => M -> (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) (LinearMap.hasCoeToFun.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f))) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) x) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] [_inst_6 : Module.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] (f : LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) 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_inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} 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_inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) 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_inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f))))) (DistribMulAction.toDistribSMul.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) 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_inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (Submodule.mkQ.{u2, u1} R M _inst_1 _inst_2 _inst_5 (LinearMap.ker.{u2, u2, u1, u3, max u1 u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u1, u3} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M M₂ (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u3} R R M M₂ (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R 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Consider using '#align linear_map.quot_ker_equiv_range_symm_apply_image LinearMap.quotKerEquivRange_symm_apply_imageₓ'. -/ @[simp] theorem quotKerEquivRange_symm_apply_image (x : M) (h : f x ∈ f.range) : f.quotKerEquivRange.symm ⟨f x, h⟩ = f.ker.mkQ x := f.quotKerEquivRange.symm_apply_apply (f.ker.mkQ x) #align linear_map.quot_ker_equiv_range_symm_apply_image LinearMap.quotKerEquivRange_symm_apply_image +#print LinearMap.quotientInfToSupQuotient /- /-- Canonical linear map from the quotient `p/(p ∩ p')` to `(p+p')/p'`, mapping `x + (p ∩ p')` to `x + p'`, where `p` and `p'` are submodules of an ambient module. -/ @@ -75,7 +100,9 @@ def quotientInfToSupQuotient (p p' : Submodule R M) : rw [ker_comp, of_le, comap_cod_restrict, ker_mkq, map_comap_subtype] exact comap_mono (inf_le_inf_right _ le_sup_left)) #align linear_map.quotient_inf_to_sup_quotient LinearMap.quotientInfToSupQuotient +-/ +#print LinearMap.quotientInfEquivSupQuotient /- /-- Second Isomorphism Law : the canonical map from `p/(p ∩ p')` to `(p+p')/p'` as a linear isomorphism. -/ @@ -93,13 +120,26 @@ noncomputable def quotientInfEquivSupQuotient (p p' : Submodule R M) : change y - (y + z) ∈ p' rwa [sub_add_eq_sub_sub, sub_self, zero_sub, neg_mem_iff]⟩ #align linear_map.quotient_inf_equiv_sup_quotient LinearMap.quotientInfEquivSupQuotient +-/ +/- warning: linear_map.coe_quotient_inf_to_sup_quotient -> LinearMap.coe_quotientInfToSupQuotient is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (p' : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5), Eq.{succ u2} ((HasQuotient.Quotient.{u2, u2} (coeSort.{succ u2, succ (succ u2)} 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Consider using '#align linear_map.coe_quotient_inf_to_sup_quotient LinearMap.coe_quotientInfToSupQuotientₓ'. -/ @[simp] theorem coe_quotientInfToSupQuotient (p p' : Submodule R M) : ⇑(quotientInfToSupQuotient p p') = quotientInfEquivSupQuotient p p' := rfl #align linear_map.coe_quotient_inf_to_sup_quotient LinearMap.coe_quotientInfToSupQuotient +/- warning: linear_map.quotient_inf_equiv_sup_quotient_apply_mk -> LinearMap.quotientInfEquivSupQuotient_apply_mk is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (p' : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (x : coeSort.{succ u2, succ (succ u2)} 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) p p'))) (LinearMap.hasCoeToFun.{u1, u1, u2, u2} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5)) p) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5)) (Sup.sup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) p p')) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5 p) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5 (Sup.sup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) p p')) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5 p) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5 (Sup.sup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) p p')) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.ofLe.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5 p (Sup.sup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) p p') (le_sup_left.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5)))) p p')) x)) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] (p : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (p' : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (x : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x p)), let map : LinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x p)) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p'))) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 p) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p')) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 p) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p')) := Submodule.ofLe.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 p (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p') (le_sup_left.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)))) p p'); Eq.{succ u1} (HasQuotient.Quotient.{u1, u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p'))) (Submodule.{u2, u1} R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p'))) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p')) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M 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Consider using '#align linear_map.quotient_inf_equiv_sup_quotient_apply_mk LinearMap.quotientInfEquivSupQuotient_apply_mkₓ'. -/ @[simp] theorem quotientInfEquivSupQuotient_apply_mk (p p' : Submodule R M) (x : p) : quotientInfEquivSupQuotient p p' (Submodule.Quotient.mk x) = @@ -107,6 +147,12 @@ theorem quotientInfEquivSupQuotient_apply_mk (p p' : Submodule R M) (x : p) : rfl #align linear_map.quotient_inf_equiv_sup_quotient_apply_mk LinearMap.quotientInfEquivSupQuotient_apply_mk +/- warning: linear_map.quotient_inf_equiv_sup_quotient_symm_apply_left -> LinearMap.quotientInfEquivSupQuotient_symm_apply_left is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (p' : 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(Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (Submodule.subtype.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5 (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) 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(AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) p p')) (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5)) x p) (Subtype.val.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) x (SetLike.coe.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Sup.sup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Lattice.toSemilatticeSup.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) p p'))) x) hx)) +Case conversion may be inaccurate. Consider using '#align linear_map.quotient_inf_equiv_sup_quotient_symm_apply_left LinearMap.quotientInfEquivSupQuotient_symm_apply_leftₓ'. -/ theorem quotientInfEquivSupQuotient_symm_apply_left (p p' : Submodule R M) (x : p ⊔ p') (hx : (x : M) ∈ p) : (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = @@ -114,12 +160,24 @@ theorem quotientInfEquivSupQuotient_symm_apply_left (p p' : Submodule R M) (x : (LinearEquiv.symm_apply_eq _).2 <| by simp [of_le_apply] #align linear_map.quotient_inf_equiv_sup_quotient_symm_apply_left LinearMap.quotientInfEquivSupQuotient_symm_apply_left +/- warning: linear_map.quotient_inf_equiv_sup_quotient_symm_apply_eq_zero_iff -> LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {p 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Consider using '#align linear_map.quotient_inf_equiv_sup_quotient_symm_apply_eq_zero_iff LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iffₓ'. -/ @[simp] theorem quotientInfEquivSupQuotient_symm_apply_eq_zero_iff {p p' : Submodule R M} {x : p ⊔ p'} : (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 ↔ (x : M) ∈ p' := (LinearEquiv.symm_apply_eq _).trans <| by simp [of_le_apply] #align linear_map.quotient_inf_equiv_sup_quotient_symm_apply_eq_zero_iff LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff +/- warning: linear_map.quotient_inf_equiv_sup_quotient_symm_apply_right -> LinearMap.quotientInfEquivSupQuotient_symm_apply_right is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) 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Consider using '#align linear_map.quotient_inf_equiv_sup_quotient_symm_apply_right LinearMap.quotientInfEquivSupQuotient_symm_apply_rightₓ'. -/ theorem quotientInfEquivSupQuotient_symm_apply_right (p p' : Submodule R M) {x : p ⊔ p'} (hx : (x : M) ∈ p') : (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 := quotientInfEquivSupQuotient_symm_apply_eq_zero_iff.2 hx @@ -136,6 +194,12 @@ namespace Submodule variable (S T : Submodule R M) (h : S ≤ T) +/- warning: submodule.quotient_quotient_equiv_quotient_aux -> Submodule.quotientQuotientEquivQuotientAux is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R 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(Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.hasQuotient.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) 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(RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.addCommGroup.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, 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R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T))) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.quotientQuotientEquivQuotientAux._proof_1.{u1} R _inst_1) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5), (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) S T) -> (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (HasQuotient.Quotient.{u2, u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.hasQuotient.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.hasQuotient.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} 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S) T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) +Case conversion may be inaccurate. Consider using '#align submodule.quotient_quotient_equiv_quotient_aux Submodule.quotientQuotientEquivQuotientAuxₓ'. -/ /-- The map from the third isomorphism theorem for modules: `(M / S) / (T / S) → M / T`. -/ def quotientQuotientEquivQuotientAux (h : S ≤ T) : (M ⧸ S) ⧸ T.map S.mkQ →ₗ[R] M ⧸ T := liftq _ (mapq S T LinearMap.id h) @@ -145,18 +209,36 @@ def quotientQuotientEquivQuotientAux (h : S ≤ T) : (M ⧸ S) ⧸ T.map S.mkQ exact (quotient.mk_eq_zero _).mpr hx) #align submodule.quotient_quotient_equiv_quotient_aux Submodule.quotientQuotientEquivQuotientAux +/- warning: submodule.quotient_quotient_equiv_quotient_aux_mk -> Submodule.quotientQuotientEquivQuotientAux_mk is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) 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(Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.quotientQuotientEquivQuotientAux.{u1, u2} R M _inst_1 _inst_2 _inst_5 S T h) (Submodule.Quotient.mk.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.quotientQuotientEquivQuotientAux._proof_1.{u1} R _inst_1) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R 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(Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mapQ.{u1, u2, u1, u2} R M _inst_1 _inst_2 _inst_5 S R M _inst_1 _inst_2 _inst_5 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) T (LinearMap.id.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) h) x) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (h : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) S T) (x : HasQuotient.Quotient.{u2, u2} M 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(Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) T (LinearMap.id.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) h) x) +Case conversion may be inaccurate. Consider using '#align submodule.quotient_quotient_equiv_quotient_aux_mk Submodule.quotientQuotientEquivQuotientAux_mkₓ'. -/ @[simp] theorem quotientQuotientEquivQuotientAux_mk (x : M ⧸ S) : quotientQuotientEquivQuotientAux S T h (Quotient.mk x) = mapQ S T LinearMap.id h x := liftQ_apply _ _ _ #align submodule.quotient_quotient_equiv_quotient_aux_mk Submodule.quotientQuotientEquivQuotientAux_mk +/- warning: submodule.quotient_quotient_equiv_quotient_aux_mk_mk -> Submodule.quotientQuotientEquivQuotientAux_mk_mk is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) 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_inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T) (Submodule.Quotient.mk.{u1, u2} R M _inst_1 _inst_2 _inst_5 S x))) (Submodule.Quotient.mk.{u1, u2} R M _inst_1 _inst_2 _inst_5 T x) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (h : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M 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Consider using '#align submodule.quotient_quotient_equiv_quotient_aux_mk_mk Submodule.quotientQuotientEquivQuotientAux_mk_mkₓ'. -/ @[simp] theorem quotientQuotientEquivQuotientAux_mk_mk (x : M) : quotientQuotientEquivQuotientAux S T h (Quotient.mk (Quotient.mk x)) = Quotient.mk x := by rw [quotient_quotient_equiv_quotient_aux_mk, mapq_apply, LinearMap.id_apply] #align submodule.quotient_quotient_equiv_quotient_aux_mk_mk Submodule.quotientQuotientEquivQuotientAux_mk_mk +/- warning: submodule.quotient_quotient_equiv_quotient -> Submodule.quotientQuotientEquivQuotient is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R 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_inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M 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S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.addCommGroup.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.quotientQuotientEquivQuotient._proof_3.{u1} R _inst_1) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T))) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.quotientQuotientEquivQuotient._proof_3.{u1} R _inst_1) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5), (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5))))) S T) -> (LinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{u2, u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.hasQuotient.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (Submodule.hasQuotient.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 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_inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 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(HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_1 (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 S)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 S) T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) +Case conversion may be inaccurate. Consider using '#align submodule.quotient_quotient_equiv_quotient Submodule.quotientQuotientEquivQuotientₓ'. -/ /-- **Noether's third isomorphism theorem** for modules: `(M / S) / (T / S) ≃ M / T`. -/ def quotientQuotientEquivQuotient : ((M ⧸ S) ⧸ T.map S.mkQ) ≃ₗ[R] M ⧸ T := { @@ -168,6 +250,12 @@ def quotientQuotientEquivQuotient : ((M ⧸ S) ⧸ T.map S.mkQ) ≃ₗ[R] M ⧸ right_inv := fun x => Quotient.inductionOn' x fun x => by simp } #align submodule.quotient_quotient_equiv_quotient Submodule.quotientQuotientEquivQuotient +/- warning: submodule.card_quotient_mul_card_quotient -> Submodule.card_quotient_mul_card_quotient is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (T 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) S))] [_inst_9 : Fintype.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S)] [_inst_10 : Fintype.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T)], Eq.{1} Nat (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) (Fintype.card.{u2} (coeSort.{succ u2, 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M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.setLike.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T))) x (Submodule.map.{u1, u1, u2, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) (LinearMap.semilinearMapClass.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_5 T) S)) (fun (a : HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) => _inst_8 a) _inst_10)) (Fintype.card.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) S) _inst_9)) (Fintype.card.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_5) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_5) T) _inst_10)) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] (S : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (T : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5), (LE.le.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5))))) T S) -> (forall [_inst_8 : DecidablePred.{succ u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (fun (x : HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.setLike.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T))) x (Submodule.map.{u2, u2, u1, u1, u1} R R M (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u1, u1} R R M (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) _inst_5 (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (Submodule.mkQ.{u2, u1} R M _inst_1 _inst_2 _inst_5 T) S))] [_inst_9 : Fintype.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) S)] [_inst_10 : Fintype.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T)], Eq.{1} Nat (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (Fintype.card.{u1} (Subtype.{succ u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (fun (x : HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) => Membership.mem.{u1, u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) 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(AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.setLike.{u2, u1} R (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M 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T)) _inst_5 (Submodule.Quotient.module.{u2, u1} R M _inst_1 _inst_2 _inst_5 T) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) (Submodule.Quotient.addCommGroup.{u2, u1} R M _inst_1 _inst_2 _inst_5 T)) 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T) S)) (fun (a : HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) => _inst_8 a) _inst_10)) (Fintype.card.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) S) _inst_9)) (Fintype.card.{u1} (HasQuotient.Quotient.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_5) (Submodule.hasQuotient.{u2, u1} R M _inst_1 _inst_2 _inst_5) T) _inst_10)) +Case conversion may be inaccurate. Consider using '#align submodule.card_quotient_mul_card_quotient Submodule.card_quotient_mul_card_quotientₓ'. -/ /-- Corollary of the third isomorphism theorem: `[S : T] [M : S] = [M : T]` -/ theorem card_quotient_mul_card_quotient (S T : Submodule R M) (hST : T ≤ S) [DecidablePred fun x => x ∈ S.map T.mkQ] [Fintype (M ⧸ S)] [Fintype (M ⧸ T)] : diff --git a/Mathbin/Topology/Algebra/Module/Basic.lean b/Mathbin/Topology/Algebra/Module/Basic.lean index 599fa3e907..e6f4e4322f 100644 --- a/Mathbin/Topology/Algebra/Module/Basic.lean +++ b/Mathbin/Topology/Algebra/Module/Basic.lean @@ -45,6 +45,12 @@ section variable {R : Type _} {M : Type _} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [Module R M] +/- warning: has_continuous_smul.of_nhds_zero -> ContinuousSMul.of_nhds_zero is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : TopologicalRing.{u1} R _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))] [_inst_7 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)], (Filter.Tendsto.{max u1 u2, u2} (Prod.{u1, u2} R M) M (fun (p : Prod.{u1, u2} R M) => SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) (Prod.fst.{u1, u2} R M p) (Prod.snd.{u1, u2} R M p)) (Filter.prod.{u1, u2} R M (nhds.{u1} R _inst_2 (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (nhds.{u2} M _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))))))) (nhds.{u2} M _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))))))) -> (forall (m : M), Filter.Tendsto.{u1, u2} R M (fun (a : R) => SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) a m) (nhds.{u1} R _inst_2 (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (nhds.{u2} M _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))))))) -> (forall (a : R), Filter.Tendsto.{u2, u2} M M (fun (m : M) => SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) a m) (nhds.{u2} M _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))))))) (nhds.{u2} M _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))))))) -> (ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) _inst_2 _inst_3) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : TopologicalSpace.{u2} R] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : TopologicalRing.{u2} R _inst_2 (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))] [_inst_7 : TopologicalAddGroup.{u1} M _inst_3 (AddCommGroup.toAddGroup.{u1} M _inst_4)], (Filter.Tendsto.{max u2 u1, u1} (Prod.{u2, u1} R M) M (fun (p : Prod.{u2, u1} R M) => HSMul.hSMul.{u2, u1, u1} R M M (instHSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) (Prod.fst.{u2, u1} R M p) (Prod.snd.{u2, u1} R M p)) (Filter.prod.{u2, u1} R M (nhds.{u2} R _inst_2 (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) (nhds.{u1} M _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))))))) (nhds.{u1} M _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))))))) -> (forall (m : M), Filter.Tendsto.{u2, u1} R M (fun (a : R) => HSMul.hSMul.{u2, u1, u1} R M M (instHSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) a m) (nhds.{u2} R _inst_2 (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) (nhds.{u1} M _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))))))) -> (forall (a : R), Filter.Tendsto.{u1, u1} M M (fun (m : M) => HSMul.hSMul.{u2, u1, u1} R M M (instHSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) a m) (nhds.{u1} M _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4)))))))) (nhds.{u1} M _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))))))) -> (ContinuousSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) _inst_2 _inst_3) +Case conversion may be inaccurate. Consider using '#align has_continuous_smul.of_nhds_zero ContinuousSMul.of_nhds_zeroₓ'. -/ theorem ContinuousSMul.of_nhds_zero [TopologicalRing R] [TopologicalAddGroup M] (hmul : Tendsto (fun p : R × M => p.1 • p.2) (𝓝 0 ×ᶠ 𝓝 0) (𝓝 0)) (hmulleft : ∀ m : M, Tendsto (fun a : R => a • m) (𝓝 0) (𝓝 0)) @@ -105,6 +111,12 @@ section variable {R : Type _} {M : Type _} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] +/- warning: submodule.eq_top_of_nonempty_interior' -> Submodule.eq_top_of_nonempty_interior' is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_7 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6)))) _inst_2 _inst_3] [_inst_8 : Filter.NeBot.{u1} R (nhdsWithin.{u1} R _inst_2 (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (setOf.{u1} R (fun (x : R) => IsUnit.{u1} R (Ring.toMonoid.{u1} R _inst_1) x)))] (s : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6), (Set.Nonempty.{u2} M (interior.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6)))) s))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) s (Top.top.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) (Submodule.hasTop.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6))) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : TopologicalSpace.{u2} R] [_inst_3 : TopologicalSpace.{u1} M] [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : ContinuousAdd.{u1} M _inst_3 (AddZeroClass.toAdd.{u1} M (AddMonoid.toAddZeroClass.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_4)))))] [_inst_6 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_7 : ContinuousSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6)))) _inst_2 _inst_3] [_inst_8 : Filter.NeBot.{u2} R (nhdsWithin.{u2} R _inst_2 (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))))) (setOf.{u2} R (fun (x : R) => IsUnit.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1))) x)))] (s : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6), (Set.Nonempty.{u1} M (interior.{u1} M _inst_3 (SetLike.coe.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6) s))) -> (Eq.{succ u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6) s (Top.top.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6) (Submodule.instTopSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_6))) +Case conversion may be inaccurate. Consider using '#align submodule.eq_top_of_nonempty_interior' Submodule.eq_top_of_nonempty_interior'ₓ'. -/ /-- If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then `⊤` is the only submodule of `M` with a nonempty interior. This is the case, e.g., if `R` is a nontrivially normed field. -/ @@ -125,6 +137,12 @@ theorem Submodule.eq_top_of_nonempty_interior' [NeBot (𝓝[{ x : R | IsUnit x } variable (R M) +/- warning: module.punctured_nhds_ne_bot -> Module.punctured_nhds_neBot is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : Ring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_7 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6)))) _inst_2 _inst_3] [_inst_8 : Nontrivial.{u2} M] [_inst_9 : Filter.NeBot.{u1} R (nhdsWithin.{u1} R _inst_2 (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) (HasCompl.compl.{u1} (Set.{u1} R) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} R) (Set.booleanAlgebra.{u1} R)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))))))] [_inst_10 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6))))] (x : M), Filter.NeBot.{u2} M (nhdsWithin.{u2} M _inst_3 x (HasCompl.compl.{u2} (Set.{u2} M) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} M) (Set.booleanAlgebra.{u2} M)) (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) x))) +but is expected to have type + forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : Ring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_7 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6)))) _inst_2 _inst_3] [_inst_8 : Nontrivial.{u2} M] [_inst_9 : Filter.NeBot.{u1} R (nhdsWithin.{u1} R _inst_2 (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (HasCompl.compl.{u1} (Set.{u1} R) (BooleanAlgebra.toHasCompl.{u1} (Set.{u1} R) (Set.instBooleanAlgebraSet.{u1} R)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))))))] [_inst_10 : NoZeroSMulDivisors.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6))))] (x : M), Filter.NeBot.{u2} M (nhdsWithin.{u2} M _inst_3 x (HasCompl.compl.{u2} (Set.{u2} M) (BooleanAlgebra.toHasCompl.{u2} (Set.{u2} M) (Set.instBooleanAlgebraSet.{u2} M)) (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) +Case conversion may be inaccurate. Consider using '#align module.punctured_nhds_ne_bot Module.punctured_nhds_neBotₓ'. -/ /-- Let `R` be a topological ring such that zero is not an isolated point (e.g., a nontrivially normed field, see `normed_field.punctured_nhds_ne_bot`). Let `M` be a nontrivial module over `R` such that `c • x = 0` implies `c = 0 ∨ x = 0`. Then `M` has no isolated points. We formulate this @@ -154,6 +172,12 @@ variable {ι R M₁ M₂ : Type _} [Semiring R] [AddCommMonoid M₁] [AddCommMon [Module R M₂] [u : TopologicalSpace R] {t : TopologicalSpace M₂} [ContinuousSMul R M₂] (f : M₁ →ₗ[R] M₂) +/- warning: has_continuous_smul_induced -> continuousSMul_induced is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M₁ : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Semiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M₁] [_inst_3 : AddCommMonoid.{u3} M₂] [_inst_4 : Module.{u1, u2} R M₁ _inst_1 _inst_2] [_inst_5 : Module.{u1, u3} R M₂ _inst_1 _inst_3] [u : TopologicalSpace.{u1} R] {t : TopologicalSpace.{u3} M₂} [_inst_6 : ContinuousSMul.{u1, u3} R M₂ (SMulZeroClass.toHasSmul.{u1, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_3))) (SMulWithZero.toSmulZeroClass.{u1, u3} R M₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_3))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_3))) (Module.toMulActionWithZero.{u1, u3} R M₂ _inst_1 _inst_3 _inst_5)))) u t] (f : LinearMap.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₁ M₂ _inst_2 _inst_3 _inst_4 _inst_5), ContinuousSMul.{u1, u2} R M₁ (SMulZeroClass.toHasSmul.{u1, u2} R M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M₁ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M₁ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M₁ _inst_1 _inst_2 _inst_4)))) u (TopologicalSpace.induced.{u2, u3} M₁ M₂ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₁ M₂ _inst_2 _inst_3 _inst_4 _inst_5) (fun (_x : LinearMap.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₁ M₂ _inst_2 _inst_3 _inst_4 _inst_5) => M₁ -> M₂) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R M₁ M₂ _inst_1 _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) f) t) +but is expected to have type + forall {R : Type.{u3}} {M₁ : Type.{u2}} {M₂ : Type.{u1}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M₁] [_inst_3 : AddCommMonoid.{u1} M₂] [_inst_4 : Module.{u3, u2} R M₁ _inst_1 _inst_2] [_inst_5 : Module.{u3, u1} R M₂ _inst_1 _inst_3] [u : TopologicalSpace.{u3} R] {t : TopologicalSpace.{u1} M₂} [_inst_6 : ContinuousSMul.{u3, u1} R M₂ (SMulZeroClass.toSMul.{u3, u1} R M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_3)) (SMulWithZero.toSMulZeroClass.{u3, u1} R M₂ (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_3)) (MulActionWithZero.toSMulWithZero.{u3, u1} R M₂ (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_3)) (Module.toMulActionWithZero.{u3, u1} R M₂ _inst_1 _inst_3 _inst_5)))) u t] (f : LinearMap.{u3, u3, u2, u1} R R _inst_1 _inst_1 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)) M₁ M₂ _inst_2 _inst_3 _inst_4 _inst_5), ContinuousSMul.{u3, u2} R M₁ (SMulZeroClass.toSMul.{u3, u2} R M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} R M₁ (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} R M₁ (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_2)) (Module.toMulActionWithZero.{u3, u2} R M₁ _inst_1 _inst_2 _inst_4)))) u (TopologicalSpace.induced.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (LinearMap.{u3, u3, u2, u1} R R _inst_1 _inst_1 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)) M₁ M₂ _inst_2 _inst_3 _inst_4 _inst_5) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) _x) (LinearMap.instFunLikeLinearMap.{u3, u3, u2, u1} R R M₁ M₂ _inst_1 _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))) f) t) +Case conversion may be inaccurate. Consider using '#align has_continuous_smul_induced continuousSMul_inducedₓ'. -/ theorem continuousSMul_induced : @ContinuousSMul R M₁ _ u (t.induced f) := { continuous_smul := by @@ -216,6 +240,12 @@ theorem Submodule.closure_smul_self_eq (s : Submodule R M) : variable [ContinuousAdd M] +/- warning: submodule.topological_closure -> Submodule.topologicalClosure is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))], (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) -> (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))], (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) -> (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) +Case conversion may be inaccurate. Consider using '#align submodule.topological_closure Submodule.topologicalClosureₓ'. -/ /-- The (topological-space) closure of a submodule of a topological `R`-module `M` is itself a submodule. -/ def Submodule.topologicalClosure (s : Submodule R M) : Submodule R M := @@ -225,36 +255,78 @@ def Submodule.topologicalClosure (s : Submodule R M) : Submodule R M := smul_mem' := fun c x hx => s.closure_smul_self_subset ⟨⟨c, x⟩, ⟨Set.mem_univ _, hx⟩, rfl⟩ } #align submodule.topological_closure Submodule.topologicalClosure +/- warning: submodule.topological_closure_coe -> Submodule.topologicalClosure_coe is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] (s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5), Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s)) (closure.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] (_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4), Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)) (closure.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7)) +Case conversion may be inaccurate. Consider using '#align submodule.topological_closure_coe Submodule.topologicalClosure_coeₓ'. -/ @[simp] theorem Submodule.topologicalClosure_coe (s : Submodule R M) : (s.topologicalClosure : Set M) = closure (s : Set M) := rfl #align submodule.topological_closure_coe Submodule.topologicalClosure_coe +/- warning: submodule.le_topological_closure -> Submodule.le_topologicalClosure is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] (s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5), LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] (_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4), LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4))))) _inst_7 (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) +Case conversion may be inaccurate. Consider using '#align submodule.le_topological_closure Submodule.le_topologicalClosureₓ'. -/ theorem Submodule.le_topologicalClosure (s : Submodule R M) : s ≤ s.topologicalClosure := subset_closure #align submodule.le_topological_closure Submodule.le_topologicalClosure +/- warning: submodule.is_closed_topological_closure -> Submodule.isClosed_topologicalClosure is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] (s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5), IsClosed.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] (_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4), IsClosed.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7)) +Case conversion may be inaccurate. Consider using '#align submodule.is_closed_topological_closure Submodule.isClosed_topologicalClosureₓ'. -/ theorem Submodule.isClosed_topologicalClosure (s : Submodule R M) : IsClosed (s.topologicalClosure : Set M) := by convert isClosed_closure #align submodule.is_closed_topological_closure Submodule.isClosed_topologicalClosure +/- warning: submodule.topological_closure_minimal -> Submodule.topologicalClosure_minimal is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] (s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) {t : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s t) -> (IsClosed.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) t)) -> (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s) t) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] (_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) {s : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4}, (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4))))) _inst_7 s) -> (IsClosed.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) s)) -> (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4))))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) s) +Case conversion may be inaccurate. Consider using '#align submodule.topological_closure_minimal Submodule.topologicalClosure_minimalₓ'. -/ theorem Submodule.topologicalClosure_minimal (s : Submodule R M) {t : Submodule R M} (h : s ≤ t) (ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t := closure_minimal h ht #align submodule.topological_closure_minimal Submodule.topologicalClosure_minimal +/- warning: submodule.topological_closure_mono -> Submodule.topologicalClosure_mono is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] {s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5} {t : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s t) -> (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 t)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] {_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4} {s : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4}, (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4))))) _inst_7 s) -> (LE.le.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4))))) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 s)) +Case conversion may be inaccurate. Consider using '#align submodule.topological_closure_mono Submodule.topologicalClosure_monoₓ'. -/ theorem Submodule.topologicalClosure_mono {s : Submodule R M} {t : Submodule R M} (h : s ≤ t) : s.topologicalClosure ≤ t.topologicalClosure := s.topologicalClosure_minimal (h.trans t.le_topologicalClosure) t.isClosed_topologicalClosure #align submodule.topological_closure_mono Submodule.topologicalClosure_mono +/- warning: is_closed.submodule_topological_closure_eq -> IsClosed.submodule_topologicalClosure_eq is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] {s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5}, (IsClosed.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s)) -> (Eq.{succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s) s) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] {_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4}, (IsClosed.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7)) -> (Eq.{succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) _inst_7) +Case conversion may be inaccurate. Consider using '#align is_closed.submodule_topological_closure_eq IsClosed.submodule_topologicalClosure_eqₓ'. -/ /-- The topological closure of a closed submodule `s` is equal to `s`. -/ theorem IsClosed.submodule_topologicalClosure_eq {s : Submodule R M} (hs : IsClosed (s : Set M)) : s.topologicalClosure = s := le_antisymm (s.topologicalClosure_minimal rfl.le hs) s.le_topologicalClosure #align is_closed.submodule_topological_closure_eq IsClosed.submodule_topologicalClosure_eq +/- warning: submodule.dense_iff_topological_closure_eq_top -> Submodule.dense_iff_topologicalClosure_eq_top is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] {s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5}, Iff (Dense.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s)) (Eq.{succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 s) (Top.top.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Submodule.hasTop.{u1, u2} R M _inst_1 _inst_4 _inst_5))) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] {_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4}, Iff (Dense.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7)) (Eq.{succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.topologicalClosure.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7) (Top.top.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.instTopSubmodule.{u1, u2} R M _inst_1 _inst_3 _inst_4))) +Case conversion may be inaccurate. Consider using '#align submodule.dense_iff_topological_closure_eq_top Submodule.dense_iff_topologicalClosure_eq_topₓ'. -/ /-- A subspace is dense iff its topological closure is the entire space. -/ theorem Submodule.dense_iff_topologicalClosure_eq_top {s : Submodule R M} : Dense (s : Set M) ↔ s.topologicalClosure = ⊤ := @@ -268,6 +340,12 @@ instance {M' : Type _} [AddCommMonoid M'] [Module R M'] [UniformSpace M'] [Conti CompleteSpace U.topologicalClosure := isClosed_closure.completeSpace_coe +/- warning: submodule.is_closed_or_dense_of_is_coatom -> Submodule.isClosed_or_dense_of_isCoatom is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u1} R] [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommMonoid.{u2} M] [_inst_5 : Module.{u1, u2} R M _inst_1 _inst_4] [_inst_6 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) _inst_2 _inst_3] [_inst_7 : ContinuousAdd.{u2} M _inst_3 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_4)))] (s : Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5), (IsCoatom.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5))) (Submodule.orderTop.{u1, u2} R M _inst_1 _inst_4 _inst_5) s) -> (Or (IsClosed.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s)) (Dense.{u2} M _inst_3 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_4 _inst_5) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_4 _inst_5)))) s))) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] [_inst_5 : ContinuousConstSMul.{u1, u2} R M _inst_2 (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_3 _inst_4))))] [_inst_6 : ContinuousAdd.{u2} M _inst_2 (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))] (_inst_7 : Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4), (IsCoatom.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) (Submodule.completeLattice.{u1, u2} R M _inst_1 _inst_3 _inst_4)))) (Submodule.instOrderTopSubmoduleToLEToPreorderInstPartialOrderSetLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7) -> (Or (IsClosed.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7)) (Dense.{u2} M _inst_2 (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M _inst_1 _inst_3 _inst_4) M (Submodule.setLike.{u1, u2} R M _inst_1 _inst_3 _inst_4) _inst_7))) +Case conversion may be inaccurate. Consider using '#align submodule.is_closed_or_dense_of_is_coatom Submodule.isClosed_or_dense_of_isCoatomₓ'. -/ /-- A maximal proper subspace of a topological module (i.e a `submodule` satisfying `is_coatom`) is either closed or dense. -/ theorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) : @@ -280,6 +358,12 @@ end closure section Pi +/- warning: linear_map.continuous_on_pi -> LinearMap.continuous_on_pi is a dubious translation: +lean 3 declaration is + forall {ι : Type.{u1}} {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Finite.{succ u1} ι] [_inst_2 : Semiring.{u2} R] [_inst_3 : TopologicalSpace.{u2} R] [_inst_4 : AddCommMonoid.{u3} M] [_inst_5 : Module.{u2, u3} R M _inst_2 _inst_4] [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : ContinuousAdd.{u3} M _inst_6 (AddZeroClass.toHasAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_4)))] [_inst_8 : ContinuousSMul.{u2, u3} R M (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_4))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_2)))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_4))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R _inst_2) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_4))) (Module.toMulActionWithZero.{u2, u3} R M _inst_2 _inst_4 _inst_5)))) _inst_3 _inst_6] (f : LinearMap.{u2, u2, max u1 u2, u3} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (ι -> R) M (Pi.addCommMonoid.{u1, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.Function.module.{u1, u2, u2} ι R R _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (Semiring.toModule.{u2} R _inst_2)) _inst_5), Continuous.{max u1 u2, u3} (ι -> R) M (Pi.topologicalSpace.{u1, u2} ι (fun (ᾰ : ι) => R) (fun (a : ι) => _inst_3)) _inst_6 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (LinearMap.{u2, u2, max u1 u2, u3} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (ι -> R) M (Pi.addCommMonoid.{u1, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.Function.module.{u1, u2, u2} ι R R _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (Semiring.toModule.{u2} R _inst_2)) _inst_5) (fun (_x : LinearMap.{u2, u2, max u1 u2, u3} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (ι -> R) M (Pi.addCommMonoid.{u1, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.Function.module.{u1, u2, u2} ι R R _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (Semiring.toModule.{u2} R _inst_2)) _inst_5) => (ι -> R) -> M) (LinearMap.hasCoeToFun.{u2, u2, max u1 u2, u3} R R (ι -> R) M _inst_2 _inst_2 (Pi.addCommMonoid.{u1, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.Function.module.{u1, u2, u2} ι R R _inst_2 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (Semiring.toModule.{u2} R _inst_2)) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) f) +but is expected to have type + forall {ι : Type.{u3}} {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : Finite.{succ u3} ι] [_inst_2 : Semiring.{u2} R] [_inst_3 : TopologicalSpace.{u2} R] [_inst_4 : AddCommMonoid.{u1} M] [_inst_5 : Module.{u2, u1} R M _inst_2 _inst_4] [_inst_6 : TopologicalSpace.{u1} M] [_inst_7 : ContinuousAdd.{u1} M _inst_6 (AddZeroClass.toAdd.{u1} M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_4)))] [_inst_8 : ContinuousSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_4)) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_2)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_4)) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R _inst_2) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_4)) (Module.toMulActionWithZero.{u2, u1} R M _inst_2 _inst_4 _inst_5)))) _inst_3 _inst_6] (f : LinearMap.{u2, u2, max u3 u2, u1} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (ι -> R) M (Pi.addCommMonoid.{u3, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.module.{u3, u2, u2} ι (fun (a._@.Mathlib.Topology.Algebra.Module.Basic._hyg.5155 : ι) => R) R _inst_2 (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (fun (i : ι) => Semiring.toModule.{u2} R _inst_2)) _inst_5), Continuous.{max u3 u2, u1} (ι -> R) M (Pi.topologicalSpace.{u3, u2} ι (fun (ᾰ : ι) => R) (fun (a : ι) => _inst_3)) _inst_6 (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), max (succ u3) (succ u2), succ u1} (LinearMap.{u2, u2, max u3 u2, u1} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (ι -> R) M (Pi.addCommMonoid.{u3, u2} ι (fun (a._@.Mathlib.Topology.Algebra.Module.Basic._hyg.5155 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.module.{u3, u2, u2} ι (fun (a._@.Mathlib.Topology.Algebra.Module.Basic._hyg.5155 : ι) => R) R _inst_2 (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (fun (i : ι) => Semiring.toModule.{u2} R _inst_2)) _inst_5) (ι -> R) (fun (_x : ι -> R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ι -> R) => M) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, max u3 u2, u1} R R (ι -> R) M _inst_2 _inst_2 (Pi.addCommMonoid.{u3, u2} ι (fun (ᾰ : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)))) _inst_4 (Pi.module.{u3, u2, u2} ι (fun (a._@.Mathlib.Topology.Algebra.Module.Basic._hyg.5155 : ι) => R) R _inst_2 (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) (fun (i : ι) => Semiring.toModule.{u2} R _inst_2)) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2))) f) +Case conversion may be inaccurate. Consider using '#align linear_map.continuous_on_pi LinearMap.continuous_on_piₓ'. -/ theorem LinearMap.continuous_on_pi {ι : Type _} {R : Type _} {M : Type _} [Finite ι] [Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f := @@ -299,6 +383,7 @@ theorem LinearMap.continuous_on_pi {ι : Type _} {R : Type _} {M : Type _} [Fini end Pi +#print ContinuousLinearMap /- /-- Continuous linear maps between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M₂` will be topological modules over the topological ring `R`. -/ @@ -307,6 +392,7 @@ structure ContinuousLinearMap {R : Type _} {S : Type _} [Semiring R] [Semiring S [AddCommMonoid M₂] [Module R M] [Module S M₂] extends M →ₛₗ[σ] M₂ where cont : Continuous to_fun := by continuity #align continuous_linear_map ContinuousLinearMap +-/ -- mathport name: «expr →SL[ ] » notation:25 M " →SL[" σ "] " M₂ => ContinuousLinearMap σ M M₂ @@ -317,6 +403,7 @@ notation:25 M " →L[" R "] " M₂ => ContinuousLinearMap (RingHom.id R) M M₂ -- mathport name: «expr →L⋆[ ] » notation:25 M " →L⋆[" R "] " M₂ => ContinuousLinearMap (starRingEnd R) M M₂ +#print ContinuousSemilinearMapClass /- /-- `continuous_semilinear_map_class F σ M M₂` asserts `F` is a type of bundled continuous `σ`-semilinear maps `M → M₂`. See also `continuous_linear_map_class F R M M₂` for the case where `σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring @@ -327,10 +414,12 @@ class ContinuousSemilinearMapClass (F : Type _) {R S : outParam (Type _)} [Semir (M₂ : outParam (Type _)) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends SemilinearMapClass F σ M M₂, ContinuousMapClass F M M₂ #align continuous_semilinear_map_class ContinuousSemilinearMapClass +-/ -- `σ`, `R` and `S` become metavariables, but they are all outparams so it's OK attribute [nolint dangerous_instance] ContinuousSemilinearMapClass.toContinuousMapClass +#print ContinuousLinearMapClass /- /-- `continuous_linear_map_class F R M M₂` asserts `F` is a type of bundled continuous `R`-linear maps `M → M₂`. This is an abbreviation for `continuous_semilinear_map_class F (ring_hom.id R) M M₂`. -/ @@ -339,7 +428,9 @@ abbrev ContinuousLinearMapClass (F : Type _) (R : outParam (Type _)) [Semiring R [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] := ContinuousSemilinearMapClass F (RingHom.id R) M M₂ #align continuous_linear_map_class ContinuousLinearMapClass +-/ +#print ContinuousLinearEquiv /- /-- Continuous linear equivalences between modules. We only put the type classes that are necessary for the definition, although in applications `M` and `M₂` will be topological modules over the topological semiring `R`. -/ @@ -351,6 +442,7 @@ structure ContinuousLinearEquiv {R : Type _} {S : Type _} [Semiring R] [Semiring continuous_toFun : Continuous to_fun := by continuity continuous_invFun : Continuous inv_fun := by continuity #align continuous_linear_equiv ContinuousLinearEquiv +-/ -- mathport name: «expr ≃SL[ ] » notation:50 M " ≃SL[" σ "] " M₂ => ContinuousLinearEquiv σ M M₂ @@ -361,6 +453,7 @@ notation:50 M " ≃L[" R "] " M₂ => ContinuousLinearEquiv (RingHom.id R) M M -- mathport name: «expr ≃L⋆[ ] » notation:50 M " ≃L⋆[" R "] " M₂ => ContinuousLinearEquiv (starRingEnd R) M M₂ +#print ContinuousSemilinearEquivClass /- /-- `continuous_semilinear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous `σ`-semilinear equivs `M → M₂`. See also `continuous_linear_equiv_class F R M M₂` for the case where `σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring @@ -374,7 +467,9 @@ class ContinuousSemilinearEquivClass (F : Type _) {R : outParam (Type _)} {S : o map_continuous : ∀ f : F, Continuous f := by continuity inv_continuous : ∀ f : F, Continuous (inv f) := by continuity #align continuous_semilinear_equiv_class ContinuousSemilinearEquivClass +-/ +#print ContinuousLinearEquivClass /- /-- `continuous_linear_equiv_class F σ M M₂` asserts `F` is a type of bundled continuous `R`-linear equivs `M → M₂`. This is an abbreviation for `continuous_semilinear_equiv_class F (ring_hom.id) M M₂`. -/ @@ -383,6 +478,7 @@ abbrev ContinuousLinearEquivClass (F : Type _) (R : outParam (Type _)) [Semiring [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] := ContinuousSemilinearEquivClass F (RingHom.id R) M M₂ #align continuous_linear_equiv_class ContinuousLinearEquivClass +-/ namespace ContinuousSemilinearEquivClass @@ -420,12 +516,18 @@ theorem isClosed_setOf_map_smul : IsClosed { f : M₁ → M₂ | ∀ c x, f (c exact isClosed_interᵢ fun c => isClosed_interᵢ fun x => isClosed_eq (continuous_apply _) ((continuous_apply _).const_smul _) -#align is_closed_set_of_map_smul isClosed_setOf_map_smul +#align is_closed_set_of_map_smul isClosed_setOf_map_smulₓ end variable [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} +/- warning: linear_map_of_mem_closure_range_coe -> linearMapOfMemClosureRangeCoe is a dubious translation: +lean 3 declaration is + forall {M₁ : Type.{u1}} {M₂ : Type.{u2}} {R : Type.{u3}} {S : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} M₂] [_inst_2 : T2Space.{u2} M₂ _inst_1] [_inst_3 : Semiring.{u3} R] [_inst_4 : Semiring.{u4} S] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u3, u1} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u4, u2} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u4, u2} S M₂ _inst_1 (SMulZeroClass.toHasSmul.{u4, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (SMulWithZero.toSmulZeroClass.{u4, u2} S M₂ (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_4)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_4) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u2} M₂ _inst_1 (AddZeroClass.toHasAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)))] {σ : RingHom.{u3, u4} R S (Semiring.toNonAssocSemiring.{u3} R _inst_3) (Semiring.toNonAssocSemiring.{u4} S _inst_4)} (f : M₁ -> M₂), (Membership.Mem.{max u1 u2, max u1 u2} (M₁ -> M₂) (Set.{max u1 u2} (M₁ -> M₂)) (Set.hasMem.{max u1 u2} (M₁ -> M₂)) f (closure.{max u1 u2} (M₁ -> M₂) (Pi.topologicalSpace.{u1, u2} M₁ (fun (ᾰ : M₁) => M₂) (fun (a : M₁) => _inst_1)) (Set.range.{max u1 u2, max (succ u1) (succ u2)} (M₁ -> M₂) (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (fun (ᾰ : LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) => M₁ -> M₂) (LinearMap.hasCoeToFun.{u3, u4, u1, u2} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ))))) -> (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) +but is expected to have type + forall {M₁ : Type.{u1}} {M₂ : Type.{u2}} {R : Type.{u3}} {S : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} M₂] [_inst_2 : T2Space.{u2} M₂ _inst_1] [_inst_3 : Semiring.{u3} R] [_inst_4 : Semiring.{u4} S] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u3, u1} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u4, u2} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u4, u2} S M₂ _inst_1 (SMulZeroClass.toSMul.{u4, u2} S M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (SMulWithZero.toSMulZeroClass.{u4, u2} S M₂ (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_4)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_4) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u2} M₂ _inst_1 (AddZeroClass.toAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)))] {σ : RingHom.{u3, u4} R S (Semiring.toNonAssocSemiring.{u3} R _inst_3) (Semiring.toNonAssocSemiring.{u4} S _inst_4)} (f : M₁ -> M₂), (Membership.mem.{max u1 u2, max u1 u2} (M₁ -> M₂) (Set.{max u1 u2} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ)) (Set.instMembershipSet.{max u1 u2} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ)) f (closure.{max u1 u2} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (Pi.topologicalSpace.{u1, u2} M₁ (fun (ᾰ : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (fun (a : M₁) => _inst_1)) (Set.range.{max u1 u2, max (succ u1) (succ u2)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) M₁ (fun (ᾰ : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (LinearMap.instFunLikeLinearMap.{u3, u4, u1, u2} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ))))) -> (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) +Case conversion may be inaccurate. Consider using '#align linear_map_of_mem_closure_range_coe linearMapOfMemClosureRangeCoeₓ'. -/ /-- Constructs a bundled linear map from a function and a proof that this function belongs to the closure of the set of linear maps. -/ @[simps (config := { fullyApplied := false })] @@ -438,6 +540,12 @@ def linearMapOfMemClosureRangeCoe (f : M₁ → M₂) (Set.range_subset_iff.2 LinearMap.map_smulₛₗ) hf } #align linear_map_of_mem_closure_range_coe linearMapOfMemClosureRangeCoe +/- warning: linear_map_of_tendsto -> linearMapOfTendsto is a dubious translation: +lean 3 declaration is + forall {M₁ : Type.{u1}} {M₂ : Type.{u2}} {α : Type.{u3}} {R : Type.{u4}} {S : Type.{u5}} [_inst_1 : TopologicalSpace.{u2} M₂] [_inst_2 : T2Space.{u2} M₂ _inst_1] [_inst_3 : Semiring.{u4} R] [_inst_4 : Semiring.{u5} S] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u4, u1} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u5, u2} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u5, u2} S M₂ _inst_1 (SMulZeroClass.toHasSmul.{u5, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (SMulWithZero.toSmulZeroClass.{u5, u2} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S _inst_4)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (MulActionWithZero.toSMulWithZero.{u5, u2} S M₂ (Semiring.toMonoidWithZero.{u5} S _inst_4) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (Module.toMulActionWithZero.{u5, u2} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u2} M₂ _inst_1 (AddZeroClass.toHasAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)))] {σ : RingHom.{u4, u5} R S (Semiring.toNonAssocSemiring.{u4} R _inst_3) (Semiring.toNonAssocSemiring.{u5} S _inst_4)} {l : Filter.{u3} α} (f : M₁ -> M₂) (g : α -> (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8)) [_inst_11 : Filter.NeBot.{u3} α l], (Filter.Tendsto.{u3, max u1 u2} α (M₁ -> M₂) (fun (a : α) (x : M₁) => coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (fun (_x : LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) => M₁ -> M₂) (LinearMap.hasCoeToFun.{u4, u5, u1, u2} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ) (g a) x) l (nhds.{max u1 u2} (M₁ -> M₂) (Pi.topologicalSpace.{u1, u2} M₁ (fun (x : M₁) => M₂) (fun (a : M₁) => _inst_1)) f)) -> (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) +but is expected to have type + forall {M₁ : Type.{u1}} {M₂ : Type.{u2}} {α : Type.{u3}} {R : Type.{u4}} {S : Type.{u5}} [_inst_1 : TopologicalSpace.{u2} M₂] [_inst_2 : T2Space.{u2} M₂ _inst_1] [_inst_3 : Semiring.{u4} R] [_inst_4 : Semiring.{u5} S] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u4, u1} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u5, u2} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u5, u2} S M₂ _inst_1 (SMulZeroClass.toSMul.{u5, u2} S M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (SMulWithZero.toSMulZeroClass.{u5, u2} S M₂ (MonoidWithZero.toZero.{u5} S (Semiring.toMonoidWithZero.{u5} S _inst_4)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (MulActionWithZero.toSMulWithZero.{u5, u2} S M₂ (Semiring.toMonoidWithZero.{u5} S _inst_4) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)) (Module.toMulActionWithZero.{u5, u2} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u2} M₂ _inst_1 (AddZeroClass.toAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)))] {σ : RingHom.{u4, u5} R S (Semiring.toNonAssocSemiring.{u4} R _inst_3) (Semiring.toNonAssocSemiring.{u5} S _inst_4)} {l : Filter.{u3} α} (f : M₁ -> M₂) (g : α -> (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8)) [_inst_11 : Filter.NeBot.{u3} α l], (Filter.Tendsto.{u3, max u1 u2} α (forall (x : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) x) (fun (a : α) (x : M₁) => FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) _x) (LinearMap.instFunLikeLinearMap.{u4, u5, u1, u2} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ) (g a) x) l (nhds.{max u1 u2} (forall (x : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) x) (Pi.topologicalSpace.{u1, u2} M₁ (fun (x : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) x) (fun (a : M₁) => _inst_1)) f)) -> (LinearMap.{u4, u5, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) +Case conversion may be inaccurate. Consider using '#align linear_map_of_tendsto linearMapOfTendstoₓ'. -/ /-- Construct a bundled linear map from a pointwise limit of linear maps -/ @[simps (config := { fullyApplied := false })] def linearMapOfTendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.ne_bot] @@ -448,6 +556,12 @@ def linearMapOfTendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) variable (M₁ M₂ σ) +/- warning: linear_map.is_closed_range_coe -> LinearMap.isClosed_range_coe is a dubious translation: +lean 3 declaration is + forall (M₁ : Type.{u1}) (M₂ : Type.{u2}) {R : Type.{u3}} {S : Type.{u4}} [_inst_1 : TopologicalSpace.{u2} M₂] [_inst_2 : T2Space.{u2} M₂ _inst_1] [_inst_3 : Semiring.{u3} R] [_inst_4 : Semiring.{u4} S] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u3, u1} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u4, u2} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u4, u2} S M₂ _inst_1 (SMulZeroClass.toHasSmul.{u4, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (SMulWithZero.toSmulZeroClass.{u4, u2} S M₂ (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_4)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_4) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6))) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u2} M₂ _inst_1 (AddZeroClass.toHasAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_6)))] (σ : RingHom.{u3, u4} R S (Semiring.toNonAssocSemiring.{u3} R _inst_3) (Semiring.toNonAssocSemiring.{u4} S _inst_4)), IsClosed.{max u1 u2} (M₁ -> M₂) (Pi.topologicalSpace.{u1, u2} M₁ (fun (ᾰ : M₁) => M₂) (fun (a : M₁) => _inst_1)) (Set.range.{max u1 u2, max (succ u1) (succ u2)} (M₁ -> M₂) (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (fun (ᾰ : LinearMap.{u3, u4, u1, u2} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) => M₁ -> M₂) (LinearMap.hasCoeToFun.{u3, u4, u1, u2} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ))) +but is expected to have type + forall (M₁ : Type.{u4}) (M₂ : Type.{u3}) {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : TopologicalSpace.{u3} M₂] [_inst_2 : T2Space.{u3} M₂ _inst_1] [_inst_3 : Semiring.{u2} R] [_inst_4 : Semiring.{u1} S] [_inst_5 : AddCommMonoid.{u4} M₁] [_inst_6 : AddCommMonoid.{u3} M₂] [_inst_7 : Module.{u2, u4} R M₁ _inst_3 _inst_5] [_inst_8 : Module.{u1, u3} S M₂ _inst_4 _inst_6] [_inst_9 : ContinuousConstSMul.{u1, u3} S M₂ _inst_1 (SMulZeroClass.toSMul.{u1, u3} S M₂ (AddMonoid.toZero.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_6)) (SMulWithZero.toSMulZeroClass.{u1, u3} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S _inst_4)) (AddMonoid.toZero.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_6)) (MulActionWithZero.toSMulWithZero.{u1, u3} S M₂ (Semiring.toMonoidWithZero.{u1} S _inst_4) (AddMonoid.toZero.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_6)) (Module.toMulActionWithZero.{u1, u3} S M₂ _inst_4 _inst_6 _inst_8))))] [_inst_10 : ContinuousAdd.{u3} M₂ _inst_1 (AddZeroClass.toAdd.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_6)))] (σ : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R _inst_3) (Semiring.toNonAssocSemiring.{u1} S _inst_4)), IsClosed.{max u4 u3} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (Pi.topologicalSpace.{u4, u3} M₁ (fun (ᾰ : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (fun (a : M₁) => _inst_1)) (Set.range.{max u4 u3, max (succ u4) (succ u3)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (LinearMap.{u2, u1, u4, u3} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (LinearMap.{u2, u1, u4, u3} R S _inst_3 _inst_4 σ M₁ M₂ _inst_5 _inst_6 _inst_7 _inst_8) M₁ (fun (ᾰ : M₁) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₁) => M₂) ᾰ) (LinearMap.instFunLikeLinearMap.{u2, u1, u4, u3} R S M₁ M₂ _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 σ))) +Case conversion may be inaccurate. Consider using '#align linear_map.is_closed_range_coe LinearMap.isClosed_range_coeₓ'. -/ theorem LinearMap.isClosed_range_coe : IsClosed (Set.range (coeFn : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂)) := isClosed_of_closure_subset fun f hf => ⟨linearMapOfMemClosureRangeCoe f hf, rfl⟩ #align linear_map.is_closed_range_coe LinearMap.isClosed_range_coe @@ -474,12 +588,25 @@ variable {R₁ : Type _} {R₂ : Type _} {R₃ : Type _} [Semiring R₁] [Semiri instance : Coe (M₁ →SL[σ₁₂] M₂) (M₁ →ₛₗ[σ₁₂] M₂) := ⟨toLinearMap⟩ +/- warning: continuous_linear_map.to_linear_map_eq_coe clashes with [anonymous] -> [anonymous] +warning: continuous_linear_map.to_linear_map_eq_coe -> [anonymous] is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u_1}} {R₂ : Type.{u_2}} [_inst_1 : Semiring.{u_1} R₁] [_inst_2 : Semiring.{u_2} R₂] {σ₁₂ : RingHom.{u_1, u_2} R₁ R₂ (Semiring.toNonAssocSemiring.{u_1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u_2} R₂ _inst_2)} {M₁ : Type.{u_4}} [_inst_4 : TopologicalSpace.{u_4} M₁] [_inst_5 : AddCommMonoid.{u_4} M₁] {M₂ : Type.{u_6}} [_inst_8 : TopologicalSpace.{u_6} M₂] [_inst_9 : AddCommMonoid.{u_6} M₂] [_inst_14 : Module.{u_1, u_4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u_2, u_6} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u_4) (succ u_6)} (LinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) ((fun (a : Sort.{max (succ u_4) (succ u_6)}) (b : Sort.{max (succ u_4) (succ u_6)}) [self : HasLiftT.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} a b] => self.0) (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (CoeTCₓ.coe.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (coeBase.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f) +but is expected to have type + forall {R₁ : Type.{u}} {R₂ : Type.{v}}, (Nat -> R₁ -> R₂) -> Nat -> (List.{u} R₁) -> (List.{v} R₂) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.to_linear_map_eq_coe [anonymous]ₓ'. -/ -- make the coercion the preferred form @[simp] -theorem toLinearMap_eq_coe (f : M₁ →SL[σ₁₂] M₂) : f.toLinearMap = f := +theorem [anonymous] (f : M₁ →SL[σ₁₂] M₂) : f.toLinearMap = f := rfl -#align continuous_linear_map.to_linear_map_eq_coe ContinuousLinearMap.toLinearMap_eq_coe - +#align continuous_linear_map.to_linear_map_eq_coe [anonymous] + +/- warning: continuous_linear_map.coe_injective -> ContinuousLinearMap.coe_injective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9], Function.Injective.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_9], Function.Injective.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_injective ContinuousLinearMap.coe_injectiveₓ'. -/ theorem coe_injective : Function.Injective (coe : (M₁ →SL[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₂] M₂) := by intro f g H @@ -502,58 +629,116 @@ instance toFun : CoeFun (M₁ →SL[σ₁₂] M₂) fun _ => M₁ → M₂ := ⟨fun f => f.toFun⟩ #align continuous_linear_map.to_fun ContinuousLinearMap.toFun +/- warning: continuous_linear_map.coe_mk -> ContinuousLinearMap.coe_mk is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : 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Consider using '#align continuous_linear_map.coe_mk ContinuousLinearMap.coe_mkₓ'. -/ @[simp] theorem coe_mk (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ →ₛₗ[σ₁₂] M₂) = f := rfl #align continuous_linear_map.coe_mk ContinuousLinearMap.coe_mk +/- warning: continuous_linear_map.coe_mk' -> ContinuousLinearMap.coe_mk' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (h : autoParamₓ.{0} (Continuous.{u3, u4} 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Consider using '#align continuous_linear_map.coe_mk' ContinuousLinearMap.coe_mk'ₓ'. -/ @[simp] theorem coe_mk' (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ → M₂) = f := rfl #align continuous_linear_map.coe_mk' ContinuousLinearMap.coe_mk' +/- warning: continuous_linear_map.continuous -> ContinuousLinearMap.continuous is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Continuous.{u3, u4} M₁ M₂ _inst_4 _inst_8 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Continuous.{u2, u1} M₁ M₂ _inst_4 _inst_8 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.continuous ContinuousLinearMap.continuousₓ'. -/ @[continuity] protected theorem continuous (f : M₁ →SL[σ₁₂] M₂) : Continuous f := f.2 #align continuous_linear_map.continuous ContinuousLinearMap.continuous +/- warning: continuous_linear_map.uniform_continuous -> ContinuousLinearMap.uniformContinuous is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {E₁ : Type.{u3}} {E₂ : Type.{u4}} [_inst_18 : UniformSpace.{u3} E₁] [_inst_19 : UniformSpace.{u4} E₂] [_inst_20 : AddCommGroup.{u3} E₁] [_inst_21 : AddCommGroup.{u4} E₂] [_inst_22 : Module.{u1, u3} R₁ E₁ _inst_1 (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_20)] [_inst_23 : Module.{u2, u4} R₂ E₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_21)] [_inst_24 : 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(x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E₁) => E₂) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_18) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_20) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_19) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_21) _inst_22 _inst_23) E₁ E₂ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_18) (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_19) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u2, u1, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_18) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_20) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_19) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_21) _inst_22 _inst_23) R₁ R₂ _inst_1 _inst_2 σ₁₂ E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_18) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_20) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_19) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_21) _inst_22 _inst_23 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_18) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_20) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_19) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_21) _inst_22 _inst_23))) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.uniform_continuous ContinuousLinearMap.uniformContinuousₓ'. -/ protected theorem uniformContinuous {E₁ E₂ : Type _} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (f : E₁ →SL[σ₁₂] E₂) : UniformContinuous f := uniformContinuous_addMonoidHom_of_continuous f.Continuous #align continuous_linear_map.uniform_continuous ContinuousLinearMap.uniformContinuous +/- warning: continuous_linear_map.coe_inj -> ContinuousLinearMap.coe_inj is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, Iff (Eq.{max (succ u3) (succ u4)} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 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u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) g)) (Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f g) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] {f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, Iff (Eq.{max (succ u2) (succ u1)} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 g)) (Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f g) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_inj ContinuousLinearMap.coe_injₓ'. -/ @[simp, norm_cast] theorem coe_inj {f g : M₁ →SL[σ₁₂] M₂} : (f : M₁ →ₛₗ[σ₁₂] M₂) = g ↔ f = g := coe_injective.eq_iff #align continuous_linear_map.coe_inj ContinuousLinearMap.coe_inj +/- warning: continuous_linear_map.coe_fn_injective -> ContinuousLinearMap.coeFn_injective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9], Function.Injective.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (M₁ -> M₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (ᾰ : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u2, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u4} R₂ M₂ _inst_2 _inst_9], Function.Injective.{max (succ u4) (succ u3), max (succ u3) (succ u4)} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (M₁ -> M₂) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (ᾰ : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u2, u1, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_fn_injective ContinuousLinearMap.coeFn_injectiveₓ'. -/ theorem coeFn_injective : @Function.Injective (M₁ →SL[σ₁₂] M₂) (M₁ → M₂) coeFn := FunLike.coe_injective #align continuous_linear_map.coe_fn_injective ContinuousLinearMap.coeFn_injective +#print ContinuousLinearMap.Simps.apply /- /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections. -/ def Simps.apply (h : M₁ →SL[σ₁₂] M₂) : M₁ → M₂ := h #align continuous_linear_map.simps.apply ContinuousLinearMap.Simps.apply +-/ +#print ContinuousLinearMap.Simps.coe /- /-- See Note [custom simps projection]. -/ def Simps.coe (h : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂ := h #align continuous_linear_map.simps.coe ContinuousLinearMap.Simps.coe +-/ initialize_simps_projections ContinuousLinearMap (to_linear_map_to_fun → apply, toLinearMap → coe) +/- warning: continuous_linear_map.ext -> ContinuousLinearMap.ext is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : 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Consider using '#align continuous_linear_map.ext ContinuousLinearMap.extₓ'. -/ @[ext] theorem ext {f g : M₁ →SL[σ₁₂] M₂} (h : ∀ x, f x = g x) : f = g := FunLike.ext f g h #align continuous_linear_map.ext ContinuousLinearMap.ext +/- warning: continuous_linear_map.ext_iff -> ContinuousLinearMap.ext_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : 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u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) g x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ext_iff ContinuousLinearMap.ext_iffₓ'. -/ theorem ext_iff {f g : M₁ →SL[σ₁₂] M₂} : f = g ↔ ∀ x, f x = g x := FunLike.ext_iff #align continuous_linear_map.ext_iff ContinuousLinearMap.ext_iff +/- warning: continuous_linear_map.copy -> ContinuousLinearMap.copy is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f' : M₁ -> M₂), (Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) f' (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f)) -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) +but is expected to have type + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f' : M₁ -> M₂), (Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) f' (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (f : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) f) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f)) -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.copy ContinuousLinearMap.copyₓ'. -/ /-- Copy of a `continuous_linear_map` with a new `to_fun` equal to the old one. Useful to fix definitional equalities. -/ protected def copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : M₁ →SL[σ₁₂] M₂ @@ -562,34 +747,76 @@ protected def copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' cont := show Continuous f' from h.symm ▸ f.Continuous #align continuous_linear_map.copy ContinuousLinearMap.copy +/- warning: continuous_linear_map.coe_copy -> ContinuousLinearMap.coe_copy is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ 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_inst_14 _inst_16 f f' h)) f' +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_copy ContinuousLinearMap.coe_copyₓ'. -/ @[simp] theorem coe_copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : ⇑(f.copy f' h) = f' := rfl #align continuous_linear_map.coe_copy ContinuousLinearMap.coe_copy +/- warning: continuous_linear_map.copy_eq -> ContinuousLinearMap.copy_eq is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f' : M₁ -> M₂) (h : Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) f' (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f)), Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.copy.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f f' h) f +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f' : M₁ -> M₂) (h : Eq.{max (succ u2) (succ u1)} (M₁ -> M₂) f' (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (f : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) f) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f)), Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.copy.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f f' h) f +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.copy_eq ContinuousLinearMap.copy_eqₓ'. -/ theorem copy_eq (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : f.copy f' h = f := FunLike.ext' h #align continuous_linear_map.copy_eq ContinuousLinearMap.copy_eq +/- warning: continuous_linear_map.map_zero -> ContinuousLinearMap.map_zero is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f (OfNat.ofNat.{u3} M₁ 0 (OfNat.mk.{u3} M₁ 0 (Zero.zero.{u3} M₁ (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))))))) (OfNat.ofNat.{u4} M₂ 0 (OfNat.mk.{u4} M₂ 0 (Zero.zero.{u4} M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) (AddMonoid.toZero.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) (AddCommMonoid.toAddMonoid.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))))) _inst_9)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.map_zero ContinuousLinearMap.map_zeroₓ'. -/ -- make some straightforward lemmas available to `simp`. protected theorem map_zero (f : M₁ →SL[σ₁₂] M₂) : f (0 : M₁) = 0 := map_zero f #align continuous_linear_map.map_zero ContinuousLinearMap.map_zero +/- warning: continuous_linear_map.map_add -> ContinuousLinearMap.map_add is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 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Consider using '#align continuous_linear_map.map_add ContinuousLinearMap.map_addₓ'. -/ protected theorem map_add (f : M₁ →SL[σ₁₂] M₂) (x y : M₁) : f (x + y) = f x + f y := map_add f x y #align continuous_linear_map.map_add ContinuousLinearMap.map_add +/- warning: continuous_linear_map.map_smulₛₗ -> ContinuousLinearMap.map_smulₛₗ is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 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(x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_2 _inst_9 _inst_16))))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ (fun (_x : R₁) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₁) => R₂) _x) (MulHomClass.toFunLike.{max u4 u3, u4, u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ R₂ (NonUnitalNonAssocSemiring.toMul.{u4} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u3} R₂ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ R₂ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (RingHom.instRingHomClassRingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2))))) σ₁₂ c) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.map_smulₛₗ ContinuousLinearMap.map_smulₛₗₓ'. -/ @[simp] protected theorem map_smulₛₗ (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) : f (c • x) = σ₁₂ c • f x := (toLinearMap _).map_smulₛₗ _ _ #align continuous_linear_map.map_smulₛₗ ContinuousLinearMap.map_smulₛₗ +/- warning: continuous_linear_map.map_smul -> ContinuousLinearMap.map_smul is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_18 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] (f : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_18) (c : R₁) (x : M₁), Eq.{succ u3} M₂ (coeFn.{max (succ 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_inst_1)))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) c x)) (SMul.smul.{u1, u3} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u3} R₁ M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u1, u3} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u1, u3} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) 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M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_18 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_9] (f : ContinuousLinearMap.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_18) (c : R₁) (x : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (HSMul.hSMul.{u3, u1, u1} R₁ M₁ M₁ (instHSMul.{u3, u1} R₁ M₁ (SMulZeroClass.toSMul.{u3, u1} R₁ M₁ (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₁ M₁ (MonoidWithZero.toZero.{u3} R₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₁ M₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1) (AddMonoid.toZero.{u1} M₁ 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_inst_9)) (MulActionWithZero.toSMulWithZero.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (Semiring.toMonoidWithZero.{u3} R₁ _inst_1) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9)) (Module.toMulActionWithZero.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_1 _inst_9 _inst_18))))) c (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousLinearMap.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_18) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousLinearMap.{u3, u3, u1, u2} R₁ R₁ _inst_1 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Consider using '#align continuous_linear_map.map_smul ContinuousLinearMap.map_smulₓ'. -/ @[simp] protected theorem map_smul [Module R₁ M₂] (f : M₁ →L[R₁] M₂) (c : R₁) (x : M₁) : f (c • x) = c • f x := by simp only [RingHom.id_apply, ContinuousLinearMap.map_smulₛₗ] #align continuous_linear_map.map_smul ContinuousLinearMap.map_smul +/- warning: continuous_linear_map.map_smul_of_tower -> ContinuousLinearMap.map_smul_of_tower is a dubious translation: +lean 3 declaration is + forall {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] {R : Type.{u3}} {S : Type.{u4}} [_inst_18 : Semiring.{u4} S] [_inst_19 : SMul.{u3, u1} R M₁] [_inst_20 : Module.{u4, u1} S M₁ _inst_18 _inst_5] [_inst_21 : SMul.{u3, u2} R M₂] [_inst_22 : Module.{u4, u2} S M₂ _inst_18 _inst_9] [_inst_23 : LinearMap.CompatibleSMul.{u1, u2, u3, u4} M₁ M₂ _inst_5 _inst_9 R S _inst_18 _inst_19 _inst_20 _inst_21 _inst_22] (f : ContinuousLinearMap.{u4, u4, u1, u2} S S _inst_18 _inst_18 (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) (c : R) (x : M₁), Eq.{succ u2} M₂ (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} S S _inst_18 _inst_18 (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) (fun (_x : ContinuousLinearMap.{u4, u4, u1, u2} S S _inst_18 _inst_18 (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u4, u4, u1, u2} S S _inst_18 _inst_18 (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) f (SMul.smul.{u3, u1} R M₁ _inst_19 c x)) (SMul.smul.{u3, u2} R M₂ _inst_21 c (coeFn.{max (succ u1) (succ u2), max (succ u1) 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Module.{u3, u1} S M₂ _inst_18 _inst_9] [_inst_23 : LinearMap.CompatibleSMul.{u2, u1, u4, u3} M₁ M₂ _inst_5 _inst_9 R S _inst_18 _inst_19 _inst_20 _inst_21 _inst_22] (f : ContinuousLinearMap.{u3, u3, u2, u1} S S _inst_18 _inst_18 (RingHom.id.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) (c : R) (x : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (HSMul.hSMul.{u4, u2, u2} R M₁ M₁ (instHSMul.{u4, u2} R M₁ _inst_19) c x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} S S _inst_18 _inst_18 (RingHom.id.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} S S _inst_18 _inst_18 (RingHom.id.{u3} 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_inst_18 (RingHom.id.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} S S _inst_18 _inst_18 (RingHom.id.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_18)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_20 _inst_22))) f x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.map_smul_of_tower ContinuousLinearMap.map_smul_of_towerₓ'. -/ @[simp] theorem map_smul_of_tower {R S : Type _} [Semiring S] [SMul R M₁] [Module S M₁] [SMul R M₂] [Module S M₂] [LinearMap.CompatibleSMul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) : @@ -597,25 +824,55 @@ theorem map_smul_of_tower {R S : Type _} [Semiring S] [SMul R M₁] [Module S M LinearMap.CompatibleSMul.map_smul f c x #align continuous_linear_map.map_smul_of_tower ContinuousLinearMap.map_smul_of_tower +/- warning: continuous_linear_map.map_sum -> ContinuousLinearMap.map_sum is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {ι : Type.{u5}} (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (s : Finset.{u5} ι) (g : ι -> M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f (Finset.sum.{u3, u5} M₁ ι _inst_5 s (fun (i : ι) => g i))) (Finset.sum.{u4, u5} M₂ ι _inst_9 s (fun (i : ι) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, 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_inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f (g i))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.map_sum ContinuousLinearMap.map_sumₓ'. -/ protected theorem map_sum {ι : Type _} (f : M₁ →SL[σ₁₂] M₂) (s : Finset ι) (g : ι → M₁) : f (∑ i in s, g i) = ∑ i in s, f (g i) := f.toLinearMap.map_sum #align continuous_linear_map.map_sum ContinuousLinearMap.map_sum +/- warning: continuous_linear_map.coe_coe -> ContinuousLinearMap.coe_coe is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ 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Consider using '#align continuous_linear_map.coe_coe ContinuousLinearMap.coe_coeₓ'. -/ @[simp, norm_cast] theorem coe_coe (f : M₁ →SL[σ₁₂] M₂) : ⇑(f : M₁ →ₛₗ[σ₁₂] M₂) = f := rfl #align continuous_linear_map.coe_coe ContinuousLinearMap.coe_coe +/- warning: continuous_linear_map.ext_ring -> ContinuousLinearMap.ext_ring is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_18 : TopologicalSpace.{u1} R₁] {f : ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_18 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₁ _inst_4 _inst_5 (Semiring.toModule.{u1} R₁ _inst_1) _inst_14} {g : ContinuousLinearMap.{u1, u1, u1, 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Consider using '#align continuous_linear_map.ext_ring ContinuousLinearMap.ext_ringₓ'. -/ @[ext] theorem ext_ring [TopologicalSpace R₁] {f g : R₁ →L[R₁] M₁} (h : f 1 = g 1) : f = g := coe_inj.1 <| LinearMap.ext_ring h #align continuous_linear_map.ext_ring ContinuousLinearMap.ext_ring +/- warning: continuous_linear_map.ext_ring_iff -> ContinuousLinearMap.ext_ring_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_18 : TopologicalSpace.{u1} R₁] {f : ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_18 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₁ _inst_4 _inst_5 (Semiring.toModule.{u1} R₁ _inst_1) 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Consider using '#align continuous_linear_map.ext_ring_iff ContinuousLinearMap.ext_ring_iffₓ'. -/ theorem ext_ring_iff [TopologicalSpace R₁] {f g : R₁ →L[R₁] M₁} : f = g ↔ f 1 = g 1 := ⟨fun h => h ▸ rfl, ext_ring⟩ #align continuous_linear_map.ext_ring_iff ContinuousLinearMap.ext_ring_iff +/- warning: continuous_linear_map.eq_on_closure_span -> ContinuousLinearMap.eqOn_closure_span is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : T2Space.{u4} M₂ _inst_8] {s : Set.{u3} M₁} {f : 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TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u2, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : T2Space.{u4} M₂ _inst_8] {s : Set.{u3} M₁} {f : ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (Set.EqOn.{u3, u4} M₁ M₂ (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ 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Consider using '#align continuous_linear_map.eq_on_closure_span ContinuousLinearMap.eqOn_closure_spanₓ'. -/ /-- If two continuous linear maps are equal on a set `s`, then they are equal on the closure of the `submodule.span` of this set. -/ theorem eqOn_closure_span [T2Space M₂] {s : Set M₁} {f g : M₁ →SL[σ₁₂] M₂} (h : Set.EqOn f g s) : @@ -623,6 +880,12 @@ theorem eqOn_closure_span [T2Space M₂] {s : Set M₁} {f g : M₁ →SL[σ₁ (LinearMap.eqOn_span' h).closure f.Continuous g.Continuous #align continuous_linear_map.eq_on_closure_span ContinuousLinearMap.eqOn_closure_span +/- warning: continuous_linear_map.ext_on -> ContinuousLinearMap.ext_on is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : T2Space.{u4} M₂ _inst_8] {s : Set.{u3} M₁}, (Dense.{u3} M₁ _inst_4 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (HasLiftT.mk.{succ u3, succ u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (CoeTCₓ.coe.{succ u3, succ u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (SetLike.Set.hasCoeT.{u3, u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14)))) (Submodule.span.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14 s))) -> (forall {f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : 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_inst_1 _inst_5 _inst_14 s))) -> (forall {f : ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16} {g : ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (Set.EqOn.{u3, u4} M₁ M₂ (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u2, u1, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ 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(ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) g) s) -> (Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ext_on ContinuousLinearMap.ext_onₓ'. -/ /-- If the submodule generated by a set `s` is dense in the ambient module, then two continuous linear maps equal on `s` are equal. -/ theorem ext_on [T2Space M₂] {s : Set M₁} (hs : Dense (Submodule.span R₁ s : Set M₁)) @@ -630,6 +893,12 @@ theorem ext_on [T2Space M₂] {s : Set M₁} (hs : Dense (Submodule.span R₁ s ext fun x => eqOn_closure_span h (hs x) #align continuous_linear_map.ext_on ContinuousLinearMap.ext_on +/- warning: submodule.topological_closure_map -> Submodule.topologicalClosure_map is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : RingHomSurjective.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂] [_inst_19 : TopologicalSpace.{u1} R₁] [_inst_20 : TopologicalSpace.{u2} R₂] [_inst_21 : ContinuousSMul.{u1, u3} R₁ M₁ (SMulZeroClass.toHasSmul.{u1, u3} R₁ M₁ (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u3} R₁ M₁ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u3} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (Module.toMulActionWithZero.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_19 _inst_4] [_inst_22 : ContinuousAdd.{u3} M₁ _inst_4 (AddZeroClass.toHasAdd.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5)))] [_inst_23 : ContinuousSMul.{u2, u4} R₂ M₂ (SMulZeroClass.toHasSmul.{u2, u4} R₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u2, u4} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u2, u4} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (Module.toMulActionWithZero.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_20 _inst_8] [_inst_24 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (s : Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14), LE.le.{u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (Preorder.toLE.{u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (PartialOrder.toPreorder.{u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.partialOrder.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (Submodule.map.{u1, u2, u3, u4, max u3 u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.semilinearMapClass.{u1, 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_inst_14 _inst_16) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f) (Submodule.topologicalClosure.{u1, u3} R₁ M₁ _inst_1 _inst_19 _inst_4 _inst_5 _inst_14 _inst_21 _inst_22 s)) (Submodule.topologicalClosure.{u2, u4} R₂ M₂ _inst_2 _inst_20 _inst_8 _inst_9 _inst_16 _inst_23 _inst_24 (Submodule.map.{u1, u2, u3, u4, max u3 u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.semilinearMapClass.{u1, u2, u3, u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂) ((fun (a : Sort.{max 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(ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f) s)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : RingHomSurjective.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂] [_inst_19 : TopologicalSpace.{u4} R₁] [_inst_20 : TopologicalSpace.{u3} R₂] [_inst_21 : ContinuousSMul.{u4, u2} R₁ M₁ (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_19 _inst_4] [_inst_22 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))] [_inst_23 : ContinuousSMul.{u3, u1} R₂ M₂ (SMulZeroClass.toSMul.{u3, u1} R₂ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₂ M₂ (MonoidWithZero.toZero.{u3} R₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₂ M₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_20 _inst_8] [_inst_24 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (s : Submodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14), LE.le.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (Preorder.toLE.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (PartialOrder.toPreorder.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (Submodule.completeLattice.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16))))) (Submodule.map.{u4, u3, u2, u1, max u2 u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instSemilinearMapClassLinearMap.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) (Submodule.topologicalClosure.{u4, u2} R₁ M₁ _inst_1 _inst_4 _inst_5 _inst_14 (ContinuousSMul.continuousConstSMul.{u4, u2} R₁ M₁ _inst_19 _inst_4 (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_21) _inst_22 s)) (Submodule.topologicalClosure.{u3, u1} R₂ M₂ _inst_2 _inst_8 _inst_9 _inst_16 (ContinuousSMul.continuousConstSMul.{u3, u1} R₂ M₂ _inst_20 _inst_8 (SMulZeroClass.toSMul.{u3, u1} R₂ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₂ M₂ (MonoidWithZero.toZero.{u3} R₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₂ M₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_23) _inst_24 (Submodule.map.{u4, u3, u2, u1, max u2 u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instSemilinearMapClassLinearMap.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) s)) +Case conversion may be inaccurate. Consider using '#align submodule.topological_closure_map Submodule.topologicalClosure_mapₓ'. -/ /-- Under a continuous linear map, the image of the `topological_closure` of a submodule is contained in the `topological_closure` of its image. -/ theorem Submodule.topologicalClosure_map [RingHomSurjective σ₁₂] [TopologicalSpace R₁] @@ -640,6 +909,12 @@ theorem Submodule.topologicalClosure_map [RingHomSurjective σ₁₂] [Topologic image_closure_subset_closure_image f.Continuous #align submodule.topological_closure_map Submodule.topologicalClosure_map +/- warning: dense_range.topological_closure_map_submodule -> DenseRange.topologicalClosure_map_submodule is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : RingHomSurjective.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂] [_inst_19 : TopologicalSpace.{u1} R₁] [_inst_20 : TopologicalSpace.{u2} R₂] [_inst_21 : ContinuousSMul.{u1, u3} R₁ M₁ (SMulZeroClass.toHasSmul.{u1, u3} R₁ M₁ (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u3} R₁ M₁ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u3} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5))) (Module.toMulActionWithZero.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_19 _inst_4] [_inst_22 : ContinuousAdd.{u3} M₁ _inst_4 (AddZeroClass.toHasAdd.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_5)))] [_inst_23 : ContinuousSMul.{u2, u4} R₂ M₂ (SMulZeroClass.toHasSmul.{u2, u4} R₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u2, u4} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u2, u4} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (Module.toMulActionWithZero.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_20 _inst_8] [_inst_24 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))] {f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (DenseRange.{u4, u3} M₂ _inst_8 M₁ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f)) -> (forall {s : Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14}, (Eq.{succ u3} 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_inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f) s)) (Top.top.{u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (Submodule.hasTop.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : RingHomSurjective.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂] [_inst_19 : TopologicalSpace.{u4} R₁] [_inst_20 : TopologicalSpace.{u3} R₂] [_inst_21 : ContinuousSMul.{u4, u2} R₁ M₁ (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_19 _inst_4] [_inst_22 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))] [_inst_23 : ContinuousSMul.{u3, u1} R₂ M₂ (SMulZeroClass.toSMul.{u3, u1} R₂ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₂ M₂ (MonoidWithZero.toZero.{u3} R₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₂ M₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_20 _inst_8] [_inst_24 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] {f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (DenseRange.{u1, u2} M₂ _inst_8 M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f)) -> (forall {s : Submodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14}, (Eq.{succ u2} (Submodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (Submodule.topologicalClosure.{u4, u2} R₁ M₁ _inst_1 _inst_4 _inst_5 _inst_14 (ContinuousSMul.continuousConstSMul.{u4, u2} R₁ M₁ _inst_19 _inst_4 (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) _inst_21) _inst_22 s) (Top.top.{u2} (Submodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (Submodule.instTopSubmodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14))) -> (Eq.{succ u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (Submodule.topologicalClosure.{u3, u1} R₂ M₂ _inst_2 _inst_8 _inst_9 _inst_16 (ContinuousSMul.continuousConstSMul.{u3, u1} R₂ M₂ _inst_20 _inst_8 (SMulZeroClass.toSMul.{u3, u1} R₂ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₂ M₂ (MonoidWithZero.toZero.{u3} R₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₂ M₂ (Semiring.toMonoidWithZero.{u3} R₂ _inst_2) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)))) _inst_23) _inst_24 (Submodule.map.{u4, u3, u2, u1, max u2 u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instSemilinearMapClassLinearMap.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) s)) (Top.top.{u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (Submodule.instTopSubmodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)))) +Case conversion may be inaccurate. Consider using '#align dense_range.topological_closure_map_submodule DenseRange.topologicalClosure_map_submoduleₓ'. -/ /-- Under a dense continuous linear map, a submodule whose `topological_closure` is `⊤` is sent to another such submodule. That is, the image of a dense set under a map with dense range is dense. -/ @@ -667,15 +942,33 @@ instance : MulAction S₂ (M₁ →SL[σ₁₂] M₂) one_smul f := ext fun x => one_smul _ _ mul_smul a b f := ext fun x => mul_smul _ _ _ +/- warning: continuous_linear_map.smul_apply -> ContinuousLinearMap.smul_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u5}} [_inst_18 : Monoid.{u5} S₂] [_inst_20 : DistribMulAction.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u2, u5, u4} R₂ S₂ M₂ (SMulZeroClass.toHasSmul.{u2, u4} R₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u2, u4} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u2, u4} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (Module.toMulActionWithZero.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u5, u4} S₂ M₂ _inst_8 (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (x : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (SMul.smul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toHasSmul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u1, u2, u3, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22)) c f) x) (SMul.smul.{u5, u4} S₂ M₂ (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20))) c (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f x)) +but is expected to have type + forall {R₁ : Type.{u5}} {R₂ : Type.{u4}} [_inst_1 : Semiring.{u5} R₁] [_inst_2 : Semiring.{u4} R₂] {σ₁₂ : RingHom.{u5, u4} R₁ R₂ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u5, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u4, u2} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u1}} [_inst_18 : Monoid.{u1} S₂] [_inst_20 : DistribMulAction.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u4, u1, u2} R₂ S₂ M₂ (SMulZeroClass.toSMul.{u4, u2} R₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₂ M₂ (MonoidWithZero.toZero.{u4} R₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₂ M₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (Module.toMulActionWithZero.{u4, u2} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u1, u2} S₂ M₂ _inst_8 (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (x : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u5, u4, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u5, u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22))) c f) x) (HSMul.hSMul.{u1, u2, u2} S₂ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (instHSMul.{u1, u2} S₂ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (SMulZeroClass.toSMul.{u1, u2} S₂ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toAddZeroClass.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_18 (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9) _inst_20)))) c (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u5, u4, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_apply ContinuousLinearMap.smul_applyₓ'. -/ theorem smul_apply (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (c • f) x = c • f x := rfl #align continuous_linear_map.smul_apply ContinuousLinearMap.smul_apply +/- warning: continuous_linear_map.coe_smul -> ContinuousLinearMap.coe_smul is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u5}} [_inst_18 : Monoid.{u5} S₂] [_inst_20 : DistribMulAction.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u2, u5, u4} R₂ S₂ M₂ (SMulZeroClass.toHasSmul.{u2, u4} R₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u2, u4} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u2, u4} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (Module.toMulActionWithZero.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u5, u4} S₂ M₂ _inst_8 (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u3) (succ u4)} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) ((fun (a : Type.{max u3 u4}) (b : Sort.{max (succ u3) (succ 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_inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) (SMul.smul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toHasSmul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u1, u2, u3, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22)) c f)) (SMul.smul.{u5, max u3 u4} S₂ (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.hasSmul.{u1, u2, u5, u3, u4} R₁ R₂ S₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 _inst_20 _inst_21) c ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f)) +but is expected to have type + forall {R₁ : Type.{u5}} {R₂ : Type.{u4}} [_inst_1 : Semiring.{u5} R₁] [_inst_2 : Semiring.{u4} R₂] {σ₁₂ : RingHom.{u5, u4} R₁ R₂ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u5, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u4, u2} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u1}} [_inst_18 : Monoid.{u1} S₂] [_inst_20 : DistribMulAction.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u4, u1, u2} R₂ S₂ M₂ (SMulZeroClass.toSMul.{u4, u2} R₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₂ M₂ (MonoidWithZero.toZero.{u4} R₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₂ M₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (Module.toMulActionWithZero.{u4, u2} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u1, u2} S₂ M₂ _inst_8 (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u3) (succ u2)} (LinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (HSMul.hSMul.{u1, max u3 u2, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u5, u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22))) c f)) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} S₂ (LinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (instHSMul.{u1, max u3 u2} S₂ (LinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instSMulLinearMap.{u5, u4, u1, u3, u2} R₁ R₂ S₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ _inst_18 _inst_20 _inst_21)) c (ContinuousLinearMap.toLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_smul ContinuousLinearMap.coe_smulₓ'. -/ @[simp, norm_cast] theorem coe_smul (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : (↑(c • f) : M₁ →ₛₗ[σ₁₂] M₂) = c • f := rfl #align continuous_linear_map.coe_smul ContinuousLinearMap.coe_smul +/- warning: continuous_linear_map.coe_smul' -> ContinuousLinearMap.coe_smul' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u5}} [_inst_18 : Monoid.{u5} S₂] [_inst_20 : DistribMulAction.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u2, u5, u4} R₂ S₂ M₂ (SMulZeroClass.toHasSmul.{u2, u4} R₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u2, u4} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u2, u4} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (Module.toMulActionWithZero.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u5, u4} S₂ M₂ _inst_8 (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{succ (max u3 u4)} (M₁ -> M₂) (coeFn.{succ (max u3 u4), succ (max u3 u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (SMul.smul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toHasSmul.{u5, max u3 u4} S₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u1, u2, u3, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22)) c f)) (SMul.smul.{u5, max u3 u4} S₂ (M₁ -> M₂) (Function.hasSMul.{u3, u5, u4} M₁ S₂ M₂ (SMulZeroClass.toHasSmul.{u5, u4} S₂ M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))) (DistribSMul.toSmulZeroClass.{u5, u4} S₂ M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u5, u4} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9) _inst_20)))) c (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f)) +but is expected to have type + forall {R₁ : Type.{u5}} {R₂ : Type.{u4}} [_inst_1 : Semiring.{u5} R₁] [_inst_2 : Semiring.{u4} R₂] {σ₁₂ : RingHom.{u5, u4} R₁ R₂ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u5, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u4, u2} R₂ M₂ _inst_2 _inst_9] {S₂ : Type.{u1}} [_inst_18 : Monoid.{u1} S₂] [_inst_20 : DistribMulAction.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)] [_inst_21 : SMulCommClass.{u4, u1, u2} R₂ S₂ M₂ (SMulZeroClass.toSMul.{u4, u2} R₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₂ M₂ (MonoidWithZero.toZero.{u4} R₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₂ M₂ (Semiring.toMonoidWithZero.{u4} R₂ _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (Module.toMulActionWithZero.{u4, u2} R₂ M₂ _inst_2 _inst_9 _inst_16)))) (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] [_inst_22 : ContinuousConstSMul.{u1, u2} S₂ M₂ _inst_8 (SMulZeroClass.toSMul.{u1, u2} S₂ M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribSMul.toSMulZeroClass.{u1, u2} S₂ M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9)) (DistribMulAction.toDistribSMul.{u1, u2} S₂ M₂ _inst_18 (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9) _inst_20)))] (c : S₂) (f : ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u3) (succ u2)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u5, u4, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (MulAction.toSMul.{u1, max u3 u2} S₂ (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) _inst_18 (ContinuousLinearMap.mulAction.{u5, u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 S₂ _inst_18 _inst_20 _inst_21 _inst_22))) c f)) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} S₂ (forall (a : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) a) (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (instHSMul.{u1, max u3 u2} S₂ (forall (a : M₁), (fun 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(x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) i) _inst_18 (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) i) _inst_9) _inst_20))))) c (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u5, u4, u3, u2} (ContinuousLinearMap.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u5, u4, u3, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_smul' ContinuousLinearMap.coe_smul'ₓ'. -/ @[simp, norm_cast] theorem coe_smul' (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : ⇑(c • f) = c • f := rfl @@ -696,21 +989,45 @@ instance : Zero (M₁ →SL[σ₁₂] M₂) := instance : Inhabited (M₁ →SL[σ₁₂] M₂) := ⟨0⟩ +/- warning: continuous_linear_map.default_def -> ContinuousLinearMap.default_def is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9], Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (Inhabited.default.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.inhabited.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_9], Eq.{max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (Inhabited.default.{max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.inhabited.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) (OfNat.ofNat.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.default_def ContinuousLinearMap.default_defₓ'. -/ @[simp] theorem default_def : (default : M₁ →SL[σ₁₂] M₂) = 0 := rfl #align continuous_linear_map.default_def ContinuousLinearMap.default_def +/- warning: continuous_linear_map.zero_apply -> ContinuousLinearMap.zero_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (x : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) x) (OfNat.ofNat.{u4} M₂ 0 (OfNat.mk.{u4} M₂ 0 (Zero.zero.{u4} M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u2, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u4} R₂ M₂ _inst_2 _inst_9] (x : M₁), Eq.{succ u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u2, u1, u3, u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u3 u4} (ContinuousLinearMap.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u2, u1, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) x) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.zero_apply ContinuousLinearMap.zero_applyₓ'. -/ @[simp] theorem zero_apply (x : M₁) : (0 : M₁ →SL[σ₁₂] M₂) x = 0 := rfl #align continuous_linear_map.zero_apply ContinuousLinearMap.zero_apply +/- warning: continuous_linear_map.coe_zero -> ContinuousLinearMap.coe_zero is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9], Eq.{max (succ u3) (succ u4)} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.LinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))))) (OfNat.ofNat.{max u3 u4} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u3 u4} (LinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.hasZero.{u1, u2, u3, u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂)))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_9], Eq.{max (succ u4) (succ u3)} (LinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (OfNat.ofNat.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) (OfNat.ofNat.{max u4 u3} (LinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u4 u3} (LinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instZeroLinearMap.{u2, u1, u4, u3} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_zero ContinuousLinearMap.coe_zeroₓ'. -/ @[simp, norm_cast] theorem coe_zero : ((0 : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂) = 0 := rfl #align continuous_linear_map.coe_zero ContinuousLinearMap.coe_zero +/- warning: continuous_linear_map.coe_zero' -> ContinuousLinearMap.coe_zero' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9], Eq.{succ (max u3 u4)} (M₁ -> M₂) (coeFn.{max (succ u3) (succ u4), succ (max u3 u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))))) (OfNat.ofNat.{max u3 u4} (M₁ -> M₂) 0 (OfNat.mk.{max u3 u4} (M₁ -> M₂) 0 (Zero.zero.{max u3 u4} (M₁ -> M₂) (Pi.instZero.{u3, u4} M₁ (fun (ᾰ : M₁) => M₂) (fun (i : M₁) => AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9))))))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_9], Eq.{max (succ u4) (succ u3)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u2, u1, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (OfNat.ofNat.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u4 u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) (OfNat.ofNat.{max u4 u3} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) 0 (Zero.toOfNat0.{max u4 u3} (forall (a : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) a) (Pi.instZero.{u4, u3} M₁ (fun (a : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) a) (fun (i : M₁) => AddMonoid.toZero.{u3} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) i) (AddCommMonoid.toAddMonoid.{u3} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) i) _inst_9))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_zero' ContinuousLinearMap.coe_zero'ₓ'. -/ /- no simp attribute on the next line as simp does not always simplify `0 x` to `0` when `0` is the zero function, while it does for the zero continuous linear map, and this is the most important property we care about. -/ @@ -719,14 +1036,24 @@ theorem coe_zero' : ⇑(0 : M₁ →SL[σ₁₂] M₂) = 0 := rfl #align continuous_linear_map.coe_zero' ContinuousLinearMap.coe_zero' +#print ContinuousLinearMap.uniqueOfLeft /- instance uniqueOfLeft [Subsingleton M₁] : Unique (M₁ →SL[σ₁₂] M₂) := coe_injective.unique #align continuous_linear_map.unique_of_left ContinuousLinearMap.uniqueOfLeft +-/ +#print ContinuousLinearMap.uniqueOfRight /- instance uniqueOfRight [Subsingleton M₂] : Unique (M₁ →SL[σ₁₂] M₂) := coe_injective.unique #align continuous_linear_map.unique_of_right ContinuousLinearMap.uniqueOfRight +-/ +/- warning: continuous_linear_map.exists_ne_zero -> ContinuousLinearMap.exists_ne_zero is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (Ne.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f (OfNat.ofNat.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u3 u4} 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M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))))))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] {f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16}, (Ne.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f (OfNat.ofNat.{max u2 u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u2 u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) -> (Exists.{succ u2} M₁ (fun (x : M₁) => Ne.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_9)))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.exists_ne_zero ContinuousLinearMap.exists_ne_zeroₓ'. -/ theorem exists_ne_zero {f : M₁ →SL[σ₁₂] M₂} (hf : f ≠ 0) : ∃ x, f x ≠ 0 := by by_contra' h @@ -737,39 +1064,69 @@ section variable (R₁ M₁) +#print ContinuousLinearMap.id /- /-- the identity map as a continuous linear map. -/ def id : M₁ →L[R₁] M₁ := ⟨LinearMap.id, continuous_id⟩ #align continuous_linear_map.id ContinuousLinearMap.id +-/ end instance : One (M₁ →L[R₁] M₁) := ⟨id R₁ M₁⟩ +#print ContinuousLinearMap.one_def /- theorem one_def : (1 : M₁ →L[R₁] M₁) = id R₁ M₁ := rfl #align continuous_linear_map.one_def ContinuousLinearMap.one_def +-/ +/- warning: continuous_linear_map.id_apply -> ContinuousLinearMap.id_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (x : M₁), Eq.{succ u2} M₁ (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.id.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) x) x +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (x : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) x) (FunLike.coe.{succ u2, succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u2, u1, u1, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) (ContinuousLinearMap.id.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) x) x +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.id_apply ContinuousLinearMap.id_applyₓ'. -/ theorem id_apply (x : M₁) : id R₁ M₁ x = x := rfl #align continuous_linear_map.id_apply ContinuousLinearMap.id_apply +#print ContinuousLinearMap.coe_id /- @[simp, norm_cast] theorem coe_id : (id R₁ M₁ : M₁ →ₗ[R₁] M₁) = LinearMap.id := rfl #align continuous_linear_map.coe_id ContinuousLinearMap.coe_id +-/ +/- warning: continuous_linear_map.coe_id' -> ContinuousLinearMap.coe_id' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5], Eq.{succ u2} (M₁ -> M₁) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.id.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) (id.{succ u2} M₁) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5], Eq.{succ u2} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) ᾰ) (FunLike.coe.{succ u2, succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u2, u1, u1, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) (ContinuousLinearMap.id.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) (id.{succ u2} M₁) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_id' ContinuousLinearMap.coe_id'ₓ'. -/ @[simp, norm_cast] theorem coe_id' : ⇑(id R₁ M₁) = id := rfl #align continuous_linear_map.coe_id' ContinuousLinearMap.coe_id' +/- warning: continuous_linear_map.coe_eq_id -> ContinuousLinearMap.coe_eq_id is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] {f : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14}, Iff (Eq.{succ u2} (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (HasLiftT.mk.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (CoeTCₓ.coe.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (coeBase.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.LinearMap.coe.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14)))) f) (LinearMap.id.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) (Eq.{succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) f (ContinuousLinearMap.id.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] {f : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14}, Iff (Eq.{succ u1} (LinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.toLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 f) (LinearMap.id.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) (Eq.{succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) f (ContinuousLinearMap.id.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_eq_id ContinuousLinearMap.coe_eq_idₓ'. -/ @[simp, norm_cast] theorem coe_eq_id {f : M₁ →L[R₁] M₁} : (f : M₁ →ₗ[R₁] M₁) = LinearMap.id ↔ f = id _ _ := by rw [← coe_id, coe_inj] #align continuous_linear_map.coe_eq_id ContinuousLinearMap.coe_eq_id +/- warning: continuous_linear_map.one_apply -> ContinuousLinearMap.one_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (x : M₁), Eq.{succ u2} M₁ (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (OfNat.ofNat.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) 1 (OfNat.mk.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) 1 (One.one.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.one.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)))) x) x +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (x : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) x) (FunLike.coe.{succ u2, succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u2, u1, u1, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) (OfNat.ofNat.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) 1 (One.toOfNat1.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.one.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14))) x) x +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.one_apply ContinuousLinearMap.one_applyₓ'. -/ @[simp] theorem one_apply (x : M₁) : (1 : M₁ →L[R₁] M₁) x = x := rfl @@ -782,16 +1139,34 @@ variable [ContinuousAdd M₂] instance : Add (M₁ →SL[σ₁₂] M₂) := ⟨fun f g => ⟨f + g, f.2.add g.2⟩⟩ +/- warning: continuous_linear_map.add_apply -> ContinuousLinearMap.add_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (g : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (x : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (HAdd.hAdd.{max u3 u4, max u3 u4, max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u1, 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_inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) g x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.add_apply ContinuousLinearMap.add_applyₓ'. -/ @[simp] theorem add_apply (f g : M₁ →SL[σ₁₂] M₂) (x : M₁) : (f + g) x = f x + g x := rfl #align continuous_linear_map.add_apply ContinuousLinearMap.add_apply +/- warning: continuous_linear_map.coe_add -> ContinuousLinearMap.coe_add is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ 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_inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHAdd.{max u2 u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.add.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_18)) f g)) (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (instHAdd.{max u2 u1} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (LinearMap.instAddLinearMap.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂)) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_add ContinuousLinearMap.coe_addₓ'. -/ @[simp, norm_cast] theorem coe_add (f g : M₁ →SL[σ₁₂] M₂) : (↑(f + g) : M₁ →ₛₗ[σ₁₂] M₂) = f + g := rfl #align continuous_linear_map.coe_add ContinuousLinearMap.coe_add +/- warning: continuous_linear_map.coe_add' -> ContinuousLinearMap.coe_add' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ 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Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (g : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u2) (succ u1)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) 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(ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_add' ContinuousLinearMap.coe_add'ₓ'. -/ @[norm_cast] theorem coe_add' (f g : M₁ →SL[σ₁₂] M₂) : ⇑(f + g) = f + g := rfl @@ -817,17 +1192,35 @@ instance : AddCommMonoid (M₁ →SL[σ₁₂] M₂) ext simp [Nat.succ_eq_one_add, add_smul] +/- warning: continuous_linear_map.coe_sum -> ContinuousLinearMap.coe_sum is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ 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[_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] {ι : Type.{u5}} (t : Finset.{u5} ι) (f : ι -> (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)), Eq.{max (succ u2) (succ u1)} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (Finset.sum.{max u2 u1, u5} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) ι (ContinuousLinearMap.addCommMonoid.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_18) t (fun (d : ι) => f d))) (Finset.sum.{max u2 u1, u5} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ M₂ _inst_5 _inst_9 _inst_14 _inst_16) ι (LinearMap.addCommMonoid.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂) t (fun (d : ι) => ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (f d))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_sum ContinuousLinearMap.coe_sumₓ'. -/ @[simp, norm_cast] theorem coe_sum {ι : Type _} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ↑(∑ d in t, f d) = (∑ d in t, f d : M₁ →ₛₗ[σ₁₂] M₂) := (AddMonoidHom.mk (coe : (M₁ →SL[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₂] M₂) rfl fun _ _ => rfl).map_sum _ _ #align continuous_linear_map.coe_sum ContinuousLinearMap.coe_sum +/- warning: continuous_linear_map.coe_sum' -> ContinuousLinearMap.coe_sum' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))] {ι : Type.{u5}} (t : Finset.{u5} ι) (f : ι -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)), Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) (coeFn.{succ (max u3 u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (Finset.sum.{max u3 u4, u5} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) ι (ContinuousLinearMap.addCommMonoid.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_18) t (fun (d : ι) => f d))) (Finset.sum.{max u3 u4, u5} (M₁ -> M₂) ι (Pi.addCommMonoid.{u3, u4} M₁ (fun (ᾰ : M₁) => M₂) (fun (i : M₁) => _inst_9)) t (fun (d : ι) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f d))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] {ι : Type.{u5}} (t : Finset.{u5} ι) (f : ι -> (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)), Eq.{max (succ u2) (succ u1)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun 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Consider using '#align continuous_linear_map.coe_sum' ContinuousLinearMap.coe_sum'ₓ'. -/ @[simp, norm_cast] theorem coe_sum' {ι : Type _} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) : ⇑(∑ d in t, f d) = ∑ d in t, f d := by simp only [← coe_coe, coe_sum, LinearMap.coeFn_sum] #align continuous_linear_map.coe_sum' ContinuousLinearMap.coe_sum' +/- warning: continuous_linear_map.sum_apply -> ContinuousLinearMap.sum_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u4} M₂ _inst_8 (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_9)))] {ι : Type.{u5}} (t : Finset.{u5} ι) (f : ι -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) (b : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (Finset.sum.{max u3 u4, u5} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) ι (ContinuousLinearMap.addCommMonoid.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_18) t (fun (d : ι) => f d)) b) (Finset.sum.{u4, u5} M₂ ι _inst_9 t (fun (d : ι) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f d) b)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] [_inst_18 : ContinuousAdd.{u1} M₂ _inst_8 (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)))] {ι : Type.{u5}} (t : Finset.{u5} ι) (f : ι -> (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) (b : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) b) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (Finset.sum.{max u2 u1, u5} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) ι (ContinuousLinearMap.addCommMonoid.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_18) t (fun (d : ι) => f d)) b) (Finset.sum.{u1, u5} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) b) ι _inst_9 t (fun (d : ι) => FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) (f d) b)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.sum_apply ContinuousLinearMap.sum_applyₓ'. -/ theorem sum_apply {ι : Type _} (t : Finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) (b : M₁) : (∑ d in t, f d) b = ∑ d in t, f d b := by simp only [coe_sum', Finset.sum_apply] #align continuous_linear_map.sum_apply ContinuousLinearMap.sum_apply @@ -836,16 +1229,24 @@ end Add variable [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] +#print ContinuousLinearMap.comp /- /-- Composition of bounded linear maps. -/ def comp (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : M₁ →SL[σ₁₃] M₃ := ⟨(g : M₂ →ₛₗ[σ₂₃] M₃).comp ↑f, g.2.comp f.2⟩ #align continuous_linear_map.comp ContinuousLinearMap.comp +-/ -- mathport name: «expr ∘L » infixr:80 " ∘L " => @ContinuousLinearMap.comp _ _ _ _ _ _ (RingHom.id _) (RingHom.id _) (RingHom.id _) _ _ _ _ _ _ _ _ _ _ _ _ RingHomCompTriple.ids +/- warning: continuous_linear_map.coe_comp -> ContinuousLinearMap.coe_comp is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (h : 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Consider using '#align continuous_linear_map.coe_comp ContinuousLinearMap.coe_compₓ'. -/ @[simp, norm_cast] theorem coe_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : (h.comp f : M₁ →ₛₗ[σ₁₃] M₃) = (h : M₂ →ₛₗ[σ₂₃] M₃).comp (f : M₁ →ₛₗ[σ₁₂] M₂) := @@ -854,22 +1255,46 @@ theorem coe_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) include σ₁₃ +/- warning: continuous_linear_map.coe_comp' -> ContinuousLinearMap.coe_comp' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (h : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u6)} (M₁ -> M₃) (coeFn.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (fun (_x : 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TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_10 : TopologicalSpace.{u3} M₃] [_inst_11 : AddCommMonoid.{u3} M₃] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u6, u4} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u5, u3} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u2, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (h : ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u1) (succ u3)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) ᾰ) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (ContinuousLinearMap.{u2, u5, u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 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succ u4, succ u3} M₁ M₂ M₃ (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₃) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) M₂ M₃ _inst_8 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u6, u5, u4, u3} (ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17 (ContinuousLinearMap.continuousSemilinearMapClass.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17))) h) (FunLike.coe.{max (succ u1) (succ u4), succ u1, succ 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Consider using '#align continuous_linear_map.coe_comp' ContinuousLinearMap.coe_comp'ₓ'. -/ @[simp, norm_cast] theorem coe_comp' (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : ⇑(h.comp f) = h ∘ f := rfl #align continuous_linear_map.coe_comp' ContinuousLinearMap.coe_comp' +/- warning: continuous_linear_map.comp_apply -> ContinuousLinearMap.comp_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (x : M₁), Eq.{succ u6} M₃ (coeFn.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (fun (_x : ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) => M₁ -> M₃) (ContinuousLinearMap.toFun.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f) x) (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (fun (_x : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) => M₂ -> M₃) (ContinuousLinearMap.toFun.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) g (coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f x)) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u6}} {R₃ : Type.{u5}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u6} R₂] [_inst_3 : Semiring.{u5} R₃] {σ₁₂ : RingHom.{u2, u6} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)} {σ₂₃ : RingHom.{u6, u5} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u5} R₃ _inst_3)} {σ₁₃ : RingHom.{u2, u5} R₁ R₃ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₃ _inst_3)} {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_10 : TopologicalSpace.{u3} M₃] [_inst_11 : AddCommMonoid.{u3} M₃] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u6, u4} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u5, u3} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u2, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (g : ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (x : M₁), Eq.{succ u3} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) x) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (ContinuousLinearMap.{u2, u5, u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) _x) 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_inst_16 _inst_17) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₃) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) M₂ M₃ _inst_8 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u6, u5, u4, u3} (ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17 (ContinuousLinearMap.continuousSemilinearMapClass.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17))) g (FunLike.coe.{max (succ u1) (succ u4), succ u1, succ u4} (ContinuousLinearMap.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u1 u4, u1, u4} (ContinuousLinearMap.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u4, u2, u6, u1, u4} (ContinuousLinearMap.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u6, u1, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_apply ContinuousLinearMap.comp_applyₓ'. -/ theorem comp_apply (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (g.comp f) x = g (f x) := rfl #align continuous_linear_map.comp_apply ContinuousLinearMap.comp_apply omit σ₁₃ +/- warning: continuous_linear_map.comp_id -> ContinuousLinearMap.comp_id is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.comp.{u1, u1, u2, u3, u3, u4} R₁ R₁ R₂ _inst_1 _inst_1 _inst_2 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) σ₁₂ σ₁₂ M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_14 _inst_16 (RingHomCompTriple.ids.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂) f (ContinuousLinearMap.id.{u1, u3} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.comp.{u4, u4, u3, u2, u2, u1} R₁ R₁ R₂ _inst_1 _inst_1 _inst_2 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) σ₁₂ σ₁₂ M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_14 _inst_16 (RingHomCompTriple.ids.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂) f (ContinuousLinearMap.id.{u4, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_id ContinuousLinearMap.comp_idₓ'. -/ @[simp] theorem comp_id (f : M₁ →SL[σ₁₂] M₂) : f.comp (id R₁ M₁) = f := ext fun x => rfl #align continuous_linear_map.comp_id ContinuousLinearMap.comp_id +/- warning: continuous_linear_map.id_comp -> ContinuousLinearMap.id_comp is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.comp.{u1, u2, u2, u3, u4, u4} R₁ R₂ R₂ _inst_1 _inst_2 _inst_2 σ₁₂ (RingHom.id.{u2} R₂ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)) σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_16 (RingHomCompTriple.right_ids.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂) (ContinuousLinearMap.id.{u2, u4} R₂ _inst_2 M₂ _inst_8 _inst_9 _inst_16) f) f +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.comp.{u4, u3, u3, u2, u1, u1} R₁ R₂ R₂ _inst_1 _inst_2 _inst_2 σ₁₂ (RingHom.id.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_16 (RingHomCompTriple.right_ids.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂) (ContinuousLinearMap.id.{u3, u1} R₂ _inst_2 M₂ _inst_8 _inst_9 _inst_16) f) f +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.id_comp ContinuousLinearMap.id_compₓ'. -/ @[simp] theorem id_comp (f : M₁ →SL[σ₁₂] M₂) : (id R₂ M₂).comp f = f := ext fun x => rfl @@ -877,6 +1302,12 @@ theorem id_comp (f : M₁ →SL[σ₁₂] M₂) : (id R₂ M₂).comp f = f := include σ₁₃ +/- warning: continuous_linear_map.comp_zero -> ContinuousLinearMap.comp_zero is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g (OfNat.ofNat.{max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (OfNat.mk.{max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.zero.{max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))))) (OfNat.ofNat.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (OfNat.mk.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (Zero.zero.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.zero.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17)))) +but is expected to have type + forall {R₁ : Type.{u1}} {R₂ : Type.{u6}} {R₃ : Type.{u5}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u6} R₂] [_inst_3 : Semiring.{u5} R₃] {σ₁₂ : RingHom.{u1, u6} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)} {σ₂₃ : RingHom.{u6, u5} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u5} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u5} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₃ _inst_3)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_10 : TopologicalSpace.{u3} M₃] [_inst_11 : AddCommMonoid.{u3} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u6, u4} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u5, u3} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (g : ContinuousLinearMap.{u6, u5, u4, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17), Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u5, u2, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u6, u5, u2, u4, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g (OfNat.ofNat.{max u2 u4} (ContinuousLinearMap.{u1, u6, u2, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) 0 (Zero.toOfNat0.{max u2 u4} (ContinuousLinearMap.{u1, u6, u2, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.zero.{u1, u6, u2, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) (OfNat.ofNat.{max u2 u3} (ContinuousLinearMap.{u1, u5, u2, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (Zero.toOfNat0.{max u2 u3} (ContinuousLinearMap.{u1, u5, u2, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.zero.{u1, u5, u2, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_zero ContinuousLinearMap.comp_zeroₓ'. -/ @[simp] theorem comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁₂] M₂) = 0 := by @@ -884,6 +1315,12 @@ theorem comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁ simp #align continuous_linear_map.comp_zero ContinuousLinearMap.comp_zero +/- warning: continuous_linear_map.zero_comp -> ContinuousLinearMap.zero_comp is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (f : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 (OfNat.ofNat.{max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) 0 (OfNat.mk.{max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) 0 (Zero.zero.{max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.zero.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17)))) f) (OfNat.ofNat.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (OfNat.mk.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (Zero.zero.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.zero.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17)))) +but is expected to have type + forall {R₁ : Type.{u6}} {R₂ : Type.{u5}} {R₃ : Type.{u1}} [_inst_1 : Semiring.{u6} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u1} R₃] {σ₁₂ : RingHom.{u6, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₃ : RingHom.{u5, u1} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₃ _inst_3)} {σ₁₃ : RingHom.{u6, u1} R₁ R₃ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u6, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u5, u3} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u1, u2} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u6, u5, u1} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (f : ContinuousLinearMap.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u2)} (ContinuousLinearMap.{u6, u1, u4, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u6, u5, u1, u4, u3, u2} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 (OfNat.ofNat.{max u3 u2} (ContinuousLinearMap.{u5, u1, u3, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) 0 (Zero.toOfNat0.{max u3 u2} (ContinuousLinearMap.{u5, u1, u3, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.zero.{u5, u1, u3, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17))) f) (OfNat.ofNat.{max u4 u2} (ContinuousLinearMap.{u6, u1, u4, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) 0 (Zero.toOfNat0.{max u4 u2} (ContinuousLinearMap.{u6, u1, u4, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.zero.{u6, u1, u4, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.zero_comp ContinuousLinearMap.zero_compₓ'. -/ @[simp] theorem zero_comp (f : M₁ →SL[σ₁₂] M₂) : (0 : M₂ →SL[σ₂₃] M₃).comp f = 0 := by @@ -891,6 +1328,12 @@ theorem zero_comp (f : M₁ →SL[σ₁₂] M₂) : (0 : M₂ →SL[σ₂₃] M simp #align continuous_linear_map.zero_comp ContinuousLinearMap.zero_comp +/- warning: continuous_linear_map.comp_add -> ContinuousLinearMap.comp_add is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_19 : ContinuousAdd.{u5} M₂ _inst_8 (AddZeroClass.toHasAdd.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_9)))] [_inst_20 : ContinuousAdd.{u6} M₃ _inst_10 (AddZeroClass.toHasAdd.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_11)))] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f₁ : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f₂ : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g (HAdd.hAdd.{max u4 u5, max u4 u5, max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHAdd.{max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.add.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_19)) f₁ f₂)) (HAdd.hAdd.{max u4 u6, max u4 u6, max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (instHAdd.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.add.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17 _inst_20)) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f₁) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f₂)) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u4}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u4} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u2, u4} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₂ _inst_2)} {σ₂₃ : RingHom.{u4, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u4} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u2, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u6}} [_inst_8 : TopologicalSpace.{u6} M₂] [_inst_9 : AddCommMonoid.{u6} M₂] {M₃ : Type.{u5}} [_inst_10 : TopologicalSpace.{u5} M₃] [_inst_11 : AddCommMonoid.{u5} M₃] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u4, u6} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u5} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u2, u4, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_19 : ContinuousAdd.{u6} M₂ _inst_8 (AddZeroClass.toAdd.{u6} M₂ (AddMonoid.toAddZeroClass.{u6} M₂ (AddCommMonoid.toAddMonoid.{u6} M₂ _inst_9)))] [_inst_20 : ContinuousAdd.{u5} M₃ _inst_10 (AddZeroClass.toAdd.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_11)))] (g : ContinuousLinearMap.{u4, u3, u6, u5} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f₁ : ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (f₂ : ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u1) (succ u5)} (ContinuousLinearMap.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u2, u4, u3, u1, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g (HAdd.hAdd.{max u1 u6, max u1 u6, max u1 u6} (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (instHAdd.{max u1 u6} (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousLinearMap.add.{u2, u4, u1, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_19)) f₁ f₂)) (HAdd.hAdd.{max u1 u5, max u1 u5, max u1 u5} (ContinuousLinearMap.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (instHAdd.{max u1 u5} (ContinuousLinearMap.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.add.{u2, u3, u1, u5} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17 _inst_20)) (ContinuousLinearMap.comp.{u2, u4, u3, u1, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f₁) (ContinuousLinearMap.comp.{u2, u4, u3, u1, u6, u5} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_add ContinuousLinearMap.comp_addₓ'. -/ @[simp] theorem comp_add [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M₁ →SL[σ₁₂] M₂) : g.comp (f₁ + f₂) = g.comp f₁ + g.comp f₂ := @@ -899,6 +1342,12 @@ theorem comp_add [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂ simp #align continuous_linear_map.comp_add ContinuousLinearMap.comp_add +/- warning: continuous_linear_map.add_comp -> ContinuousLinearMap.add_comp is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_19 : ContinuousAdd.{u6} M₃ _inst_10 (AddZeroClass.toHasAdd.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_11)))] (g₁ : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (g₂ : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 (HAdd.hAdd.{max u5 u6, max u5 u6, max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (instHAdd.{max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.add.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17 _inst_19)) g₁ g₂) f) (HAdd.hAdd.{max u4 u6, max u4 u6, max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (instHAdd.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.add.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17 _inst_19)) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g₁ f) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g₂ f)) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u5}} {R₃ : Type.{u4}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u4} R₃] {σ₁₂ : RingHom.{u2, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} {σ₁₃ : RingHom.{u2, u4} R₁ R₃ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u5, u3} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u4, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u2, u5, u4} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_19 : ContinuousAdd.{u6} M₃ _inst_10 (AddZeroClass.toAdd.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_11)))] (g₁ : ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (g₂ : ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u2, u5, u1, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u1) (succ u6)} (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.comp.{u2, u5, u4, u1, u3, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 (HAdd.hAdd.{max u3 u6, max u3 u6, max u3 u6} (ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (instHAdd.{max u3 u6} (ContinuousLinearMap.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (ContinuousLinearMap.add.{u5, u4, u3, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17 _inst_19)) g₁ g₂) f) (HAdd.hAdd.{max u1 u6, max u1 u6, max u1 u6} (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (instHAdd.{max u1 u6} (ContinuousLinearMap.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17) (ContinuousLinearMap.add.{u2, u4, u1, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_17 _inst_19)) (ContinuousLinearMap.comp.{u2, u5, u4, u1, u3, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g₁ f) (ContinuousLinearMap.comp.{u2, u5, u4, u1, u3, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g₂ f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.add_comp ContinuousLinearMap.add_compₓ'. -/ @[simp] theorem add_comp [ContinuousAdd M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : (g₁ + g₂).comp f = g₁.comp f + g₂.comp f := by @@ -908,6 +1357,12 @@ theorem add_comp [ContinuousAdd M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f omit σ₁₃ +/- warning: continuous_linear_map.comp_assoc -> ContinuousLinearMap.comp_assoc is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {M₁ : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M₁] [_inst_5 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_10 : TopologicalSpace.{u6} M₃] [_inst_11 : AddCommMonoid.{u6} M₃] {M₄ : Type.{u7}} [_inst_12 : TopologicalSpace.{u7} M₄] [_inst_13 : AddCommMonoid.{u7} M₄] [_inst_14 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] {R₄ : Type.{u8}} [_inst_19 : Semiring.{u8} R₄] [_inst_20 : Module.{u8, u7} R₄ M₄ _inst_19 _inst_13] {σ₁₄ : RingHom.{u1, u8} R₁ R₄ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} {σ₂₄ : RingHom.{u2, u8} R₂ R₄ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} {σ₃₄ : RingHom.{u3, u8} R₃ R₄ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} [_inst_21 : RingHomCompTriple.{u1, u3, u8} R₁ R₃ R₄ _inst_1 _inst_3 _inst_19 σ₁₃ σ₃₄ σ₁₄] [_inst_22 : RingHomCompTriple.{u2, u3, u8} R₂ R₃ R₄ _inst_2 _inst_3 _inst_19 σ₂₃ σ₃₄ σ₂₄] [_inst_23 : RingHomCompTriple.{u1, u2, u8} R₁ R₂ R₄ _inst_1 _inst_2 _inst_19 σ₁₂ σ₂₄ σ₁₄] (h : ContinuousLinearMap.{u3, u8, u6, u7} R₃ R₄ _inst_3 _inst_19 σ₃₄ M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_17 _inst_20) (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u4) (succ u7)} (ContinuousLinearMap.{u1, u8, u4, u7} R₁ R₄ _inst_1 _inst_19 σ₁₄ M₁ _inst_4 _inst_5 M₄ _inst_12 _inst_13 _inst_14 _inst_20) (ContinuousLinearMap.comp.{u1, u2, u8, u4, u5, u7} R₁ R₂ R₄ _inst_1 _inst_2 _inst_19 σ₁₂ σ₂₄ σ₁₄ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₄ _inst_12 _inst_13 _inst_14 _inst_16 _inst_20 _inst_23 (ContinuousLinearMap.comp.{u2, u3, u8, u5, u6, u7} R₂ R₃ R₄ _inst_2 _inst_3 _inst_19 σ₂₃ σ₃₄ σ₂₄ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_16 _inst_17 _inst_20 _inst_22 h g) f) (ContinuousLinearMap.comp.{u1, u3, u8, u4, u6, u7} R₁ R₃ R₄ _inst_1 _inst_3 _inst_19 σ₁₃ σ₃₄ σ₁₄ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_14 _inst_17 _inst_20 _inst_21 h (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f)) +but is expected to have type + forall {R₁ : Type.{u6}} {R₂ : Type.{u5}} {R₃ : Type.{u4}} [_inst_1 : Semiring.{u6} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u4} R₃] {σ₁₂ : RingHom.{u6, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} {σ₁₃ : RingHom.{u6, u4} R₁ R₃ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] {M₃ : Type.{u3}} [_inst_10 : TopologicalSpace.{u3} M₃] [_inst_11 : AddCommMonoid.{u3} M₃] {M₄ : Type.{u7}} [_inst_12 : TopologicalSpace.{u7} M₄] [_inst_13 : AddCommMonoid.{u7} M₄] [_inst_14 : Module.{u6, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u5, u2} R₂ M₂ _inst_2 _inst_9] [_inst_17 : Module.{u4, u3} R₃ M₃ _inst_3 _inst_11] [_inst_18 : RingHomCompTriple.{u6, u5, u4} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] {R₄ : Type.{u8}} [_inst_19 : Semiring.{u8} R₄] [_inst_20 : Module.{u8, u7} R₄ M₄ _inst_19 _inst_13] {σ₁₄ : RingHom.{u6, u8} R₁ R₄ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} {σ₂₄ : RingHom.{u5, u8} R₂ R₄ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} {σ₃₄ : RingHom.{u4, u8} R₃ R₄ (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u8} R₄ _inst_19)} [_inst_21 : RingHomCompTriple.{u6, u4, u8} R₁ R₃ R₄ _inst_1 _inst_3 _inst_19 σ₁₃ σ₃₄ σ₁₄] [_inst_22 : RingHomCompTriple.{u5, u4, u8} R₂ R₃ R₄ _inst_2 _inst_3 _inst_19 σ₂₃ σ₃₄ σ₂₄] [_inst_23 : RingHomCompTriple.{u6, u5, u8} R₁ R₂ R₄ _inst_1 _inst_2 _inst_19 σ₁₂ σ₂₄ σ₁₄] (h : ContinuousLinearMap.{u4, u8, u3, u7} R₃ R₄ _inst_3 _inst_19 σ₃₄ M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_17 _inst_20) (g : ContinuousLinearMap.{u5, u4, u2, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_16 _inst_17) (f : ContinuousLinearMap.{u6, u5, u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16), Eq.{max (succ u1) (succ u7)} (ContinuousLinearMap.{u6, u8, u1, u7} R₁ R₄ _inst_1 _inst_19 σ₁₄ M₁ _inst_4 _inst_5 M₄ _inst_12 _inst_13 _inst_14 _inst_20) (ContinuousLinearMap.comp.{u6, u5, u8, u1, u2, u7} R₁ R₂ R₄ _inst_1 _inst_2 _inst_19 σ₁₂ σ₂₄ σ₁₄ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₄ _inst_12 _inst_13 _inst_14 _inst_16 _inst_20 _inst_23 (ContinuousLinearMap.comp.{u5, u4, u8, u2, u3, u7} R₂ R₃ R₄ _inst_2 _inst_3 _inst_19 σ₂₃ σ₃₄ σ₂₄ M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_16 _inst_17 _inst_20 _inst_22 h g) f) (ContinuousLinearMap.comp.{u6, u4, u8, u1, u3, u7} R₁ R₃ R₄ _inst_1 _inst_3 _inst_19 σ₁₃ σ₃₄ σ₁₄ M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_14 _inst_17 _inst_20 _inst_21 h (ContinuousLinearMap.comp.{u6, u5, u4, u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_16 _inst_17 _inst_18 g f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_assoc ContinuousLinearMap.comp_assocₓ'. -/ theorem comp_assoc {R₄ : Type _} [Semiring R₄] [Module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : @@ -918,15 +1373,33 @@ theorem comp_assoc {R₄ : Type _} [Semiring R₄] [Module R₄ M₄] {σ₁₄ instance : Mul (M₁ →L[R₁] M₁) := ⟨comp⟩ +/- warning: continuous_linear_map.mul_def -> ContinuousLinearMap.mul_def is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (g : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14), Eq.{succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (HMul.hMul.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u2, u2} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 _inst_14 (RingHomCompTriple.right_ids.{u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) f g) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (g : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14), Eq.{succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (HMul.hMul.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g) (ContinuousLinearMap.comp.{u2, u2, u2, u1, u1, u1} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 _inst_14 (RingHomCompTriple.ids.{u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) f g) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.mul_def ContinuousLinearMap.mul_defₓ'. -/ theorem mul_def (f g : M₁ →L[R₁] M₁) : f * g = f.comp g := rfl #align continuous_linear_map.mul_def ContinuousLinearMap.mul_def +/- warning: continuous_linear_map.coe_mul -> ContinuousLinearMap.coe_mul is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (g : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14), Eq.{succ u2} (M₁ -> M₁) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ 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(RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g)) (Function.comp.{succ u2, succ u2, succ u2} M₁ M₁ M₁ (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) f) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) g)) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ 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(ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) (HMul.hMul.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g)) (Function.comp.{succ u1, succ u1, succ u1} M₁ M₁ M₁ (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) f) (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_mul ContinuousLinearMap.coe_mulₓ'. -/ @[simp] theorem coe_mul (f g : M₁ →L[R₁] M₁) : ⇑(f * g) = f ∘ g := rfl #align continuous_linear_map.coe_mul ContinuousLinearMap.coe_mul +/- warning: continuous_linear_map.mul_apply -> ContinuousLinearMap.mul_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (g : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (x : M₁), Eq.{succ u2} M₁ (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (HMul.hMul.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g) x) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) f (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) g x)) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] (f : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (g : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (x : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) x) (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) (HMul.hMul.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (instHMul.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.mul.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14)) f g) x) (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) f (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) g x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.mul_apply ContinuousLinearMap.mul_applyₓ'. -/ theorem mul_apply (f g : M₁ →L[R₁] M₁) (x : M₁) : (f * g) x = f (g x) := rfl #align continuous_linear_map.mul_apply ContinuousLinearMap.mul_apply @@ -950,6 +1423,12 @@ instance [ContinuousAdd M₁] : Semiring (M₁ →L[R₁] M₁) := left_distrib := fun f g h => ext fun x => map_add f (g x) (h x) right_distrib := fun _ _ _ => ext fun _ => LinearMap.add_apply _ _ _ } +/- warning: continuous_linear_map.to_linear_map_ring_hom -> ContinuousLinearMap.toLinearMapRingHom is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], RingHom.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (Semiring.toNonAssocSemiring.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (Semiring.toNonAssocSemiring.{u2} (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (Module.End.semiring.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], RingHom.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (Semiring.toNonAssocSemiring.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (Semiring.toNonAssocSemiring.{u2} (LinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ M₁ _inst_5 _inst_5 _inst_14 _inst_14) (Module.End.semiring.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.to_linear_map_ring_hom ContinuousLinearMap.toLinearMapRingHomₓ'. -/ /-- `continuous_linear_map.to_linear_map` as a `ring_hom`.-/ @[simps] def toLinearMapRingHom [ContinuousAdd M₁] : (M₁ →L[R₁] M₁) →+* M₁ →ₗ[R₁] M₁ @@ -965,6 +1444,12 @@ section ApplyAction variable [ContinuousAdd M₁] +/- warning: continuous_linear_map.apply_module -> ContinuousLinearMap.applyModule is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], Module.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], Module.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.apply_module ContinuousLinearMap.applyModuleₓ'. -/ /-- The tautological action by `M₁ →L[R₁] M₁` on `M`. This generalizes `function.End.apply_mul_action`. -/ @@ -972,41 +1457,83 @@ instance applyModule : Module (M₁ →L[R₁] M₁) M₁ := Module.compHom _ toLinearMapRingHom #align continuous_linear_map.apply_module ContinuousLinearMap.applyModule +/- warning: continuous_linear_map.smul_def -> ContinuousLinearMap.smul_def is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))] (f : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (a : M₁), Eq.{succ u2} M₁ (SMul.smul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toHasSmul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (MulZeroClass.toHasZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MulZeroOneClass.toMulZeroClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MonoidWithZero.toMulZeroOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) f a) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) => M₁ -> M₁) (ContinuousLinearMap.toFun.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) f a) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u1} M₁ _inst_4 (AddZeroClass.toAdd.{u1} M₁ (AddMonoid.toAddZeroClass.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)))] (f : ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (a : M₁), Eq.{succ u1} M₁ (HSMul.hSMul.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ (instHSMul.{u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toSMul.{u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.zero.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.monoidWithZero.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_5)) (Module.toMulActionWithZero.{u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)))))) f a) (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₁) _x) (ContinuousMapClass.toFunLike.{u1, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ M₁ _inst_4 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{u1, u2, u2, u1, u1} (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14))) f a) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_def ContinuousLinearMap.smul_defₓ'. -/ @[simp] protected theorem smul_def (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a := rfl #align continuous_linear_map.smul_def ContinuousLinearMap.smul_def +/- warning: continuous_linear_map.apply_has_faithful_smul -> ContinuousLinearMap.applyFaithfulSMul is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], FaithfulSMul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toHasSmul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (MulZeroClass.toHasZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MulZeroOneClass.toMulZeroClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MonoidWithZero.toMulZeroOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], FaithfulSMul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toSMul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.zero.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.monoidWithZero.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.apply_has_faithful_smul ContinuousLinearMap.applyFaithfulSMulₓ'. -/ /-- `continuous_linear_map.apply_module` is faithful. -/ -instance apply_faithfulSMul : FaithfulSMul (M₁ →L[R₁] M₁) M₁ := +instance applyFaithfulSMul : FaithfulSMul (M₁ →L[R₁] M₁) M₁ := ⟨fun _ _ => ContinuousLinearMap.ext⟩ -#align continuous_linear_map.apply_has_faithful_smul ContinuousLinearMap.apply_faithfulSMul - -instance apply_sMulCommClass : SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁ +#align continuous_linear_map.apply_has_faithful_smul ContinuousLinearMap.applyFaithfulSMul + +/- warning: continuous_linear_map.apply_smul_comm_class -> ContinuousLinearMap.applySMulCommClass is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], SMulCommClass.{u1, u2, u2} R₁ (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₁ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) (SMulZeroClass.toHasSmul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (MulZeroClass.toHasZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MulZeroOneClass.toMulZeroClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MonoidWithZero.toMulZeroOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], SMulCommClass.{u1, u2, u2} R₁ (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (SMulZeroClass.toSMul.{u1, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u2} R₁ M₁ (MonoidWithZero.toZero.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) (SMulZeroClass.toSMul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.zero.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.monoidWithZero.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.apply_smul_comm_class ContinuousLinearMap.applySMulCommClassₓ'. -/ +instance applySMulCommClass : SMulCommClass R₁ (M₁ →L[R₁] M₁) M₁ where smul_comm r e m := (e.map_smul r m).symm -#align continuous_linear_map.apply_smul_comm_class ContinuousLinearMap.apply_sMulCommClass - -instance apply_smul_comm_class' : SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁ +#align continuous_linear_map.apply_smul_comm_class ContinuousLinearMap.applySMulCommClass + +/- warning: continuous_linear_map.apply_smul_comm_class' -> ContinuousLinearMap.applySMulCommClass' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toHasAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], SMulCommClass.{u2, u1, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ M₁ (SMulZeroClass.toHasSmul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (MulZeroClass.toHasZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MulZeroOneClass.toMulZeroClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (MonoidWithZero.toMulZeroOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (Semiring.toMonoidWithZero.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19)) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₁ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5))) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : ContinuousAdd.{u2} M₁ _inst_4 (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))], SMulCommClass.{u2, u1, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) R₁ M₁ (SMulZeroClass.toSMul.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.zero.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.monoidWithZero.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₁ _inst_4 _inst_5 _inst_14 _inst_14) M₁ (ContinuousLinearMap.semiring.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19) _inst_5 (ContinuousLinearMap.applyModule.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 _inst_19))))) (SMulZeroClass.toSMul.{u1, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u2} R₁ M₁ (MonoidWithZero.toZero.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.apply_smul_comm_class' ContinuousLinearMap.applySMulCommClass'ₓ'. -/ +instance applySMulCommClass' : SMulCommClass (M₁ →L[R₁] M₁) R₁ M₁ where smul_comm := ContinuousLinearMap.map_smul -#align continuous_linear_map.apply_smul_comm_class' ContinuousLinearMap.apply_smul_comm_class' +#align continuous_linear_map.apply_smul_comm_class' ContinuousLinearMap.applySMulCommClass' instance : ContinuousConstSMul (M₁ →L[R₁] M₁) M₁ := ⟨ContinuousLinearMap.continuous⟩ end ApplyAction +/- warning: continuous_linear_map.prod -> ContinuousLinearMap.prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11], (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) -> (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) -> (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11], (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) -> (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) -> (ContinuousLinearMap.{u1, u1, u2, max u4 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (instTopologicalSpaceProd.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prod ContinuousLinearMap.prodₓ'. -/ /-- The cartesian product of two bounded linear maps, as a bounded linear map. -/ protected def prod [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) : M₁ →L[R₁] M₂ × M₃ := ⟨(f₁ : M₁ →ₗ[R₁] M₂).Prod f₂, f₁.2.prod_mk f₂.2⟩ #align continuous_linear_map.prod ContinuousLinearMap.prod +/- warning: continuous_linear_map.coe_prod -> ContinuousLinearMap.coe_prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] (f₁ : 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Consider using '#align continuous_linear_map.coe_prod ContinuousLinearMap.coe_prodₓ'. -/ @[simp, norm_cast] theorem coe_prod [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) : (f₁.Prod f₂ : M₁ →ₗ[R₁] M₂ × M₃) = LinearMap.prod f₁ f₂ := rfl #align continuous_linear_map.coe_prod ContinuousLinearMap.coe_prod +/- warning: continuous_linear_map.prod_apply -> ContinuousLinearMap.prod_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] (f₁ : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 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(Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) f₂ x)) +but is expected to have type + forall {R₁ : Type.{u4}} [_inst_1 : Semiring.{u4} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u4, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u4, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u4, u2} R₁ M₃ _inst_1 _inst_11] (f₁ : ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (f₂ : ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (x : M₁), Eq.{max (succ u3) (succ u2)} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => Prod.{u3, u2} M₂ M₃) x) (FunLike.coe.{max (max (succ u1) (succ u3)) (succ u2), succ u1, max (succ u3) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => Prod.{u3, u2} M₂ M₃) _x) (ContinuousMapClass.toFunLike.{max (max u1 u3) u2, u1, max u3 u2} (ContinuousLinearMap.{u4, u4, u1, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) M₁ (Prod.{u3, u2} M₂ M₃) _inst_4 (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (ContinuousSemilinearMapClass.toContinuousMapClass.{max (max u1 u3) u2, u4, u4, u1, max u3 u2} (ContinuousLinearMap.{u4, u4, u1, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 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_inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u1 u3, u1, u3} (ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u3, u4, u4, u1, u3} (ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19))) f₁ x) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) M₁ M₃ _inst_4 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u4, u4, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20))) f₂ x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prod_apply ContinuousLinearMap.prod_applyₓ'. -/ @[simp, norm_cast] theorem prod_apply [Module R₁ M₂] [Module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) (x : M₁) : f₁.Prod f₂ x = (f₁ x, f₂ x) := @@ -1017,11 +1544,23 @@ section variable (R₁ M₁ M₂) +/- warning: continuous_linear_map.inl -> ContinuousLinearMap.inl is a dubious translation: +lean 3 declaration is + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, u2, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_14 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +but is expected to have type + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, u2, max u3 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u2, u3} M₁ M₂) (instTopologicalSpaceProd.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_14 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.inl ContinuousLinearMap.inlₓ'. -/ /-- The left injection into a product is a continuous linear map. -/ def inl [Module R₁ M₂] : M₁ →L[R₁] M₁ × M₂ := (id R₁ M₁).Prod 0 #align continuous_linear_map.inl ContinuousLinearMap.inl +/- warning: continuous_linear_map.inr -> ContinuousLinearMap.inr is a dubious translation: +lean 3 declaration is + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +but is expected to have type + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, u3, max u3 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (instTopologicalSpaceProd.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.inr ContinuousLinearMap.inrₓ'. -/ /-- The right injection into a product is a continuous linear map. -/ def inr [Module R₁ M₂] : M₂ →L[R₁] M₁ × M₂ := (0 : M₂ →L[R₁] M₁).Prod (id R₁ M₂) @@ -1031,49 +1570,99 @@ end variable {F : Type _} +/- warning: continuous_linear_map.inl_apply -> ContinuousLinearMap.inl_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] (x : M₁), Eq.{max (succ u2) (succ u3)} (Prod.{u2, u3} M₁ M₂) (coeFn.{max (succ u2) (succ (max u2 u3)), max (succ u2) (succ (max u2 u3))} (ContinuousLinearMap.{u1, u1, u2, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ 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Consider using '#align continuous_linear_map.inl_apply ContinuousLinearMap.inl_applyₓ'. -/ @[simp] theorem inl_apply [Module R₁ M₂] (x : M₁) : inl R₁ M₁ M₂ x = (x, 0) := rfl #align continuous_linear_map.inl_apply ContinuousLinearMap.inl_apply +/- warning: continuous_linear_map.inr_apply -> ContinuousLinearMap.inr_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] (x : M₂), Eq.{max (succ u2) (succ u3)} (Prod.{u2, u3} M₁ M₂) (coeFn.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} (ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 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_inst_19)) (ContinuousLinearMap.inr.{u1, u2, u3} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) x) (Prod.mk.{u2, u3} M₁ M₂ (OfNat.ofNat.{u2} M₁ 0 (OfNat.mk.{u2} M₁ 0 (Zero.zero.{u2} M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_5)))))) x) +but is expected to have type + forall {R₁ : Type.{u3}} [_inst_1 : Semiring.{u3} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_9] (x : M₂), Eq.{max (succ u1) (succ u2)} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => Prod.{u1, u2} M₁ M₂) x) (FunLike.coe.{max (succ u1) (succ u2), succ u2, max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u2, max u2 u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ 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Consider using '#align continuous_linear_map.inr_apply ContinuousLinearMap.inr_applyₓ'. -/ @[simp] theorem inr_apply [Module R₁ M₂] (x : M₂) : inr R₁ M₁ M₂ x = (0, x) := rfl #align continuous_linear_map.inr_apply ContinuousLinearMap.inr_apply +/- warning: continuous_linear_map.coe_inl -> ContinuousLinearMap.coe_inl is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ u2) (succ (max u2 u3))} (LinearMap.{u1, u1, u2, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ (Prod.{u2, u3} M₁ M₂) _inst_5 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_14 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 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AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ u1) (succ u2)} (LinearMap.{u3, u3, u1, max u1 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ (Prod.{u1, u2} M₁ M₂) _inst_5 (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) _inst_14 (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.toLinearMap.{u3, u3, u1, max u1 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) _inst_14 (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.inl.{u3, u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (LinearMap.inl.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_inl ContinuousLinearMap.coe_inlₓ'. -/ @[simp, norm_cast] theorem coe_inl [Module R₁ M₂] : (inl R₁ M₁ M₂ : M₁ →ₗ[R₁] M₁ × M₂) = LinearMap.inl R₁ M₁ M₂ := rfl #align continuous_linear_map.coe_inl ContinuousLinearMap.coe_inl +/- warning: continuous_linear_map.coe_inr -> ContinuousLinearMap.coe_inr is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ u3) (succ (max u2 u3))} (LinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ (Prod.{u2, u3} M₁ M₂) _inst_9 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) ((fun (a : Sort.{max (succ u3) (succ (max u2 u3))}) (b : Sort.{max (succ u3) (succ (max u2 u3))}) [self : HasLiftT.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} a b] => self.0) (ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (LinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ (Prod.{u2, u3} M₁ M₂) _inst_9 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (HasLiftT.mk.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} (ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (LinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ (Prod.{u2, u3} M₁ M₂) _inst_9 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (CoeTCₓ.coe.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} (ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (LinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ (Prod.{u2, u3} M₁ M₂) _inst_9 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (coeBase.{max (succ u3) (succ (max u2 u3)), max (succ u3) (succ (max u2 u3))} (ContinuousLinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (LinearMap.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ (Prod.{u2, u3} M₁ M₂) _inst_9 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.LinearMap.coe.{u1, u1, u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19))))) (ContinuousLinearMap.inr.{u1, u2, u3} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (LinearMap.inr.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +but is expected to have type + forall {R₁ : Type.{u3}} [_inst_1 : Semiring.{u3} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ u1) (succ u2)} (LinearMap.{u3, u3, u2, max u1 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₂ (Prod.{u1, u2} M₁ M₂) _inst_9 (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.toLinearMap.{u3, u3, u2, max u1 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₂ _inst_8 _inst_9 (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) _inst_19 (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.inr.{u3, u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (LinearMap.inr.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_inr ContinuousLinearMap.coe_inrₓ'. -/ @[simp, norm_cast] theorem coe_inr [Module R₁ M₂] : (inr R₁ M₁ M₂ : M₂ →ₗ[R₁] M₁ × M₂) = LinearMap.inr R₁ M₁ M₂ := rfl #align continuous_linear_map.coe_inr ContinuousLinearMap.coe_inr +/- warning: continuous_linear_map.is_closed_ker -> ContinuousLinearMap.isClosed_ker is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] {F : Type.{u5}} [_inst_19 : T1Space.{u4} M₂ _inst_8] [_inst_20 : ContinuousSemilinearMapClass.{u5, u1, u2, u3, u4} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16] (f : F), IsClosed.{u3} M₁ _inst_4 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (HasLiftT.mk.{succ u3, succ u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (CoeTCₓ.coe.{succ u3, succ u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (Set.{u3} M₁) (SetLike.Set.hasCoeT.{u3, u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14)))) (LinearMap.ker.{u1, u2, u3, u4, u5} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ F (ContinuousSemilinearMapClass.toSemilinearMapClass.{u5, u1, u2, u3, u4} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_20) f)) +but is expected to have type + forall {R₁ : Type.{u3}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u3} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u3, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₂] [_inst_9 : AddCommMonoid.{u5} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_9] {F : Type.{u4}} [_inst_19 : T1Space.{u5} M₂ _inst_8] [_inst_20 : ContinuousSemilinearMapClass.{u4, u3, u2, u1, u5} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16] (f : F), IsClosed.{u1} M₁ _inst_4 (SetLike.coe.{u1, u1} (Submodule.{u3, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u3, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u3, u2, u1, u5, u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ F (ContinuousSemilinearMapClass.toSemilinearMapClass.{u4, u3, u2, u1, u5} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 _inst_20) f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.is_closed_ker ContinuousLinearMap.isClosed_kerₓ'. -/ theorem isClosed_ker [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M₁ M₂] (f : F) : IsClosed (ker f : Set M₁) := continuous_iff_isClosed.1 (map_continuous f) _ isClosed_singleton #align continuous_linear_map.is_closed_ker ContinuousLinearMap.isClosed_ker +/- warning: continuous_linear_map.is_complete_ker -> ContinuousLinearMap.isComplete_ker is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_16 : Module.{u2, u3} R₂ M₂ _inst_2 _inst_9] {F : Type.{u4}} {M' : Type.{u5}} [_inst_19 : UniformSpace.{u5} M'] [_inst_20 : CompleteSpace.{u5} M' _inst_19] [_inst_21 : AddCommMonoid.{u5} M'] [_inst_22 : Module.{u1, u5} R₁ M' _inst_1 _inst_21] [_inst_23 : T1Space.{u3} M₂ _inst_8] [_inst_24 : ContinuousSemilinearMapClass.{u4, u1, u2, u5, u3} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M' (UniformSpace.toTopologicalSpace.{u5} M' _inst_19) _inst_21 M₂ _inst_8 _inst_9 _inst_22 _inst_16] (f : F), IsComplete.{u5} M' _inst_19 ((fun (a : Type.{u5}) (b : Type.{u5}) [self : HasLiftT.{succ u5, succ u5} a b] => self.0) (Submodule.{u1, u5} R₁ M' _inst_1 _inst_21 _inst_22) (Set.{u5} M') (HasLiftT.mk.{succ u5, succ u5} (Submodule.{u1, u5} R₁ M' _inst_1 _inst_21 _inst_22) (Set.{u5} M') (CoeTCₓ.coe.{succ u5, succ u5} (Submodule.{u1, u5} R₁ M' _inst_1 _inst_21 _inst_22) (Set.{u5} M') (SetLike.Set.hasCoeT.{u5, u5} (Submodule.{u1, u5} R₁ M' _inst_1 _inst_21 _inst_22) M' (Submodule.setLike.{u1, u5} R₁ M' _inst_1 _inst_21 _inst_22)))) (LinearMap.ker.{u1, u2, u5, u3, u4} R₁ R₂ M' M₂ _inst_1 _inst_2 _inst_21 _inst_9 _inst_22 _inst_16 σ₁₂ F (ContinuousSemilinearMapClass.toSemilinearMapClass.{u4, u1, u2, u5, u3} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M' (UniformSpace.toTopologicalSpace.{u5} M' _inst_19) _inst_21 M₂ _inst_8 _inst_9 _inst_22 _inst_16 _inst_24) f)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u4, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_16 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_9] {F : Type.{u2}} {M' : Type.{u5}} [_inst_19 : UniformSpace.{u5} M'] [_inst_20 : CompleteSpace.{u5} M' _inst_19] [_inst_21 : AddCommMonoid.{u5} M'] [_inst_22 : Module.{u4, u5} R₁ M' _inst_1 _inst_21] [_inst_23 : T1Space.{u3} M₂ _inst_8] [_inst_24 : ContinuousSemilinearMapClass.{u2, u4, u1, u5, u3} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M' (UniformSpace.toTopologicalSpace.{u5} M' _inst_19) _inst_21 M₂ _inst_8 _inst_9 _inst_22 _inst_16] (f : F), IsComplete.{u5} M' _inst_19 (SetLike.coe.{u5, u5} (Submodule.{u4, u5} R₁ M' _inst_1 _inst_21 _inst_22) M' (Submodule.setLike.{u4, u5} R₁ M' _inst_1 _inst_21 _inst_22) (LinearMap.ker.{u4, u1, u5, u3, u2} R₁ R₂ M' M₂ _inst_1 _inst_2 _inst_21 _inst_9 _inst_22 _inst_16 σ₁₂ F (ContinuousSemilinearMapClass.toSemilinearMapClass.{u2, u4, u1, u5, u3} F R₁ R₂ _inst_1 _inst_2 σ₁₂ M' (UniformSpace.toTopologicalSpace.{u5} M' _inst_19) _inst_21 M₂ _inst_8 _inst_9 _inst_22 _inst_16 _inst_24) f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.is_complete_ker ContinuousLinearMap.isComplete_kerₓ'. -/ theorem isComplete_ker {M' : Type _} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) : IsComplete (ker f : Set M') := (isClosed_ker f).IsComplete #align continuous_linear_map.is_complete_ker ContinuousLinearMap.isComplete_ker +#print ContinuousLinearMap.completeSpace_ker /- instance (priority := 100) completeSpace_ker {M' : Type _} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M'] [Module R₁ M'] [T1Space M₂] [ContinuousSemilinearMapClass F σ₁₂ M' M₂] (f : F) : CompleteSpace (ker f) := (isClosed_ker f).completeSpace_coe #align continuous_linear_map.complete_space_ker ContinuousLinearMap.completeSpace_ker +-/ +/- warning: continuous_linear_map.ker_prod -> ContinuousLinearMap.ker_prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{succ u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u1, u1, u2, max u3 u4, max u2 u3 u4} R₁ R₁ M₁ (Prod.{u3, u4} M₂ M₃) _inst_1 _inst_1 _inst_5 (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u3 u4, u1, u1, u2, max u3 u4} (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, max u3 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20))) (ContinuousLinearMap.prod.{u1, u2, u3, u4} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) (Inf.inf.{u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (Submodule.hasInf.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R₁ R₁ M₁ M₂ _inst_1 _inst_1 _inst_5 _inst_9 _inst_14 _inst_19 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u3, u1, u1, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) f) (LinearMap.ker.{u1, u1, u2, u4, max u2 u4} R₁ R₁ M₁ M₃ _inst_1 _inst_1 _inst_5 _inst_11 _inst_14 _inst_20 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20)) g)) +but is expected to have type + forall {R₁ : Type.{u4}} [_inst_1 : Semiring.{u4} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u4, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u4, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u4, u2} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{succ u1} (Submodule.{u4, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u4, u4, u1, max u3 u2, max (max u1 u3) u2} R₁ R₁ M₁ (Prod.{u3, u2} M₂ M₃) _inst_1 _inst_1 _inst_5 (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, u1, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max (max u1 u3) u2, u4, u4, u1, max u3 u2} (ContinuousLinearMap.{u4, u4, u1, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20)) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, max u3 u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20))) (ContinuousLinearMap.prod.{u4, u1, u3, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) (Inf.inf.{u1} (Submodule.{u4, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (Submodule.instInfSubmodule.{u4, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u4, u4, u1, u3, max u1 u3} R₁ R₁ M₁ M₂ _inst_1 _inst_1 _inst_5 _inst_9 _inst_14 _inst_19 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u1 u3, u4, u4, u1, u3} (ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) f) (LinearMap.ker.{u4, u4, u1, u2, max u1 u2} R₁ R₁ M₁ M₃ _inst_1 _inst_1 _inst_5 _inst_11 _inst_14 _inst_20 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u1 u2, u4, u4, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20)) g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ker_prod ContinuousLinearMap.ker_prodₓ'. -/ @[simp] theorem ker_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : ker (f.Prod g) = ker f ⊓ ker g := LinearMap.ker_prod f g #align continuous_linear_map.ker_prod ContinuousLinearMap.ker_prod +/- warning: continuous_linear_map.cod_restrict -> ContinuousLinearMap.codRestrict is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (p : Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16), (forall (x : M₁), Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f x) p) -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) (Subtype.topologicalSpace.{u4} M₂ (fun (x : M₂) => Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) +but is expected to have type + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (p : Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16), (forall (x : M₁), Membership.mem.{u4, u4} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x) p) -> (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u4} M₂ (fun (x : M₂) => Membership.mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u4} M₂ (fun (x : M₂) => Membership.mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.cod_restrict ContinuousLinearMap.codRestrictₓ'. -/ /-- Restrict codomain of a continuous linear map. -/ def codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) : M₁ →SL[σ₁₂] p where @@ -1081,51 +1670,101 @@ def codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : toLinearMap := (f : M₁ →ₛₗ[σ₁₂] M₂).codRestrict p h #align continuous_linear_map.cod_restrict ContinuousLinearMap.codRestrict +/- warning: continuous_linear_map.coe_cod_restrict -> ContinuousLinearMap.coe_codRestrict is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} 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_inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)))) f) h) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (p : Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (h : forall (x : M₁), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} 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_inst_14 _inst_16))) f x) p), Eq.{max (succ u2) (succ u1)} (LinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) _inst_5 (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, 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Consider using '#align continuous_linear_map.coe_cod_restrict ContinuousLinearMap.coe_codRestrictₓ'. -/ @[norm_cast] theorem coe_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) : (f.codRestrict p h : M₁ →ₛₗ[σ₁₂] p) = (f : M₁ →ₛₗ[σ₁₂] M₂).codRestrict p h := rfl #align continuous_linear_map.coe_cod_restrict ContinuousLinearMap.coe_codRestrict +/- warning: continuous_linear_map.coe_cod_restrict_apply -> ContinuousLinearMap.coe_codRestrict_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} 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_inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) M₁ (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) _inst_4 (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)))) (ContinuousLinearMap.codRestrict.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f p h) x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_cod_restrict_apply ContinuousLinearMap.coe_codRestrict_applyₓ'. -/ @[simp] theorem coe_codRestrict_apply (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) (x) : (f.codRestrict p h x : M₂) = f x := rfl #align continuous_linear_map.coe_cod_restrict_apply ContinuousLinearMap.coe_codRestrict_apply +/- warning: continuous_linear_map.ker_cod_restrict -> ContinuousLinearMap.ker_codRestrict is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] [_inst_14 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (p : Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (h : forall (x : M₁), Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) f x) p), Eq.{succ u3} (Submodule.{u1, u3} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u1, u2, u3, u4, max u3 u4} R₁ R₂ M₁ (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) _inst_1 _inst_2 _inst_5 (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) σ₁₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) (Subtype.topologicalSpace.{u4} M₂ (fun (x : M₂) => Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) (Subtype.topologicalSpace.{u4} M₂ (fun (x : M₂) => Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) (Subtype.topologicalSpace.{u4} M₂ (fun (x : M₂) => Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (coeSort.{succ u4, succ (succ u4)} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) Type.{u4} (SetLike.hasCoeToSort.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) p) (Subtype.topologicalSpace.{u4} M₂ (fun (x : M₂) => Membership.Mem.{u4, u4} M₂ (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.hasMem.{u4, u4} (Submodule.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u2, u4} R₂ M₂ _inst_2 _inst_9 _inst_16 p))) (ContinuousLinearMap.codRestrict.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f p h)) (LinearMap.ker.{u1, u2, u3, u4, max u3 u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) f) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_14 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_5] [_inst_16 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_9] (f : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (p : Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (h : forall (x : M₁), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) M₁ M₂ _inst_4 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16))) f x) p), Eq.{succ u2} (Submodule.{u4, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.ker.{u4, u3, u2, u1, max u2 u1} R₁ R₂ M₁ (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) _inst_1 _inst_2 _inst_5 (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) σ₁₂ (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ 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(Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p) _inst_8) (Submodule.addCommMonoid.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p)) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 (Subtype.{succ u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) M₂ (Submodule.setLike.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16)) x p)) (instTopologicalSpaceSubtype.{u1} M₂ (fun (x : M₂) => Membership.mem.{u1, u1} M₂ (Submodule.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16) (SetLike.instMembership.{u1, u1} 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_inst_9 _inst_16 p) _inst_14 (Submodule.module.{u3, u1} R₂ M₂ _inst_2 _inst_9 _inst_16 p))) (ContinuousLinearMap.codRestrict.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 f p h)) (LinearMap.ker.{u4, u3, u2, u1, max u2 u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_5 _inst_9 _inst_14 _inst_16 σ₁₂ (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_16)) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ker_cod_restrict ContinuousLinearMap.ker_codRestrictₓ'. -/ @[simp] theorem ker_codRestrict (f : M₁ →SL[σ₁₂] M₂) (p : Submodule R₂ M₂) (h : ∀ x, f x ∈ p) : ker (f.codRestrict p h) = ker f := (f : M₁ →ₛₗ[σ₁₂] M₂).ker_codRestrict p h #align continuous_linear_map.ker_cod_restrict ContinuousLinearMap.ker_codRestrict +#print Submodule.subtypeL /- /-- `submodule.subtype` as a `continuous_linear_map`. -/ def Submodule.subtypeL (p : Submodule R₁ M₁) : p →L[R₁] M₁ where cont := continuous_subtype_val toLinearMap := p.Subtype #align submodule.subtypeL Submodule.subtypeL +-/ +/- warning: submodule.coe_subtypeL -> Submodule.coe_subtypeL is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14), Eq.{succ u2} 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u2} R₁ M₁ _inst_1 _inst_5 _inst_14 p) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] [_inst_14 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14), Eq.{succ u1} (LinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (Subtype.{succ u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p)) M₁ (Submodule.addCommMonoid.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_5 (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_14) (ContinuousLinearMap.toLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ 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Consider using '#align submodule.coe_subtypeL Submodule.coe_subtypeLₓ'. -/ @[simp, norm_cast] theorem Submodule.coe_subtypeL (p : Submodule R₁ M₁) : (p.subtypeL : p →ₗ[R₁] M₁) = p.Subtype := rfl #align submodule.coe_subtypeL Submodule.coe_subtypeL +/- warning: submodule.coe_subtypeL' -> Submodule.coe_subtypeL' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14), Eq.{succ u2} ((coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) p) -> M₁) (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ 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Consider using '#align submodule.coe_subtypeL' Submodule.coe_subtypeL'ₓ'. -/ @[simp] theorem Submodule.coe_subtypeL' (p : Submodule R₁ M₁) : ⇑p.subtypeL = p.Subtype := rfl #align submodule.coe_subtypeL' Submodule.coe_subtypeL' +/- warning: submodule.subtypeL_apply -> Submodule.subtypeL_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (x : coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) p), Eq.{succ u2} M₁ (coeFn.{succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} 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(Submodule.addCommMonoid.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) M₁ _inst_4 _inst_5 (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (Subtype.{succ u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p)) (instTopologicalSpaceSubtype.{u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p) _inst_4) (Submodule.addCommMonoid.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) M₁ _inst_4 _inst_5 (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_14))) (Submodule.subtypeL.{u2, u1} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 p) x) (Subtype.val.{succ u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Set.{u1} M₁) (Set.instMembershipSet.{u1} M₁) x (SetLike.coe.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) p)) x) +Case conversion may be inaccurate. Consider using '#align submodule.subtypeL_apply Submodule.subtypeL_applyₓ'. -/ @[simp, norm_cast] theorem Submodule.subtypeL_apply (p : Submodule R₁ M₁) (x : p) : p.subtypeL x = x := rfl #align submodule.subtypeL_apply Submodule.subtypeL_apply +/- warning: submodule.range_subtypeL -> Submodule.range_subtypeL is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14), Eq.{succ u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (LinearMap.range.{u1, u1, u2, u2, u2} R₁ R₁ (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) p) M₁ _inst_1 _inst_1 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Consider using '#align submodule.range_subtypeL Submodule.range_subtypeLₓ'. -/ @[simp] theorem Submodule.range_subtypeL (p : Submodule R₁ M₁) : range p.subtypeL = p := Submodule.range_subtype _ #align submodule.range_subtypeL Submodule.range_subtypeL +/- warning: submodule.ker_subtypeL -> Submodule.ker_subtypeL is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] (p : Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14), Eq.{succ u2} (Submodule.{u1, u2} R₁ (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) p) _inst_1 (Submodule.addCommMonoid.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14 p) (Submodule.module.{u1, u2} 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M₁ _inst_1 _inst_5 _inst_14 p) _inst_14 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) p) (Subtype.topologicalSpace.{u2} M₁ (fun (x : M₁) => Membership.Mem.{u2, u2} M₁ (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p) _inst_4) (Submodule.addCommMonoid.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14 p) M₁ _inst_4 _inst_5 (Submodule.module.{u1, u2} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_14)) (Submodule.subtypeL.{u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 _inst_14 p)) (Bot.bot.{u2} (Submodule.{u1, u2} R₁ 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p) _inst_14 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (ContinuousLinearMap.{u2, u2, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (Subtype.{succ u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p)) (instTopologicalSpaceSubtype.{u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p) _inst_4) (Submodule.addCommMonoid.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) M₁ _inst_4 _inst_5 (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) _inst_14) (ContinuousSemilinearMapClass.toSemilinearMapClass.{u1, u2, u2, u1, 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R₁ M₁ _inst_1 _inst_5 _inst_14 p) (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p)) (Submodule.instBotSubmodule.{u2, u1} R₁ (Subtype.{succ u1} M₁ (fun (x : M₁) => Membership.mem.{u1, u1} M₁ (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14) M₁ (Submodule.setLike.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14)) x p)) _inst_1 (Submodule.addCommMonoid.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p) (Submodule.module.{u2, u1} R₁ M₁ _inst_1 _inst_5 _inst_14 p))) +Case conversion may be inaccurate. Consider using '#align submodule.ker_subtypeL Submodule.ker_subtypeLₓ'. -/ @[simp] theorem Submodule.ker_subtypeL (p : Submodule R₁ M₁) : ker p.subtypeL = ⊥ := Submodule.ker_subtype _ @@ -1133,6 +1772,12 @@ theorem Submodule.ker_subtypeL (p : Submodule R₁ M₁) : ker p.subtypeL = ⊥ variable (R₁ M₁ M₂) +/- warning: continuous_linear_map.fst -> ContinuousLinearMap.fst is a dubious translation: +lean 3 declaration is + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, max u2 u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₁ _inst_4 _inst_5 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_14 +but is expected to have type + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, max u3 u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (instTopologicalSpaceProd.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₁ _inst_4 _inst_5 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_14 +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.fst ContinuousLinearMap.fstₓ'. -/ /-- `prod.fst` as a `continuous_linear_map`. -/ def fst [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₁ where @@ -1140,6 +1785,12 @@ def fst [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₁ toLinearMap := LinearMap.fst R₁ M₁ M₂ #align continuous_linear_map.fst ContinuousLinearMap.fst +/- warning: continuous_linear_map.snd -> ContinuousLinearMap.snd is a dubious translation: +lean 3 declaration is + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, max u2 u3, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₂ _inst_8 _inst_9 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_19 +but is expected to have type + forall (R₁ : Type.{u1}) [_inst_1 : Semiring.{u1} R₁] (M₁ : Type.{u2}) [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] (M₂ : Type.{u3}) [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], ContinuousLinearMap.{u1, u1, max u3 u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (instTopologicalSpaceProd.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₂ _inst_8 _inst_9 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_19 +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.snd ContinuousLinearMap.sndₓ'. -/ /-- `prod.snd` as a `continuous_linear_map`. -/ def snd [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₂ where @@ -1149,79 +1800,163 @@ def snd [Module R₁ M₂] : M₁ × M₂ →L[R₁] M₂ variable {R₁ M₁ M₂} +/- warning: continuous_linear_map.coe_fst -> ContinuousLinearMap.coe_fst is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ (max u2 u3)) (succ u2)} (LinearMap.{u1, u1, max u2 u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) M₁ (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_5 (Prod.module.{u1, u2, u3} R₁ M₁ 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Consider using '#align continuous_linear_map.coe_fst ContinuousLinearMap.coe_fstₓ'. -/ @[simp, norm_cast] theorem coe_fst [Module R₁ M₂] : ↑(fst R₁ M₁ M₂) = LinearMap.fst R₁ M₁ M₂ := rfl #align continuous_linear_map.coe_fst ContinuousLinearMap.coe_fst +/- warning: continuous_linear_map.coe_fst' -> ContinuousLinearMap.coe_fst' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ (max u2 u3)) (succ u2)} ((Prod.{u2, u3} M₁ M₂) -> M₁) (coeFn.{max (succ (max u2 u3)) (succ u2), max (succ (max u2 u3)) (succ u2)} (ContinuousLinearMap.{u1, u1, max u2 u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, 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Consider using '#align continuous_linear_map.coe_fst' ContinuousLinearMap.coe_fst'ₓ'. -/ @[simp, norm_cast] theorem coe_fst' [Module R₁ M₂] : ⇑(fst R₁ M₁ M₂) = Prod.fst := rfl #align continuous_linear_map.coe_fst' ContinuousLinearMap.coe_fst' +/- warning: continuous_linear_map.coe_snd -> ContinuousLinearMap.coe_snd is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ (max u2 u3)) (succ u3)} (LinearMap.{u1, u1, max u2 u3, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) M₂ (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_9 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 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_inst_1 _inst_5 _inst_9 _inst_14 _inst_19) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_snd ContinuousLinearMap.coe_sndₓ'. -/ @[simp, norm_cast] theorem coe_snd [Module R₁ M₂] : ↑(snd R₁ M₁ M₂) = LinearMap.snd R₁ M₁ M₂ := rfl #align continuous_linear_map.coe_snd ContinuousLinearMap.coe_snd +/- warning: continuous_linear_map.coe_snd' -> ContinuousLinearMap.coe_snd' is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ (max u2 u3)) (succ u3)} ((Prod.{u2, u3} M₁ M₂) -> M₂) (coeFn.{max (succ (max u2 u3)) (succ u3), max (succ (max u2 u3)) (succ u3)} (ContinuousLinearMap.{u1, u1, max u2 u3, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, 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_inst_9 _inst_14 _inst_19) _inst_19))) (ContinuousLinearMap.snd.{u3, u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (Prod.snd.{u1, u2} M₁ M₂) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_snd' ContinuousLinearMap.coe_snd'ₓ'. -/ @[simp, norm_cast] theorem coe_snd' [Module R₁ M₂] : ⇑(snd R₁ M₁ M₂) = Prod.snd := rfl #align continuous_linear_map.coe_snd' ContinuousLinearMap.coe_snd' +/- warning: continuous_linear_map.fst_prod_snd -> ContinuousLinearMap.fst_prod_snd is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9], Eq.{succ (max u2 u3)} (ContinuousLinearMap.{u1, u1, max u2 u3, max u2 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.prod.{u1, max u2 u3, u2, u3} R₁ _inst_1 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_14 _inst_19 (ContinuousLinearMap.fst.{u1, u2, u3} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.snd.{u1, u2, u3} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.id.{u1, max u2 u3} R₁ _inst_1 (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) +but is expected to have type + forall {R₁ : Type.{u3}} [_inst_1 : Semiring.{u3} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_14 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_9], Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, max u1 u2, max u2 u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.prod.{u3, max u1 u2, u1, u2} R₁ _inst_1 (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_14 _inst_19 (ContinuousLinearMap.fst.{u3, u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.snd.{u3, u1, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19)) (ContinuousLinearMap.id.{u3, max u2 u1} R₁ _inst_1 (Prod.{u1, u2} M₁ M₂) (instTopologicalSpaceProd.{u1, u2} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u2} M₁ M₂ _inst_5 _inst_9) (Prod.module.{u3, u1, u2} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.fst_prod_snd ContinuousLinearMap.fst_prod_sndₓ'. -/ @[simp] theorem fst_prod_snd [Module R₁ M₂] : (fst R₁ M₁ M₂).Prod (snd R₁ M₁ M₂) = id R₁ (M₁ × M₂) := ext fun ⟨x, y⟩ => rfl #align continuous_linear_map.fst_prod_snd ContinuousLinearMap.fst_prod_snd +/- warning: continuous_linear_map.fst_comp_prod -> ContinuousLinearMap.fst_comp_prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.comp.{u1, u1, u1, u2, max u3 u4, u3} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) M₂ _inst_8 _inst_9 _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) _inst_19 (RingHomCompTriple.right_ids.{u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (ContinuousLinearMap.fst.{u1, u3, u4} R₁ _inst_1 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.prod.{u1, u2, u3, u4} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) f +but is expected to have type + forall {R₁ : Type.{u4}} [_inst_1 : Semiring.{u4} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u4, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u4, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u4, u2} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{max (succ u1) (succ u3)} (ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (ContinuousLinearMap.comp.{u4, u4, u4, u1, max u3 u2, u3} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) M₂ _inst_8 _inst_9 _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) _inst_19 (RingHomCompTriple.ids.{u4, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1))) (ContinuousLinearMap.fst.{u4, u3, u2} R₁ _inst_1 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.prod.{u4, u1, u3, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) f +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.fst_comp_prod ContinuousLinearMap.fst_comp_prodₓ'. -/ @[simp] theorem fst_comp_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : (fst R₁ M₂ M₃).comp (f.Prod g) = f := ext fun x => rfl #align continuous_linear_map.fst_comp_prod ContinuousLinearMap.fst_comp_prod +/- warning: continuous_linear_map.snd_comp_prod -> ContinuousLinearMap.snd_comp_prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousLinearMap.comp.{u1, u1, u1, u2, max u3 u4, u4} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_8 _inst_10) (Prod.addCommMonoid.{u3, u4} M₂ M₃ _inst_9 _inst_11) M₃ _inst_10 _inst_11 _inst_14 (Prod.module.{u1, u3, u4} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (ContinuousLinearMap.snd.{u1, u3, u4} R₁ _inst_1 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.prod.{u1, u2, u3, u4} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) g +but is expected to have type + forall {R₁ : Type.{u4}} [_inst_1 : Semiring.{u4} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u4, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u4, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u4, u2} R₁ M₃ _inst_1 _inst_11] (f : ContinuousLinearMap.{u4, u4, u1, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (g : ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20), Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousLinearMap.comp.{u4, u4, u4, u1, max u3 u2, u2} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 (Prod.{u3, u2} M₂ M₃) (instTopologicalSpaceProd.{u3, u2} M₂ M₃ _inst_8 _inst_10) (Prod.instAddCommMonoidSum.{u3, u2} M₂ M₃ _inst_9 _inst_11) M₃ _inst_10 _inst_11 _inst_14 (Prod.module.{u4, u3, u2} R₁ M₂ M₃ _inst_1 _inst_9 _inst_11 _inst_19 _inst_20) _inst_20 (RingHomCompTriple.ids.{u4, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1))) (ContinuousLinearMap.snd.{u4, u3, u2} R₁ _inst_1 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousLinearMap.prod.{u4, u1, u3, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 f g)) g +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.snd_comp_prod ContinuousLinearMap.snd_comp_prodₓ'. -/ @[simp] theorem snd_comp_prod [Module R₁ M₂] [Module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) : (snd R₁ M₂ M₃).comp (f.Prod g) = g := ext fun x => rfl #align continuous_linear_map.snd_comp_prod ContinuousLinearMap.snd_comp_prod +/- warning: continuous_linear_map.prod_map -> ContinuousLinearMap.prodMap is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₄] [_inst_13 : AddCommMonoid.{u5} M₄] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_13], (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) -> (ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21) -> (ContinuousLinearMap.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_4 _inst_10) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_8 _inst_12) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₄] [_inst_13 : AddCommMonoid.{u5} M₄] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_13], (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) -> (ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21) -> (ContinuousLinearMap.{u1, u1, max u4 u2, max u5 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (instTopologicalSpaceProd.{u2, u4} M₁ M₃ _inst_4 _inst_10) (Prod.instAddCommMonoidSum.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.{u3, u5} M₂ M₄) (instTopologicalSpaceProd.{u3, u5} M₂ M₄ _inst_8 _inst_12) (Prod.instAddCommMonoidSum.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prod_map ContinuousLinearMap.prodMapₓ'. -/ /-- `prod.map` of two continuous linear maps. -/ def prodMap [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : M₁ × M₃ →L[R₁] M₂ × M₄ := (f₁.comp (fst R₁ M₁ M₃)).Prod (f₂.comp (snd R₁ M₁ M₃)) #align continuous_linear_map.prod_map ContinuousLinearMap.prodMap +/- warning: continuous_linear_map.coe_prod_map -> ContinuousLinearMap.coe_prodMap is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₄] [_inst_13 : AddCommMonoid.{u5} M₄] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_13] (f₁ : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (f₂ : ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21), Eq.{max (succ (max u2 u4)) (succ (max u3 u5))} (LinearMap.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (Prod.{u3, u5} M₂ M₄) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) 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f₁) (ContinuousLinearMap.toLinearMap.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21 f₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_prod_map ContinuousLinearMap.coe_prodMapₓ'. -/ @[simp, norm_cast] theorem coe_prodMap [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : ↑(f₁.Prod_map f₂) = (f₁ : M₁ →ₗ[R₁] M₂).Prod_map (f₂ : M₃ →ₗ[R₁] M₄) := rfl #align continuous_linear_map.coe_prod_map ContinuousLinearMap.coe_prodMap +/- warning: continuous_linear_map.coe_prod_map' -> ContinuousLinearMap.coe_prod_map is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₄] [_inst_13 : AddCommMonoid.{u5} M₄] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_13] (f₁ : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (f₂ : ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21), Eq.{max (succ (max u2 u4)) (succ (max u3 u5))} ((Prod.{u2, u4} M₁ M₃) -> (Prod.{u3, u5} M₂ M₄)) (coeFn.{max (succ (max u2 u4)) (succ (max u3 u5)), max (succ (max u2 u4)) (succ (max u3 u5))} (ContinuousLinearMap.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_4 _inst_10) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_8 _inst_12) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) (fun (_x : ContinuousLinearMap.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_4 _inst_10) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_8 _inst_12) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) => (Prod.{u2, u4} M₁ M₃) -> (Prod.{u3, u5} M₂ M₄)) (ContinuousLinearMap.toFun.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_4 _inst_10) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_5 _inst_11) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_8 _inst_12) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) (ContinuousLinearMap.prodMap.{u1, u2, u3, u4, u5} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_14 _inst_19 _inst_20 _inst_21 f₁ f₂)) (Prod.map.{u2, u3, u4, u5} M₁ M₂ M₃ M₄ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) f₁) (coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21) (fun (_x : ContinuousLinearMap.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21) => M₃ -> M₄) (ContinuousLinearMap.toFun.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 _inst_13 _inst_20 _inst_21) f₂)) +but is expected to have type + forall {R₁ : Type.{u5}} [_inst_1 : Semiring.{u5} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₂] [_inst_9 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_10 : TopologicalSpace.{u3} M₃] [_inst_11 : AddCommMonoid.{u3} M₃] {M₄ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₄] [_inst_13 : AddCommMonoid.{u2} M₄] [_inst_14 : Module.{u5, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u5, u4} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u5, u3} R₁ M₃ _inst_1 _inst_11] [_inst_21 : Module.{u5, u2} R₁ M₄ _inst_1 _inst_13] (f₁ : ContinuousLinearMap.{u5, u5, u1, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_14 _inst_19) (f₂ : ContinuousLinearMap.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) M₃ _inst_10 _inst_11 M₄ _inst_12 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(max u1 u4) u3) u2, u5, u5, max u1 u3, max u4 u2} (ContinuousLinearMap.{u5, u5, max u3 u1, max u2 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₃) (instTopologicalSpaceProd.{u1, u3} M₁ M₃ _inst_4 _inst_10) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₃ _inst_5 _inst_11) (Prod.{u4, u2} M₂ M₄) (instTopologicalSpaceProd.{u4, u2} M₂ M₄ _inst_8 _inst_12) (Prod.instAddCommMonoidSum.{u4, u2} M₂ M₄ _inst_9 _inst_13) (Prod.module.{u5, u1, u3} R₁ M₁ M₃ _inst_1 _inst_5 _inst_11 _inst_14 _inst_20) (Prod.module.{u5, u4, u2} R₁ M₂ M₄ _inst_1 _inst_9 _inst_13 _inst_19 _inst_21)) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₃) (instTopologicalSpaceProd.{u1, u3} M₁ M₃ _inst_4 _inst_10) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₃ _inst_5 _inst_11) (Prod.{u4, u2} M₂ M₄) (instTopologicalSpaceProd.{u4, u2} M₂ M₄ _inst_8 _inst_12) (Prod.instAddCommMonoidSum.{u4, u2} M₂ M₄ _inst_9 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Consider using '#align continuous_linear_map.coe_prod_map' ContinuousLinearMap.coe_prod_mapₓ'. -/ @[simp, norm_cast] -theorem coe_prod_map' [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) +theorem coe_prod_map [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) : ⇑(f₁.Prod_map f₂) = Prod.map f₁ f₂ := rfl -#align continuous_linear_map.coe_prod_map' ContinuousLinearMap.coe_prod_map' - +#align continuous_linear_map.coe_prod_map' ContinuousLinearMap.coe_prod_map + +/- warning: continuous_linear_map.coprod -> ContinuousLinearMap.coprod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : ContinuousAdd.{u4} M₃ _inst_10 (AddZeroClass.toHasAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_11)))], (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) -> (ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) -> (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : ContinuousAdd.{u4} M₃ _inst_10 (AddZeroClass.toAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_11)))], (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) -> (ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) -> (ContinuousLinearMap.{u1, u1, max u3 u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (instTopologicalSpaceProd.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coprod ContinuousLinearMap.coprodₓ'. -/ /-- The continuous linear map given by `(x, y) ↦ f₁ x + f₂ y`. -/ def coprod [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : M₁ × M₂ →L[R₁] M₃ := ⟨LinearMap.coprod f₁ f₂, (f₁.cont.comp continuous_fst).add (f₂.cont.comp continuous_snd)⟩ #align continuous_linear_map.coprod ContinuousLinearMap.coprod +/- warning: continuous_linear_map.coe_coprod -> ContinuousLinearMap.coe_coprod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : 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M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20 f₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_coprod ContinuousLinearMap.coe_coprodₓ'. -/ @[norm_cast, simp] theorem coe_coprod [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : (f₁.coprod f₂ : M₁ × M₂ →ₗ[R₁] M₃) = LinearMap.coprod f₁ f₂ := rfl #align continuous_linear_map.coe_coprod ContinuousLinearMap.coe_coprod +/- warning: continuous_linear_map.coprod_apply -> ContinuousLinearMap.coprod_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : ContinuousAdd.{u4} M₃ _inst_10 (AddZeroClass.toHasAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_11)))] (f₁ : ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (f₂ : ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (x : Prod.{u2, u3} M₁ M₂), Eq.{succ u4} M₃ (coeFn.{max (succ (max u2 u3)) (succ u4), max (succ (max u2 u3)) (succ u4)} (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (Prod.{u2, u3} M₁ M₂) (Prod.topologicalSpace.{u2, u3} M₁ M₂ _inst_4 _inst_8) (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20) 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(x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) (Prod.fst.{u1, u3} M₁ M₂ x)) _inst_11)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) M₁ M₃ _inst_4 _inst_10 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u4, u4, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 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(ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20))) f₂ (Prod.snd.{u1, u3} M₁ M₂ x))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coprod_apply ContinuousLinearMap.coprod_applyₓ'. -/ @[simp] theorem coprod_apply [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x) : f₁.coprod f₂ x = f₁ x.1 + f₂ x.2 := rfl #align continuous_linear_map.coprod_apply ContinuousLinearMap.coprod_apply +/- warning: continuous_linear_map.range_coprod -> ContinuousLinearMap.range_coprod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M₁] [_inst_5 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_10 : TopologicalSpace.{u4} M₃] [_inst_11 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_11] [_inst_21 : ContinuousAdd.{u4} M₃ _inst_10 (AddZeroClass.toHasAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_11)))] (f₁ : ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (f₂ : ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20), Eq.{succ u4} (Submodule.{u1, u4} R₁ M₃ _inst_1 _inst_11 _inst_20) (LinearMap.range.{u1, u1, max u2 u3, u4, max (max u2 u3) u4} R₁ R₁ (Prod.{u2, u3} M₁ M₂) M₃ _inst_1 _inst_1 (Prod.addCommMonoid.{u2, u3} M₁ M₂ _inst_5 _inst_9) _inst_11 (Prod.module.{u1, u2, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ 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_inst_11 _inst_20) (CompleteLattice.toConditionallyCompleteLattice.{u4} (Submodule.{u1, u4} R₁ M₃ _inst_1 _inst_11 _inst_20) (Submodule.completeLattice.{u1, u4} R₁ M₃ _inst_1 _inst_11 _inst_20))))) (LinearMap.range.{u1, u1, u2, u4, max u2 u4} R₁ R₁ M₁ M₃ _inst_1 _inst_1 _inst_5 _inst_11 _inst_14 _inst_20 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20)) (RingHomSurjective.ids.{u1} R₁ _inst_1) f₁) (LinearMap.range.{u1, u1, u3, u4, max u3 u4} R₁ R₁ M₂ M₃ _inst_1 _inst_1 _inst_9 _inst_11 _inst_19 _inst_20 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u4, u1, u1, u3, u4} (ContinuousLinearMap.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20)) (RingHomSurjective.ids.{u1} R₁ _inst_1) f₂)) +but is expected to have type + forall {R₁ : Type.{u4}} [_inst_1 : Semiring.{u4} R₁] {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₂] [_inst_9 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u2}} [_inst_10 : TopologicalSpace.{u2} M₃] [_inst_11 : AddCommMonoid.{u2} M₃] [_inst_14 : Module.{u4, u1} R₁ M₁ _inst_1 _inst_5] [_inst_19 : Module.{u4, u3} R₁ M₂ _inst_1 _inst_9] [_inst_20 : Module.{u4, u2} R₁ M₃ _inst_1 _inst_11] [_inst_21 : ContinuousAdd.{u2} M₃ _inst_10 (AddZeroClass.toAdd.{u2} M₃ (AddMonoid.toAddZeroClass.{u2} M₃ (AddCommMonoid.toAddMonoid.{u2} M₃ _inst_11)))] (f₁ : ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (f₂ : ContinuousLinearMap.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20), Eq.{succ u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (LinearMap.range.{u4, u4, max u1 u3, u2, max (max u1 u2) u3} R₁ R₁ (Prod.{u1, u3} M₁ M₂) M₃ _inst_1 _inst_1 (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₂ _inst_5 _inst_9) _inst_11 (Prod.module.{u4, u1, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, max u3 u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₂) (instTopologicalSpaceProd.{u1, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u4, u1, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max (max u1 u3) u2, u4, u4, max u1 u3, u2} (ContinuousLinearMap.{u4, u4, max u3 u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₂) (instTopologicalSpaceProd.{u1, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u4, u1, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₂) (instTopologicalSpaceProd.{u1, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u4, u1, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, max u1 u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (Prod.{u1, u3} M₁ M₂) (instTopologicalSpaceProd.{u1, u3} M₁ M₂ _inst_4 _inst_8) (Prod.instAddCommMonoidSum.{u1, u3} M₁ M₂ _inst_5 _inst_9) M₃ _inst_10 _inst_11 (Prod.module.{u4, u1, u3} R₁ M₁ M₂ _inst_1 _inst_5 _inst_9 _inst_14 _inst_19) _inst_20)) (RingHomSurjective.ids.{u4} R₁ _inst_1) (ContinuousLinearMap.coprod.{u4, u1, u3, u2} R₁ _inst_1 M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_14 _inst_19 _inst_20 _inst_21 f₁ f₂)) (Sup.sup.{u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (SemilatticeSup.toSup.{u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (Lattice.toSemilatticeSup.{u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20) (Submodule.completeLattice.{u4, u2} R₁ M₃ _inst_1 _inst_11 _inst_20))))) (LinearMap.range.{u4, u4, u1, u2, max u1 u2} R₁ R₁ M₁ M₃ _inst_1 _inst_1 _inst_5 _inst_11 _inst_14 _inst_20 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u1 u2, u4, u4, u1, u2} (ContinuousLinearMap.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_4 _inst_5 M₃ _inst_10 _inst_11 _inst_14 _inst_20)) (RingHomSurjective.ids.{u4} R₁ _inst_1) f₁) (LinearMap.range.{u4, u4, u3, u2, max u3 u2} R₁ R₁ M₂ M₃ _inst_1 _inst_1 _inst_9 _inst_11 _inst_19 _inst_20 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (ContinuousLinearMap.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₂ _inst_8 _inst_9 M₃ _inst_10 _inst_11 _inst_19 _inst_20)) (RingHomSurjective.ids.{u4} R₁ _inst_1) f₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.range_coprod ContinuousLinearMap.range_coprodₓ'. -/ theorem range_coprod [Module R₁ M₂] [Module R₁ M₃] [ContinuousAdd M₃] (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) : range (f₁.coprod f₂) = range f₁ ⊔ range f₂ := LinearMap.range_coprod _ _ @@ -1232,13 +1967,21 @@ section variable {R S : Type _} [Semiring R] [Semiring S] [Module R M₁] [Module R M₂] [Module R S] [Module S M₂] [IsScalarTower R S M₂] [TopologicalSpace S] [ContinuousSMul S M₂] +#print ContinuousLinearMap.smulRight /- /-- The linear map `λ x, c x • f`. Associates to a scalar-valued linear map and an element of `M₂` the `M₂`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M₂`). See also `continuous_linear_map.smul_rightₗ` and `continuous_linear_map.smul_rightL`. -/ def smulRight (c : M₁ →L[R] S) (f : M₂) : M₁ →L[R] M₂ := { c.toLinearMap.smul_right f with cont := c.2.smul continuous_const } #align continuous_linear_map.smul_right ContinuousLinearMap.smulRight +-/ +/- warning: continuous_linear_map.smul_right_apply -> ContinuousLinearMap.smulRight_apply is a dubious translation: +lean 3 declaration is + forall {M₁ : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M₁] [_inst_5 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] {R : Type.{u3}} {S : Type.{u4}} [_inst_19 : Semiring.{u3} R] [_inst_20 : Semiring.{u4} S] [_inst_21 : Module.{u3, u1} R M₁ _inst_19 _inst_5] [_inst_22 : Module.{u3, u2} R M₂ _inst_19 _inst_9] [_inst_23 : Module.{u3, u4} R S _inst_19 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20)))] [_inst_24 : Module.{u4, u2} S M₂ _inst_20 _inst_9] [_inst_25 : IsScalarTower.{u3, u4, u2} R S M₂ (SMulZeroClass.toHasSmul.{u3, u4} R S (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20)))))) (SMulWithZero.toSmulZeroClass.{u3, u4} R S (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_19)))) (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20)))))) (MulActionWithZero.toSMulWithZero.{u3, u4} R S (Semiring.toMonoidWithZero.{u3} R _inst_19) (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20)))))) (Module.toMulActionWithZero.{u3, u4} R S _inst_19 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20))) _inst_23)))) (SMulZeroClass.toHasSmul.{u4, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u4, u2} S M₂ (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_20)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_20) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_20 _inst_9 _inst_24)))) (SMulZeroClass.toHasSmul.{u3, u2} R M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M₂ (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_19)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M₂ (Semiring.toMonoidWithZero.{u3} R _inst_19) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u3, u2} R M₂ _inst_19 _inst_9 _inst_22))))] [_inst_26 : TopologicalSpace.{u4} S] [_inst_27 : ContinuousSMul.{u4, u2} S M₂ (SMulZeroClass.toHasSmul.{u4, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u4, u2} S M₂ (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_20)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_20) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_20 _inst_9 _inst_24)))) _inst_26 _inst_8] {c : ContinuousLinearMap.{u3, u3, u1, u4} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20))) _inst_21 _inst_23} {f : M₂} {x : M₁}, Eq.{succ u2} M₂ (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u1, u2} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_21 _inst_22) (fun (_x : ContinuousLinearMap.{u3, u3, u1, u2} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_21 _inst_22) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u3, u3, u1, u2} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_21 _inst_22) (ContinuousLinearMap.smulRight.{u1, u2, u3, u4} M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 R S _inst_19 _inst_20 _inst_21 _inst_22 _inst_23 _inst_24 _inst_25 _inst_26 _inst_27 c f) x) (SMul.smul.{u4, u2} S M₂ (SMulZeroClass.toHasSmul.{u4, u2} S M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u4, u2} S M₂ (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S _inst_20)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u4, u2} S M₂ (Semiring.toMonoidWithZero.{u4} S _inst_20) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u4, u2} S M₂ _inst_20 _inst_9 _inst_24)))) (coeFn.{max (succ u1) (succ u4), max (succ u1) (succ u4)} (ContinuousLinearMap.{u3, u3, u1, u4} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20))) _inst_21 _inst_23) (fun (_x : ContinuousLinearMap.{u3, u3, u1, u4} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20))) _inst_21 _inst_23) => M₁ -> S) (ContinuousLinearMap.toFun.{u3, u3, u1, u4} R R _inst_19 _inst_19 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S _inst_20))) _inst_21 _inst_23) c x) f) +but is expected to have type + forall {M₁ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₁] [_inst_5 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] {R : Type.{u4}} {S : Type.{u2}} [_inst_19 : Semiring.{u4} R] [_inst_20 : Semiring.{u2} S] [_inst_21 : Module.{u4, u3} R M₁ _inst_19 _inst_5] [_inst_22 : Module.{u4, u1} R M₂ _inst_19 _inst_9] [_inst_23 : Module.{u4, u2} R S _inst_19 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20)))] [_inst_24 : Module.{u2, u1} S M₂ _inst_20 _inst_9] [_inst_25 : IsScalarTower.{u4, u2, u1} R S M₂ (SMulZeroClass.toSMul.{u4, u2} R S (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_20)) (SMulWithZero.toSMulZeroClass.{u4, u2} R S (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_19)) (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_20)) (MulActionWithZero.toSMulWithZero.{u4, u2} R S (Semiring.toMonoidWithZero.{u4} R _inst_19) (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_20)) (Module.toMulActionWithZero.{u4, u2} R S _inst_19 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_23)))) (SMulZeroClass.toSMul.{u2, u1} S M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u2, u1} S M₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_20)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u2, u1} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_20) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} S M₂ _inst_20 _inst_9 _inst_24)))) (SMulZeroClass.toSMul.{u4, u1} R M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u4, u1} R M₂ (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_19)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u4, u1} R M₂ (Semiring.toMonoidWithZero.{u4} R _inst_19) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u4, u1} R M₂ _inst_19 _inst_9 _inst_22))))] [_inst_26 : TopologicalSpace.{u2} S] [_inst_27 : ContinuousSMul.{u2, u1} S M₂ (SMulZeroClass.toSMul.{u2, u1} S M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u2, u1} S M₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_20)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u2, u1} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_20) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} S M₂ _inst_20 _inst_9 _inst_24)))) _inst_26 _inst_8] {c : ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23} {f : M₂} {x : M₁}, Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (ContinuousLinearMap.{u4, u4, u3, u1} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_21 _inst_22) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u1, u3, u1} (ContinuousLinearMap.{u4, u4, u3, u1} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 M₂ _inst_8 _inst_9 _inst_21 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(Module.toMulActionWithZero.{u2, u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => S) x) M₂ _inst_20 _inst_9 _inst_24))))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => S) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23) M₁ S _inst_4 _inst_26 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23) R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} R R _inst_19 _inst_19 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_19)) M₁ _inst_4 _inst_5 S _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_20))) _inst_21 _inst_23))) c x) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_right_apply ContinuousLinearMap.smulRight_applyₓ'. -/ @[simp] theorem smulRight_apply {c : M₁ →L[R] S} {f : M₂} {x : M₁} : (smulRight c f : M₁ → M₂) x = c x • f := @@ -1249,17 +1992,35 @@ end variable [Module R₁ M₂] [TopologicalSpace R₁] [ContinuousSMul R₁ M₂] +/- warning: continuous_linear_map.smul_right_one_one -> ContinuousLinearMap.smulRight_one_one is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_19 : Module.{u1, u2} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u1} R₁] [_inst_21 : ContinuousSMul.{u1, u2} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] (c : ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19), Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} 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_inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19) c (OfNat.ofNat.{u1} R₁ 1 (OfNat.mk.{u1} R₁ 1 (One.one.{u1} R₁ (AddMonoidWithOne.toOne.{u1} R₁ (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R₁ (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))))))))) c +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_19 : Module.{u2, u1} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u2} R₁] [_inst_21 : ContinuousSMul.{u2, u1} R₁ M₂ (SMulZeroClass.toSMul.{u2, u1} R₁ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u2, u1} R₁ M₂ (MonoidWithZero.toZero.{u2} R₁ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u2, u1} R₁ M₂ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] (c : ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19), Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : R₁) => M₂) (OfNat.ofNat.{u2} R₁ 1 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1)))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19) R₁ (fun (_x : R₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : R₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19) R₁ M₂ _inst_20 _inst_8 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u2, u1} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19))) c (OfNat.ofNat.{u2} R₁ 1 (One.toOfNat1.{u2} R₁ (Semiring.toOne.{u2} R₁ _inst_1))))) c +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_right_one_one ContinuousLinearMap.smulRight_one_oneₓ'. -/ @[simp] theorem smulRight_one_one (c : R₁ →L[R₁] M₂) : smulRight (1 : R₁ →L[R₁] R₁) (c 1) = c := by ext <;> simp [← ContinuousLinearMap.map_smul_of_tower] #align continuous_linear_map.smul_right_one_one ContinuousLinearMap.smulRight_one_one +/- warning: continuous_linear_map.smul_right_one_eq_iff -> ContinuousLinearMap.smulRight_one_eq_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_19 : Module.{u1, u2} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u1} R₁] [_inst_21 : ContinuousSMul.{u1, u2} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] {f : M₂} {f' : M₂}, Iff (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19) (ContinuousLinearMap.smulRight.{u1, u2, u1, u1} R₁ 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(RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (One.one.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) (ContinuousLinearMap.one.{u1, u1} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1))))) f')) (Eq.{succ u2} M₂ f f') +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_19 : Module.{u2, u1} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u2} R₁] [_inst_21 : ContinuousSMul.{u2, u1} R₁ M₂ (SMulZeroClass.toSMul.{u2, u1} R₁ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u2, u1} R₁ M₂ (MonoidWithZero.toZero.{u2} R₁ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u2, u1} R₁ M₂ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] {f : M₂} {f' : M₂}, Iff (Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19) (ContinuousLinearMap.smulRight.{u2, u1, u2, u2} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (IsScalarTower.left.{u2, u1} R₁ M₂ (MonoidWithZero.toMonoid.{u2} R₁ 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_inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1)) (ContinuousLinearMap.one.{u2, u2} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1)))) f) (ContinuousLinearMap.smulRight.{u2, u1, u2, u2} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (IsScalarTower.left.{u2, u1} R₁ M₂ (MonoidWithZero.toMonoid.{u2} R₁ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1)) (MulActionWithZero.toMulAction.{u2, u1} R₁ M₂ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} R₁ M₂ _inst_1 _inst_9 _inst_19))) _inst_20 _inst_21 (OfNat.ofNat.{u2} (ContinuousLinearMap.{u2, u2, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1)) 1 (One.toOfNat1.{u2} (ContinuousLinearMap.{u2, u2, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1)) (ContinuousLinearMap.one.{u2, u2} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1)))) f')) (Eq.{succ u1} M₂ f f') +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_right_one_eq_iff ContinuousLinearMap.smulRight_one_eq_iffₓ'. -/ @[simp] theorem smulRight_one_eq_iff {f f' : M₂} : smulRight (1 : R₁ →L[R₁] R₁) f = smulRight (1 : R₁ →L[R₁] R₁) f' ↔ f = f' := by simp only [ext_ring_iff, smul_right_apply, one_apply, one_smul] #align continuous_linear_map.smul_right_one_eq_iff ContinuousLinearMap.smulRight_one_eq_iff +/- warning: continuous_linear_map.smul_right_comp -> ContinuousLinearMap.smulRight_comp is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₂ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₂] [_inst_9 : AddCommMonoid.{u2} M₂] [_inst_19 : Module.{u1, u2} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u1} R₁] [_inst_21 : ContinuousSMul.{u1, u2} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] [_inst_22 : ContinuousMul.{u1} R₁ _inst_20 (Distrib.toHasMul.{u1} R₁ (NonUnitalNonAssocSemiring.toDistrib.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))))] {x : M₂} {c : R₁}, Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19) (ContinuousLinearMap.comp.{u1, u1, u1, u1, u1, u2} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1) _inst_19 (RingHomCompTriple.right_ids.{u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (ContinuousLinearMap.smulRight.{u1, u2, u1, u1} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19 (IsScalarTower.left.{u1, u2} R₁ M₂ (MonoidWithZero.toMonoid.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)) (MulActionWithZero.toMulAction.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19))) _inst_20 _inst_21 (OfNat.ofNat.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (OfNat.mk.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (One.one.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) (ContinuousLinearMap.one.{u1, u1} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1))))) x) (ContinuousLinearMap.smulRight.{u1, u1, u1, u1} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1) (IsScalarTower.left.{u1, u1} R₁ R₁ (MonoidWithZero.toMonoid.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)) (Monoid.toMulAction.{u1} R₁ (MonoidWithZero.toMonoid.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) _inst_20 (ContinuousMul.to_continuousSMul.{u1} R₁ _inst_20 (Distrib.toHasMul.{u1} R₁ (NonUnitalNonAssocSemiring.toDistrib.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)))) _inst_22) (OfNat.ofNat.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (OfNat.mk.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (One.one.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) (ContinuousLinearMap.one.{u1, u1} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1))))) c)) (ContinuousLinearMap.smulRight.{u1, u2, u1, u1} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₂ _inst_8 _inst_9 R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19 (Semiring.toModule.{u1} R₁ _inst_1) _inst_19 (IsScalarTower.left.{u1, u2} R₁ M₂ (MonoidWithZero.toMonoid.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)) (MulActionWithZero.toMulAction.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19))) _inst_20 _inst_21 (OfNat.ofNat.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (OfNat.mk.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) 1 (One.one.{u1} (ContinuousLinearMap.{u1, u1, u1, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1) (Semiring.toModule.{u1} R₁ _inst_1)) (ContinuousLinearMap.one.{u1, u1} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) (Semiring.toModule.{u1} R₁ _inst_1))))) (SMul.smul.{u1, u2} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₂ (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_9))) (Module.toMulActionWithZero.{u1, u2} R₁ M₂ _inst_1 _inst_9 _inst_19)))) c x)) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₂ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₂] [_inst_9 : AddCommMonoid.{u1} M₂] [_inst_19 : Module.{u2, u1} R₁ M₂ _inst_1 _inst_9] [_inst_20 : TopologicalSpace.{u2} R₁] [_inst_21 : ContinuousSMul.{u2, u1} R₁ M₂ (SMulZeroClass.toSMul.{u2, u1} R₁ M₂ (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (SMulWithZero.toSMulZeroClass.{u2, u1} R₁ M₂ (MonoidWithZero.toZero.{u2} R₁ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (MulActionWithZero.toSMulWithZero.{u2, u1} R₁ M₂ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} R₁ M₂ _inst_1 _inst_9 _inst_19)))) _inst_20 _inst_8] [_inst_22 : ContinuousMul.{u2} R₁ _inst_20 (NonUnitalNonAssocSemiring.toMul.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)))] {x : M₂} {c : R₁}, Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19) (ContinuousLinearMap.comp.{u2, u2, u2, u2, u2, u1} R₁ R₁ R₁ _inst_1 _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (RingHomCompTriple.ids.{u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (ContinuousLinearMap.smulRight.{u2, u1, u2, u2} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₂ _inst_8 _inst_9 R₁ R₁ _inst_1 _inst_1 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (Semiring.toModule.{u2} R₁ _inst_1) _inst_19 (IsScalarTower.left.{u2, u1} R₁ M₂ (MonoidWithZero.toMonoid.{u2} R₁ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1)) (MulActionWithZero.toMulAction.{u2, u1} R₁ M₂ (Semiring.toMonoidWithZero.{u2} R₁ _inst_1) (AddMonoid.toZero.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_9)) (Module.toMulActionWithZero.{u2, u1} R₁ M₂ _inst_1 _inst_9 _inst_19))) _inst_20 _inst_21 (OfNat.ofNat.{u2} (ContinuousLinearMap.{u2, u2, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1)) 1 (One.toOfNat1.{u2} (ContinuousLinearMap.{u2, u2, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1) (Semiring.toModule.{u2} R₁ _inst_1)) (ContinuousLinearMap.one.{u2, u2} R₁ _inst_1 R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) (Semiring.toModule.{u2} R₁ _inst_1)))) x) (ContinuousLinearMap.smulRight.{u2, u2, u2, u2} R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ _inst_20 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) R₁ R₁ _inst_1 _inst_1 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Consider using '#align continuous_linear_map.smul_right_comp ContinuousLinearMap.smulRight_compₓ'. -/ theorem smulRight_comp [ContinuousMul R₁] {x : M₂} {c : R₁} : (smulRight (1 : R₁ →L[R₁] R₁) x).comp (smulRight (1 : R₁ →L[R₁] R₁) c) = smulRight (1 : R₁ →L[R₁] R₁) (c • x) := @@ -1276,64 +2037,128 @@ variable {R : Type _} [Semiring R] {M : Type _} [TopologicalSpace M] [AddCommMon {M₂ : Type _} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M₂] {ι : Type _} {φ : ι → Type _} [∀ i, TopologicalSpace (φ i)] [∀ i, AddCommMonoid (φ i)] [∀ i, Module R (φ i)] +#print ContinuousLinearMap.pi /- /-- `pi` construction for continuous linear functions. From a family of continuous linear functions it produces a continuous linear function into a family of topological modules. -/ def pi (f : ∀ i, M →L[R] φ i) : M →L[R] ∀ i, φ i := ⟨LinearMap.pi fun i => f i, continuous_pi fun i => (f i).Continuous⟩ #align continuous_linear_map.pi ContinuousLinearMap.pi +-/ +/- warning: continuous_linear_map.coe_pi' -> ContinuousLinearMap.coe_pi' is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_2 _inst_3 (φ 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Consider using '#align continuous_linear_map.coe_pi' ContinuousLinearMap.coe_pi'ₓ'. -/ @[simp] theorem coe_pi' (f : ∀ i, M →L[R] φ i) : ⇑(pi f) = fun c i => f i c := rfl #align continuous_linear_map.coe_pi' ContinuousLinearMap.coe_pi' +/- warning: continuous_linear_map.coe_pi -> ContinuousLinearMap.coe_pi is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_2 _inst_3 (φ i) (_inst_8 i) (_inst_9 i) _inst_4 (_inst_10 i)), Eq.{max (succ u2) (succ 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Consider using '#align continuous_linear_map.coe_pi ContinuousLinearMap.coe_piₓ'. -/ @[simp] theorem coe_pi (f : ∀ i, M →L[R] φ i) : (pi f : M →ₗ[R] ∀ i, φ i) = LinearMap.pi fun i => f i := rfl #align continuous_linear_map.coe_pi ContinuousLinearMap.coe_pi +/- warning: continuous_linear_map.pi_apply -> ContinuousLinearMap.pi_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_2 _inst_3 (φ i) (_inst_8 i) (_inst_9 i) _inst_4 (_inst_10 i)) 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Consider using '#align continuous_linear_map.pi_apply ContinuousLinearMap.pi_applyₓ'. -/ theorem pi_apply (f : ∀ i, M →L[R] φ i) (c : M) (i : ι) : pi f c i = f i c := rfl #align continuous_linear_map.pi_apply ContinuousLinearMap.pi_apply +/- warning: continuous_linear_map.pi_eq_zero -> ContinuousLinearMap.pi_eq_zero is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_2 _inst_3 (φ i) (_inst_8 i) (_inst_9 i) _inst_4 (_inst_10 i)), Iff (Eq.{max 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Consider using '#align continuous_linear_map.pi_eq_zero ContinuousLinearMap.pi_eq_zeroₓ'. -/ theorem pi_eq_zero (f : ∀ i, M →L[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 := by simp only [ext_iff, pi_apply, Function.funext_iff] exact forall_swap #align continuous_linear_map.pi_eq_zero ContinuousLinearMap.pi_eq_zero +/- warning: continuous_linear_map.pi_zero -> ContinuousLinearMap.pi_zero is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)], Eq.{max (succ u2) (succ (max u3 u4))} (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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Consider using '#align continuous_linear_map.pi_zero ContinuousLinearMap.pi_zeroₓ'. -/ theorem pi_zero : pi (fun i => 0 : ∀ i, M →L[R] φ i) = 0 := ext fun _ => rfl #align continuous_linear_map.pi_zero ContinuousLinearMap.pi_zero +/- warning: continuous_linear_map.pi_comp -> ContinuousLinearMap.pi_comp is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_1 _inst_3] {M₂ : Type.{u3}} [_inst_5 : TopologicalSpace.{u3} M₂] [_inst_6 : AddCommMonoid.{u3} M₂] [_inst_7 : Module.{u1, u3} R M₂ _inst_1 _inst_6] {ι : Type.{u4}} {φ : ι -> Type.{u5}} [_inst_8 : forall (i : ι), TopologicalSpace.{u5} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u5} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u5} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R 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Consider using '#align continuous_linear_map.pi_comp ContinuousLinearMap.pi_compₓ'. -/ theorem pi_comp (f : ∀ i, M →L[R] φ i) (g : M₂ →L[R] M) : (pi f).comp g = pi fun i => (f i).comp g := rfl #align continuous_linear_map.pi_comp ContinuousLinearMap.pi_comp +#print ContinuousLinearMap.proj /- /-- The projections from a family of topological modules are continuous linear maps. -/ def proj (i : ι) : (∀ i, φ i) →L[R] φ i := ⟨LinearMap.proj i, continuous_apply _⟩ #align continuous_linear_map.proj ContinuousLinearMap.proj +-/ +/- warning: continuous_linear_map.proj_apply -> ContinuousLinearMap.proj_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {ι : Type.{u2}} {φ : ι -> Type.{u3}} [_inst_8 : forall (i : ι), TopologicalSpace.{u3} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u3} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u3} R (φ i) _inst_1 (_inst_9 i)] (i : ι) (b : forall (i : ι), φ i), Eq.{succ u3} ((fun (i : ι) => φ i) i) 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Consider using '#align continuous_linear_map.proj_apply ContinuousLinearMap.proj_applyₓ'. -/ @[simp] theorem proj_apply (i : ι) (b : ∀ i, φ i) : (proj i : (∀ i, φ i) →L[R] φ i) b = b i := rfl #align continuous_linear_map.proj_apply ContinuousLinearMap.proj_apply +/- warning: continuous_linear_map.proj_pi -> ContinuousLinearMap.proj_pi is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M₂ : Type.{u2}} [_inst_5 : TopologicalSpace.{u2} M₂] [_inst_6 : AddCommMonoid.{u2} M₂] [_inst_7 : Module.{u1, u2} R M₂ _inst_1 _inst_6] {ι : Type.{u3}} {φ : ι -> Type.{u4}} [_inst_8 : forall (i : ι), TopologicalSpace.{u4} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u4} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u4} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₂ _inst_5 _inst_6 (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (_inst_10 i)) (i : ι), Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₂ _inst_5 _inst_6 (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (_inst_10 i)) (ContinuousLinearMap.comp.{u1, u1, u1, u2, max u3 u4, u4} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M₂ _inst_5 _inst_6 (forall (i : ι), φ i) (Pi.topologicalSpace.{u3, u4} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u3, u4} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (Pi.module.{u3, u4, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i) (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (ContinuousLinearMap.proj.{u1, u3, u4} R _inst_1 ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a) (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i) i) (ContinuousLinearMap.pi.{u1, u2, u3, u4} R _inst_1 M₂ _inst_5 _inst_6 _inst_7 ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_8 i) (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i) f)) (f i) +but is expected to have type + forall {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] {M₂ : Type.{u3}} [_inst_5 : TopologicalSpace.{u3} M₂] [_inst_6 : AddCommMonoid.{u3} M₂] [_inst_7 : Module.{u4, u3} R M₂ _inst_1 _inst_6] {ι : Type.{u1}} {φ : ι -> Type.{u2}} [_inst_8 : forall (i : ι), TopologicalSpace.{u2} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u2} (φ i)] [_inst_10 : forall (i : ι), Module.{u4, u2} R (φ i) _inst_1 (_inst_9 i)] (f : forall (i : ι), ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) M₂ _inst_5 _inst_6 (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (_inst_10 i)) (i : ι), Eq.{max (succ u3) (succ u2)} (ContinuousLinearMap.{u4, u4, u3, u2} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) M₂ _inst_5 _inst_6 (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (_inst_10 i)) (ContinuousLinearMap.comp.{u4, u4, u4, u3, max u1 u2, u2} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) M₂ _inst_5 _inst_6 (forall (i : ι), φ i) (Pi.topologicalSpace.{u1, u2} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u1, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) _inst_7 (Pi.module.{u1, u2, u4} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i) (RingHomCompTriple.ids.{u4, u4} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1))) (ContinuousLinearMap.proj.{u4, u1, u2} R _inst_1 ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a) (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i) i) (ContinuousLinearMap.pi.{u4, u3, u1, u2} R _inst_1 M₂ _inst_5 _inst_6 _inst_7 ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_8 i) (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i) f)) (f i) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.proj_pi ContinuousLinearMap.proj_piₓ'. -/ theorem proj_pi (f : ∀ i, M₂ →L[R] φ i) (i : ι) : (proj i).comp (pi f) = f i := ext fun c => rfl #align continuous_linear_map.proj_pi ContinuousLinearMap.proj_pi +/- warning: continuous_linear_map.infi_ker_proj -> ContinuousLinearMap.infᵢ_ker_proj is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {ι : Type.{u2}} {φ : ι -> Type.{u3}} [_inst_8 : forall (i : ι), TopologicalSpace.{u3} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u3} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u3} R (φ i) _inst_1 (_inst_9 i)], Eq.{succ (max u2 u3)} (Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (infᵢ.{max u2 u3, succ u2} (Submodule.{u1, max u2 u3} R (forall (i : 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(Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Submodule.hasBot.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {ι : Type.{u3}} {φ : ι -> Type.{u2}} [_inst_8 : forall (i : ι), TopologicalSpace.{u2} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u2} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u2} R (φ i) _inst_1 (_inst_9 i)], Eq.{max (succ u3) (succ u2)} (Submodule.{u1, max u3 u2} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (infᵢ.{max u3 u2, succ u3} (Submodule.{u1, max u3 u2} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Submodule.instInfSetSubmodule.{u1, max u3 u2} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) ι (fun (i : ι) => LinearMap.ker.{u1, u1, max u3 u2, u2, max u3 u2} R R (forall (i : ι), φ i) (φ i) _inst_1 _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (_inst_9 i) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (ContinuousLinearMap.{u1, u1, max u3 u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (forall (i : ι), φ i) (Pi.topologicalSpace.{u3, u2} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i)) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u2, u1, u1, max u3 u2, u2} (ContinuousLinearMap.{u1, u1, max u3 u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (forall (i : ι), φ i) (Pi.topologicalSpace.{u3, u2} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i)) R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (forall (i : ι), φ i) (Pi.topologicalSpace.{u3, u2} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, max u3 u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (forall (i : ι), φ i) (Pi.topologicalSpace.{u3, u2} ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a)) (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (φ i) (_inst_8 i) (_inst_9 i) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)) (_inst_10 i))) (ContinuousLinearMap.proj.{u1, u3, u2} R _inst_1 ι (fun (i : ι) => φ i) (fun (a : ι) => _inst_8 a) (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i) i))) (Bot.bot.{max u3 u2} (Submodule.{u1, max u3 u2} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Submodule.instBotSubmodule.{u1, max u3 u2} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u3, u2} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u3, u2, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.infi_ker_proj ContinuousLinearMap.infᵢ_ker_projₓ'. -/ theorem infᵢ_ker_proj : (⨅ i, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i)) = ⊥ := LinearMap.infᵢ_ker_proj #align continuous_linear_map.infi_ker_proj ContinuousLinearMap.infᵢ_ker_proj variable (R φ) +/- warning: continuous_linear_map.infi_ker_proj_equiv -> ContinuousLinearMap.infᵢKerProjEquiv is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) [_inst_1 : Semiring.{u1} R] {ι : Type.{u2}} (φ : ι -> Type.{u3}) [_inst_8 : forall (i : ι), TopologicalSpace.{u3} (φ i)] [_inst_9 : forall (i : ι), AddCommMonoid.{u3} (φ i)] [_inst_10 : forall (i : ι), Module.{u1, u3} R (φ i) _inst_1 (_inst_9 i)] {I : Set.{u2} ι} {J : Set.{u2} ι} [_inst_11 : DecidablePred.{succ u2} ι (fun (i : ι) => Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i I)], (Disjoint.{u2} (Set.{u2} ι) (CompleteSemilatticeInf.toPartialOrder.{u2} (Set.{u2} ι) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Set.{u2} ι) (Order.Coframe.toCompleteLattice.{u2} (Set.{u2} ι) (CompleteDistribLattice.toCoframe.{u2} (Set.{u2} ι) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u2} (Set.{u2} ι) (Set.completeBooleanAlgebra.{u2} ι)))))) (GeneralizedBooleanAlgebra.toOrderBot.{u2} (Set.{u2} ι) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u2} (Set.{u2} ι) (Set.booleanAlgebra.{u2} ι))) I J) -> (HasSubset.Subset.{u2} (Set.{u2} ι) (Set.hasSubset.{u2} ι) (Set.univ.{u2} ι) (Union.union.{u2} (Set.{u2} ι) (Set.hasUnion.{u2} ι) I J)) -> (ContinuousLinearEquiv.{u1, u1, max u2 u3, max u2 u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R _inst_1) (RingHomInvPair.ids.{u1} R _inst_1) (coeSort.{succ (max u2 u3), succ (succ (max u2 u3))} (Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) Type.{max u2 u3} (SetLike.hasCoeToSort.{max u2 u3, max u2 u3} (Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (forall (i : ι), φ i) (Submodule.setLike.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i)))) (infᵢ.{max u2 u3, succ u2} (Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Submodule.hasInf.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) ι (fun (i : ι) => infᵢ.{max u2 u3, 0} (Submodule.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Submodule.hasInf.{u1, max u2 u3} R (forall (i : ι), φ i) _inst_1 (Pi.addCommMonoid.{u2, u3} ι (fun (i : ι) => φ i) (fun (i : ι) => _inst_9 i)) (Pi.module.{u2, u3, u1} ι (fun (i : ι) => φ i) R _inst_1 (fun (i : ι) => _inst_9 i) (fun (i : ι) => _inst_10 i))) (Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i J) (fun (H : Membership.Mem.{u2, u2} ι (Set.{u2} ι) (Set.hasMem.{u2} ι) i J) => LinearMap.ker.{u1, u1, max u2 u3, 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Consider using '#align continuous_linear_map.infi_ker_proj_equiv ContinuousLinearMap.infᵢKerProjEquivₓ'. -/ /-- If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections of `φ` is linearly equivalent to the product over `I`. -/ -def infiKerProjEquiv {I J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjoint I J) +def infᵢKerProjEquiv {I J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjoint I J) (hu : Set.univ ⊆ I ∪ J) : (⨅ i ∈ J, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i)) ≃L[R] ∀ i : I, φ i where @@ -1351,7 +2176,7 @@ def infiKerProjEquiv {I J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjo (continuous_pi fun i => by dsimp; split_ifs <;> [apply continuous_apply, exact continuous_zero]) _ -#align continuous_linear_map.infi_ker_proj_equiv ContinuousLinearMap.infiKerProjEquiv +#align continuous_linear_map.infi_ker_proj_equiv ContinuousLinearMap.infᵢKerProjEquiv end Pi @@ -1365,14 +2190,32 @@ variable {R : Type _} [Ring R] {R₂ : Type _} [Ring R₂] {R₃ : Type _} [Ring section +/- warning: continuous_linear_map.map_neg -> ContinuousLinearMap.map_neg is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 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Consider using '#align continuous_linear_map.map_neg ContinuousLinearMap.map_negₓ'. -/ protected theorem map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = -f x := map_neg _ _ #align continuous_linear_map.map_neg ContinuousLinearMap.map_neg +/- warning: continuous_linear_map.map_sub -> ContinuousLinearMap.map_sub is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ 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Consider using '#align continuous_linear_map.map_sub ContinuousLinearMap.map_subₓ'. -/ protected theorem map_sub (f : M →SL[σ₁₂] M₂) (x y : M) : f (x - y) = f x - f y := map_sub _ _ _ #align continuous_linear_map.map_sub ContinuousLinearMap.map_sub +/- warning: continuous_linear_map.sub_apply' -> ContinuousLinearMap.sub_apply' is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) 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Consider using '#align continuous_linear_map.sub_apply' ContinuousLinearMap.sub_apply'ₓ'. -/ @[simp] theorem sub_apply' (f g : M →SL[σ₁₂] M₂) (x : M) : ((f : M →ₛₗ[σ₁₂] M₂) - g) x = f x - g x := rfl @@ -1384,16 +2227,34 @@ section variable [Module R M₂] [Module R M₃] [Module R M₄] +/- warning: continuous_linear_map.range_prod_eq -> ContinuousLinearMap.range_prod_eq is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₃] [_inst_9 : AddCommGroup.{u4} M₃] [_inst_12 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_15 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_16 : Module.{u1, u4} R M₃ 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) f) (LinearMap.ker.{u1, u1, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) g)) (Top.top.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) _inst_12) (Submodule.hasTop.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) _inst_12))) -> (Eq.{succ (max u3 u4)} (Submodule.{u1, max u3 u4} R (Prod.{u3, u4} M₂ M₃) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) (LinearMap.range.{u1, u1, u2, max u3 u4, max u2 u3 u4} R R M (Prod.{u3, u4} M₂ M₃) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) _inst_12 (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_6 _inst_8) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) _inst_12 (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u3 u4, u1, u1, u2, max u3 u4} (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_6 _inst_8) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) _inst_12 (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_6 _inst_8) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) _inst_12 (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, max u3 u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (Prod.{u3, u4} M₂ M₃) (Prod.topologicalSpace.{u3, u4} M₂ M₃ _inst_6 _inst_8) (Prod.addCommMonoid.{u3, u4} M₂ M₃ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)) _inst_12 (Prod.module.{u1, u3, u4} R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (ContinuousLinearMap.prod.{u1, u2, u3, u4} R (Ring.toSemiring.{u1} R _inst_1) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_15 _inst_16 f g)) (Submodule.prod.{u1, u3, u4} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_15 (LinearMap.range.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u3, u1, u1, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) f) M₃ (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16 (LinearMap.range.{u1, u1, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) g))) +but is expected to have type + forall {R : Type.{u4}} [_inst_1 : Ring.{u4} R] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M₂] [_inst_7 : AddCommGroup.{u2} M₂] {M₃ : Type.{u1}} [_inst_8 : TopologicalSpace.{u1} M₃] [_inst_9 : AddCommGroup.{u1} M₃] [_inst_12 : Module.{u4, u3} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_15 : Module.{u4, u2} R M₂ (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7)] [_inst_16 : Module.{u4, u1} R M₃ (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9)] {f : ContinuousLinearMap.{u4, u4, u3, u2} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_15} {g : ContinuousLinearMap.{u4, u4, u3, u1} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16}, (Eq.{succ u3} (Submodule.{u4, u3} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) _inst_12) (Sup.sup.{u3} (Submodule.{u4, u3} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) _inst_12) 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(NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) (ContinuousLinearMap.{u4, u4, u3, u2} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_15) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_15) R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R 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_inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_15 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_15)) (RingHomSurjective.ids.{u4} R (Ring.toSemiring.{u4} R _inst_1)) f) M₃ (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_16 (LinearMap.range.{u4, u4, u3, u1, max u3 u1} R R M M₃ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) (ContinuousLinearMap.{u4, u4, u3, u1} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u1, u4, u4, u3, u1} (ContinuousLinearMap.{u4, u4, u3, u1} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u1} R R (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u1} M₃ _inst_9) _inst_12 _inst_16)) (RingHomSurjective.ids.{u4} R (Ring.toSemiring.{u4} R _inst_1)) g))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.range_prod_eq ContinuousLinearMap.range_prod_eqₓ'. -/ theorem range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃} (h : ker f ⊔ ker g = ⊤) : range (f.Prod g) = (range f).Prod (range g) := LinearMap.range_prod_eq h #align continuous_linear_map.range_prod_eq ContinuousLinearMap.range_prod_eq +/- warning: continuous_linear_map.ker_prod_ker_le_ker_coprod -> ContinuousLinearMap.ker_prod_ker_le_ker_coprod is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₃] [_inst_9 : AddCommGroup.{u4} M₃] [_inst_12 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_15 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_16 : Module.{u1, u4} R M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)] [_inst_18 : ContinuousAdd.{u4} M₃ _inst_8 (AddZeroClass.toHasAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (SubNegMonoid.toAddMonoid.{u4} M₃ (AddGroup.toSubNegMonoid.{u4} M₃ (AddCommGroup.toAddGroup.{u4} M₃ _inst_9)))))] (f : ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (g : ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16), LE.le.{max u2 u3} (Submodule.{u1, max u2 u3} R (Prod.{u2, u3} M M₂) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) (Preorder.toLE.{max u2 u3} (Submodule.{u1, max u2 u3} R (Prod.{u2, u3} M M₂) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) (PartialOrder.toPreorder.{max u2 u3} (Submodule.{u1, max u2 u3} R (Prod.{u2, u3} M M₂) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) (SetLike.partialOrder.{max u2 u3, max u2 u3} (Submodule.{u1, max u2 u3} R (Prod.{u2, u3} M M₂) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15)) (Prod.{u2, u3} M M₂) (Submodule.setLike.{u1, max u2 u3} R (Prod.{u2, u3} M M₂) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15))))) (Submodule.prod.{u1, u2, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) _inst_12 (LinearMap.ker.{u1, u1, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) f) M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_15 (LinearMap.ker.{u1, u1, u3, u4, max u3 u4} R R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u4, u1, u1, u3, u4} (ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) g)) (LinearMap.ker.{u1, u1, max u2 u3, u4, max (max u2 u3) u4} R R (Prod.{u2, u3} M M₂) M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max (max u2 u3) u4, u1, u1, max u2 u3, u4} (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16)) (ContinuousLinearMap.coprod.{u1, u2, u3, u4} R (Ring.toSemiring.{u1} R _inst_1) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_15 _inst_16 _inst_18 f g)) +but is expected to have type + forall {R : Type.{u3}} [_inst_1 : Ring.{u3} R] {M : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] {M₃ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₃] [_inst_9 : AddCommGroup.{u4} M₃] [_inst_12 : Module.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_15 : Module.{u3, u1} R M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] [_inst_16 : Module.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)] [_inst_18 : ContinuousAdd.{u4} M₃ _inst_8 (AddZeroClass.toAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (SubNegMonoid.toAddMonoid.{u4} M₃ (AddGroup.toSubNegMonoid.{u4} M₃ (AddCommGroup.toAddGroup.{u4} M₃ _inst_9)))))] (f : ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (g : ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16), LE.le.{max u2 u1} (Submodule.{u3, max u1 u2} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (Preorder.toLE.{max u2 u1} (Submodule.{u3, max u1 u2} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (PartialOrder.toPreorder.{max u2 u1} (Submodule.{u3, max u1 u2} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (OmegaCompletePartialOrder.toPartialOrder.{max u2 u1} (Submodule.{u3, max u1 u2} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (CompleteLattice.instOmegaCompletePartialOrder.{max u2 u1} (Submodule.{u3, max u1 u2} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (Submodule.completeLattice.{u3, max u2 u1} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)))))) (Submodule.prod.{u3, u2, u1} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) _inst_12 (LinearMap.ker.{u3, u3, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u3, u3, u2, u4} (ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) f) M₂ (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_15 (LinearMap.ker.{u3, u3, u1, u4, max u1 u4} R R M₂ M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R 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_inst_15) _inst_16 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, max u1 u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_1))) (Prod.{u2, u1} M M₂) (instTopologicalSpaceProd.{u2, u1} M M₂ _inst_4 _inst_6) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15) _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max (max u2 u1) u4, u3, u3, max u2 u1, u4} (ContinuousLinearMap.{u3, u3, max u1 u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} 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(AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_15 _inst_16 _inst_18 f g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ker_prod_ker_le_ker_coprod ContinuousLinearMap.ker_prod_ker_le_ker_coprodₓ'. -/ theorem ker_prod_ker_le_ker_coprod [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) : (LinearMap.ker f).Prod (LinearMap.ker g) ≤ LinearMap.ker (f.coprod g) := LinearMap.ker_prod_ker_le_ker_coprod f.toLinearMap g.toLinearMap #align continuous_linear_map.ker_prod_ker_le_ker_coprod ContinuousLinearMap.ker_prod_ker_le_ker_coprod +/- warning: continuous_linear_map.ker_coprod_of_disjoint_range -> ContinuousLinearMap.ker_coprod_of_disjoint_range is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₃] [_inst_9 : AddCommGroup.{u4} M₃] [_inst_12 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_15 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_16 : Module.{u1, u4} R M₃ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)] [_inst_18 : ContinuousAdd.{u4} M₃ _inst_8 (AddZeroClass.toHasAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (SubNegMonoid.toAddMonoid.{u4} M₃ (AddGroup.toSubNegMonoid.{u4} M₃ (AddCommGroup.toAddGroup.{u4} M₃ _inst_9)))))] (f : ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (g : ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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(AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max (max u2 u3) u4, u1, u1, max u2 u3, u4} (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, max u2 u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Prod.{u2, u3} M M₂) (Prod.topologicalSpace.{u2, u3} M M₂ _inst_4 _inst_6) (Prod.addCommMonoid.{u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u1, u2, u3} R M M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_15) _inst_16)) (ContinuousLinearMap.coprod.{u1, u2, u3, u4} R (Ring.toSemiring.{u1} R _inst_1) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_15 _inst_16 _inst_18 f g)) (Submodule.prod.{u1, u2, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) _inst_12 (LinearMap.ker.{u1, u1, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u1, u1, u2, u4} (ContinuousLinearMap.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) f) M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_15 (LinearMap.ker.{u1, u1, u3, u4, max u3 u4} R R M₂ M₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u3 u4, u1, u1, u3, u4} (ContinuousLinearMap.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u4} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) g))) +but is expected to have type + forall {R : Type.{u3}} [_inst_1 : Ring.{u3} R] {M : Type.{u2}} [_inst_4 : TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] {M₃ : Type.{u4}} [_inst_8 : TopologicalSpace.{u4} M₃] [_inst_9 : AddCommGroup.{u4} M₃] [_inst_12 : Module.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_15 : Module.{u3, u1} R M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] [_inst_16 : Module.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9)] [_inst_18 : ContinuousAdd.{u4} M₃ _inst_8 (AddZeroClass.toAdd.{u4} M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (SubNegMonoid.toAddMonoid.{u4} M₃ (AddGroup.toSubNegMonoid.{u4} M₃ (AddCommGroup.toAddGroup.{u4} M₃ _inst_9)))))] (f : ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (g : ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16), (Disjoint.{u4} (Submodule.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16) (OmegaCompletePartialOrder.toPartialOrder.{u4} (Submodule.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16) (CompleteLattice.instOmegaCompletePartialOrder.{u4} (Submodule.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16) (Submodule.completeLattice.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16))) (Submodule.instOrderBotSubmoduleToLEToPreorderInstPartialOrderSetLike.{u3, u4} R M₃ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_16) (LinearMap.range.{u3, u3, u2, u4, max u2 u4} R R M M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u4, u3, u3, u2, u4} (ContinuousLinearMap.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_12 _inst_16)) (RingHomSurjective.ids.{u3} R (Ring.toSemiring.{u3} R _inst_1)) f) (LinearMap.range.{u3, u3, u1, u4, max u1 u4} R R M₂ M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u1 u4, u3, u3, u1, u4} (ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) (RingHomSurjective.ids.{u3} R (Ring.toSemiring.{u3} R _inst_1)) g)) -> (Eq.{max (succ u2) (succ u1)} (Submodule.{u3, max u2 u1} R (Prod.{u2, u1} M M₂) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15)) (LinearMap.ker.{u3, u3, max u2 u1, u4, max (max u2 u4) u1} R R (Prod.{u2, u1} M M₂) M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (Prod.instAddCommMonoidSum.{u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) (Prod.module.{u3, u2, u1} R M M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_15) _inst_16 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_inst_12 _inst_16)) f) M₂ (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_15 (LinearMap.ker.{u3, u3, u1, u4, max u1 u4} R R M₂ M₃ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u1 u4, u3, u3, u1, u4} (ContinuousLinearMap.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u1, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_9) _inst_15 _inst_16)) g))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ker_coprod_of_disjoint_range ContinuousLinearMap.ker_coprod_of_disjoint_rangeₓ'. -/ theorem ker_coprod_of_disjoint_range [ContinuousAdd M₃] (f : M →L[R] M₃) (g : M₂ →L[R] M₃) (hd : Disjoint (range f) (range g)) : LinearMap.ker (f.coprod g) = (LinearMap.ker f).Prod (LinearMap.ker g) := @@ -1409,16 +2270,34 @@ variable [TopologicalAddGroup M₂] instance : Neg (M →SL[σ₁₂] M₂) := ⟨fun f => ⟨-f, f.2.neg⟩⟩ +/- warning: continuous_linear_map.neg_apply -> ContinuousLinearMap.neg_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u4} M₂ _inst_6 (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)] (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (x : M), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ 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TopologicalSpace.{u2} M] [_inst_5 : AddCommGroup.{u2} M] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] [_inst_12 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_13 : Module.{u3, u1} R₂ M₂ (Ring.toSemiring.{u3} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] {σ₁₂ : RingHom.{u4, u3} R R₂ (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₂ (Ring.toNonAssocRing.{u3} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u1} M₂ _inst_6 (AddCommGroup.toAddGroup.{u1} M₂ _inst_7)] (f : ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (x : M), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) 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Consider using '#align continuous_linear_map.neg_apply ContinuousLinearMap.neg_applyₓ'. -/ @[simp] theorem neg_apply (f : M →SL[σ₁₂] M₂) (x : M) : (-f) x = -f x := rfl #align continuous_linear_map.neg_apply ContinuousLinearMap.neg_apply +/- warning: continuous_linear_map.coe_neg -> ContinuousLinearMap.coe_neg is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u4} M₂ _inst_6 (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)] (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u3) (succ u4)} (LinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) ((fun (a : Type.{max u3 u4}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{succ (max u3 u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ 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R R₂ (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₂ (Ring.toNonAssocRing.{u3} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u1} M₂ _inst_6 (AddCommGroup.toAddGroup.{u1} M₂ _inst_7)] (f : ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u2) (succ u1)} (LinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousLinearMap.toLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) 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Consider using '#align continuous_linear_map.coe_neg ContinuousLinearMap.coe_negₓ'. -/ @[simp, norm_cast] theorem coe_neg (f : M →SL[σ₁₂] M₂) : (↑(-f) : M →ₛₗ[σ₁₂] M₂) = -f := rfl #align continuous_linear_map.coe_neg ContinuousLinearMap.coe_neg +/- warning: continuous_linear_map.coe_neg' -> ContinuousLinearMap.coe_neg' is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u4} M₂ _inst_6 (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)] (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13), Eq.{succ (max u3 u4)} (M -> M₂) (coeFn.{succ (max u3 u4), succ (max u3 u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) => M -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (Neg.neg.{max u3 u4} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (ContinuousLinearMap.neg.{u1, u2, u3, u4} R _inst_1 R₂ _inst_2 M _inst_4 _inst_5 M₂ _inst_6 _inst_7 _inst_12 _inst_13 σ₁₂ _inst_15) f)) (Neg.neg.{max u3 u4} (M -> M₂) (Pi.instNeg.{u3, u4} M (fun (ᾰ : M) => M₂) (fun (i : M) => SubNegMonoid.toHasNeg.{u4} M₂ (AddGroup.toSubNegMonoid.{u4} M₂ (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)))) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} 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[_inst_12 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_5)] [_inst_13 : Module.{u3, u1} R₂ M₂ (Ring.toSemiring.{u3} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] {σ₁₂ : RingHom.{u4, u3} R R₂ (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₂ (Ring.toNonAssocRing.{u3} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u1} M₂ _inst_6 (AddCommGroup.toAddGroup.{u1} M₂ _inst_7)] (f : ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u2) (succ u1)} (forall (ᾰ : M), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) ᾰ) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R R₂ 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Consider using '#align continuous_linear_map.coe_neg' ContinuousLinearMap.coe_neg'ₓ'. -/ @[norm_cast] theorem coe_neg' (f : M →SL[σ₁₂] M₂) : ⇑(-f) = -f := rfl @@ -1450,15 +2329,33 @@ instance : AddCommGroup (M →SL[σ₁₂] M₂) := by ext <;> apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm, sub_eq_add_neg] +/- warning: continuous_linear_map.sub_apply -> ContinuousLinearMap.sub_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u4} M₂ _inst_6 (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)] (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (g : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (x : M), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M 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Consider using '#align continuous_linear_map.sub_apply ContinuousLinearMap.sub_applyₓ'. -/ theorem sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x := rfl #align continuous_linear_map.sub_apply ContinuousLinearMap.sub_apply +/- warning: continuous_linear_map.coe_sub -> ContinuousLinearMap.coe_sub is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ 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Consider using '#align continuous_linear_map.coe_sub ContinuousLinearMap.coe_subₓ'. -/ @[simp, norm_cast] theorem coe_sub (f g : M →SL[σ₁₂] M₂) : (↑(f - g) : M →ₛₗ[σ₁₂] M₂) = f - g := rfl #align continuous_linear_map.coe_sub ContinuousLinearMap.coe_sub +/- warning: continuous_linear_map.coe_sub' -> ContinuousLinearMap.coe_sub' is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} [_inst_15 : TopologicalAddGroup.{u4} M₂ _inst_6 (AddCommGroup.toAddGroup.{u4} M₂ _inst_7)] (f : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (g : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13), Eq.{succ (max u3 u4)} (M -> M₂) (coeFn.{succ (max u3 u4), succ (max u3 u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) 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_inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13))) f) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) M M₂ _inst_4 _inst_6 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u2} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13))) g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_sub' ContinuousLinearMap.coe_sub'ₓ'. -/ @[simp, norm_cast] theorem coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g := rfl @@ -1466,6 +2363,12 @@ theorem coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g := end +/- warning: continuous_linear_map.comp_neg -> ContinuousLinearMap.comp_neg is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {R₃ : Type.{u3}} [_inst_3 : Ring.{u3} R₃] {M : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M] [_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u5}} [_inst_6 : TopologicalSpace.{u5} M₂] [_inst_7 : AddCommGroup.{u5} M₂] {M₃ : Type.{u6}} [_inst_8 : TopologicalSpace.{u6} M₃] [_inst_9 : AddCommGroup.{u6} M₃] [_inst_12 : Module.{u1, u4} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u5} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7)] [_inst_14 : Module.{u3, u6} R₃ M₃ (Ring.toSemiring.{u3} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} {σ₁₃ : RingHom.{u1, u3} R R₃ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u5} M₂ _inst_6 (AddCommGroup.toAddGroup.{u5} M₂ _inst_7)] [_inst_17 : TopologicalAddGroup.{u6} M₃ _inst_8 (AddCommGroup.toAddGroup.{u6} M₃ _inst_9)] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (f : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g (Neg.neg.{max u4 u5} (ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13) (ContinuousLinearMap.neg.{u1, u2, u4, u5} R _inst_1 R₂ _inst_2 M _inst_4 _inst_5 M₂ _inst_6 _inst_7 _inst_12 _inst_13 σ₁₂ _inst_16) f)) (Neg.neg.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.neg.{u1, u3, u4, u6} R _inst_1 R₃ _inst_3 M _inst_4 _inst_5 M₃ _inst_8 _inst_9 _inst_12 _inst_14 σ₁₃ _inst_17) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f)) +but is expected to have type + forall {R : Type.{u6}} [_inst_1 : Ring.{u6} R] {R₂ : Type.{u5}} [_inst_2 : Ring.{u5} R₂] {R₃ : Type.{u4}} [_inst_3 : Ring.{u4} R₃] {M : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M] [_inst_5 : AddCommGroup.{u1} M] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] {M₃ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₃] [_inst_9 : AddCommGroup.{u2} M₃] [_inst_12 : Module.{u6, u1} R M (Ring.toSemiring.{u6} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5)] [_inst_13 : Module.{u5, u3} R₂ M₂ (Ring.toSemiring.{u5} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_14 : Module.{u4, u2} R₃ M₃ (Ring.toSemiring.{u4} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9)] {σ₁₂ : RingHom.{u6, u5} R R₂ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2))} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} {σ₁₃ : RingHom.{u6, u4} R R₃ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u6, u5, u4} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u3} M₂ _inst_6 (AddCommGroup.toAddGroup.{u3} M₂ _inst_7)] [_inst_17 : TopologicalAddGroup.{u2} M₃ _inst_8 (AddCommGroup.toAddGroup.{u2} M₃ _inst_9)] (g : ContinuousLinearMap.{u5, u4, u3, u2} R₂ R₃ (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_13 _inst_14) (f : ContinuousLinearMap.{u6, u5, u1, u3} R R₂ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u6, u4, u1, u2} R R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.comp.{u6, u5, u4, u1, u3, u2} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g (Neg.neg.{max u1 u3} (ContinuousLinearMap.{u6, u5, u1, u3} R R₂ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_13) (ContinuousLinearMap.neg.{u6, u5, u1, u3} R _inst_1 R₂ _inst_2 M _inst_4 _inst_5 M₂ _inst_6 _inst_7 _inst_12 _inst_13 σ₁₂ _inst_16) f)) (Neg.neg.{max u1 u2} (ContinuousLinearMap.{u6, u4, u1, u2} R R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.neg.{u6, u4, u1, u2} R _inst_1 R₃ _inst_3 M _inst_4 _inst_5 M₃ _inst_8 _inst_9 _inst_12 _inst_14 σ₁₃ _inst_17) (ContinuousLinearMap.comp.{u6, u5, u4, u1, u3, u2} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_neg ContinuousLinearMap.comp_negₓ'. -/ @[simp] theorem comp_neg [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : g.comp (-f) = -g.comp f := @@ -1474,6 +2377,12 @@ theorem comp_neg [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddG simp #align continuous_linear_map.comp_neg ContinuousLinearMap.comp_neg +/- warning: continuous_linear_map.neg_comp -> ContinuousLinearMap.neg_comp is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {R₃ : Type.{u3}} [_inst_3 : Ring.{u3} R₃] {M : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M] [_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u5}} [_inst_6 : TopologicalSpace.{u5} M₂] [_inst_7 : AddCommGroup.{u5} M₂] {M₃ : Type.{u6}} [_inst_8 : TopologicalSpace.{u6} M₃] [_inst_9 : AddCommGroup.{u6} M₃] [_inst_12 : Module.{u1, u4} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u5} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7)] [_inst_14 : Module.{u3, u6} R₃ M₃ (Ring.toSemiring.{u3} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} {σ₁₃ : RingHom.{u1, u3} R R₃ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u6} M₃ _inst_8 (AddCommGroup.toAddGroup.{u6} M₃ _inst_9)] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (f : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 (Neg.neg.{max u5 u6} (ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (ContinuousLinearMap.neg.{u2, u3, u5, u6} R₂ _inst_2 R₃ _inst_3 M₂ _inst_6 _inst_7 M₃ _inst_8 _inst_9 _inst_13 _inst_14 σ₂₃ _inst_16) g) f) (Neg.neg.{max u4 u6} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.neg.{u1, u3, u4, u6} R _inst_1 R₃ _inst_3 M _inst_4 _inst_5 M₃ _inst_8 _inst_9 _inst_12 _inst_14 σ₁₃ _inst_16) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f)) +but is expected to have type + forall {R : Type.{u6}} [_inst_1 : Ring.{u6} R] {R₂ : Type.{u5}} [_inst_2 : Ring.{u5} R₂] {R₃ : Type.{u4}} [_inst_3 : Ring.{u4} R₃] {M : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M] [_inst_5 : AddCommGroup.{u1} M] {M₂ : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M₂] [_inst_7 : AddCommGroup.{u2} M₂] {M₃ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₃] [_inst_9 : AddCommGroup.{u3} M₃] [_inst_12 : Module.{u6, u1} R M (Ring.toSemiring.{u6} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5)] [_inst_13 : Module.{u5, u2} R₂ M₂ (Ring.toSemiring.{u5} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7)] [_inst_14 : Module.{u4, u3} R₃ M₃ (Ring.toSemiring.{u4} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9)] {σ₁₂ : RingHom.{u6, u5} R R₂ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2))} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} {σ₁₃ : RingHom.{u6, u4} R R₃ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u6, u5, u4} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u3} M₃ _inst_8 (AddCommGroup.toAddGroup.{u3} M₃ _inst_9)] (g : ContinuousLinearMap.{u5, u4, u2, u3} R₂ R₃ (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_13 _inst_14) (f : ContinuousLinearMap.{u6, u5, u1, u2} R R₂ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u1) (succ u3)} (ContinuousLinearMap.{u6, u4, u1, u3} R R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.comp.{u6, u5, u4, u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 (Neg.neg.{max u2 u3} (ContinuousLinearMap.{u5, u4, u2, u3} R₂ R₃ (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_13 _inst_14) (ContinuousLinearMap.neg.{u5, u4, u2, u3} R₂ _inst_2 R₃ _inst_3 M₂ _inst_6 _inst_7 M₃ _inst_8 _inst_9 _inst_13 _inst_14 σ₂₃ _inst_16) g) f) (Neg.neg.{max u1 u3} (ContinuousLinearMap.{u6, u4, u1, u3} R R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.neg.{u6, u4, u1, u3} R _inst_1 R₃ _inst_3 M _inst_4 _inst_5 M₃ _inst_8 _inst_9 _inst_12 _inst_14 σ₁₃ _inst_16) (ContinuousLinearMap.comp.{u6, u5, u4, u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.neg_comp ContinuousLinearMap.neg_compₓ'. -/ @[simp] theorem neg_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : (-g).comp f = -g.comp f := @@ -1482,6 +2391,12 @@ theorem neg_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddG simp #align continuous_linear_map.neg_comp ContinuousLinearMap.neg_comp +/- warning: continuous_linear_map.comp_sub -> ContinuousLinearMap.comp_sub is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {R₃ : Type.{u3}} [_inst_3 : Ring.{u3} R₃] {M : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M] [_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u5}} [_inst_6 : TopologicalSpace.{u5} M₂] [_inst_7 : AddCommGroup.{u5} M₂] {M₃ : Type.{u6}} [_inst_8 : TopologicalSpace.{u6} M₃] [_inst_9 : AddCommGroup.{u6} M₃] [_inst_12 : Module.{u1, u4} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u5} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7)] [_inst_14 : Module.{u3, u6} R₃ M₃ (Ring.toSemiring.{u3} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} {σ₁₃ : RingHom.{u1, u3} R R₃ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u5} M₂ _inst_6 (AddCommGroup.toAddGroup.{u5} M₂ _inst_7)] [_inst_17 : TopologicalAddGroup.{u6} M₃ _inst_8 (AddCommGroup.toAddGroup.{u6} M₃ _inst_9)] (g : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (f₁ : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13) (f₂ : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u4) (succ u6)} 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_inst_14 σ₁₃ _inst_17)) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f₁) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f₂)) +but is expected to have type + forall {R : Type.{u6}} [_inst_1 : Ring.{u6} R] {R₂ : Type.{u5}} [_inst_2 : Ring.{u5} R₂] {R₃ : Type.{u4}} [_inst_3 : Ring.{u4} R₃] {M : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M] [_inst_5 : AddCommGroup.{u1} M] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] {M₃ : Type.{u2}} [_inst_8 : TopologicalSpace.{u2} M₃] [_inst_9 : AddCommGroup.{u2} M₃] [_inst_12 : Module.{u6, u1} R M (Ring.toSemiring.{u6} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5)] [_inst_13 : Module.{u5, u3} R₂ M₂ (Ring.toSemiring.{u5} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_14 : Module.{u4, u2} R₃ M₃ (Ring.toSemiring.{u4} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9)] {σ₁₂ : RingHom.{u6, u5} R R₂ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2))} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} {σ₁₃ : RingHom.{u6, u4} R R₃ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u6, u5, u4} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u3} M₂ _inst_6 (AddCommGroup.toAddGroup.{u3} M₂ _inst_7)] [_inst_17 : TopologicalAddGroup.{u2} M₃ _inst_8 (AddCommGroup.toAddGroup.{u2} M₃ _inst_9)] (g : ContinuousLinearMap.{u5, u4, u3, u2} R₂ R₃ (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_13 _inst_14) (f₁ : ContinuousLinearMap.{u6, u5, u1, u3} R R₂ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_13) (f₂ : ContinuousLinearMap.{u6, u5, u1, u3} R R₂ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u6, u4, u1, u2} R R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_14) (ContinuousLinearMap.comp.{u6, u5, u4, u1, u3, u2} R R₂ R₃ (Ring.toSemiring.{u6} R _inst_1) (Ring.toSemiring.{u5} R₂ _inst_2) (Ring.toSemiring.{u4} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g (HSub.hSub.{max u1 u3, max u1 u3, max u1 u3} (ContinuousLinearMap.{u6, 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_inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u2} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g f₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_sub ContinuousLinearMap.comp_subₓ'. -/ @[simp] theorem comp_sub [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₂] [TopologicalAddGroup M₃] (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M →SL[σ₁₂] M₂) : g.comp (f₁ - f₂) = g.comp f₁ - g.comp f₂ := @@ -1490,6 +2405,12 @@ theorem comp_sub [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddG simp #align continuous_linear_map.comp_sub ContinuousLinearMap.comp_sub +/- warning: continuous_linear_map.sub_comp -> ContinuousLinearMap.sub_comp is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {R₃ : Type.{u3}} [_inst_3 : Ring.{u3} R₃] {M : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M] [_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u5}} [_inst_6 : TopologicalSpace.{u5} M₂] [_inst_7 : AddCommGroup.{u5} M₂] {M₃ : Type.{u6}} [_inst_8 : TopologicalSpace.{u6} M₃] [_inst_9 : AddCommGroup.{u6} M₃] [_inst_12 : Module.{u1, u4} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u5} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7)] [_inst_14 : Module.{u3, u6} R₃ M₃ (Ring.toSemiring.{u3} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} {σ₁₃ : RingHom.{u1, u3} R R₃ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₃ (Ring.toNonAssocRing.{u3} R₃ _inst_3))} [_inst_15 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃] [_inst_16 : TopologicalAddGroup.{u6} M₃ _inst_8 (AddCommGroup.toAddGroup.{u6} M₃ _inst_9)] (g₁ : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (g₂ : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₂₃ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_13 _inst_14) (f : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) _inst_12 _inst_13), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ (Ring.toSemiring.{u1} R 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(ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g₁ f) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u3} R₃ _inst_3) σ₁₂ σ₂₃ σ₁₃ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M₂ _inst_7) M₃ _inst_8 (AddCommGroup.toAddCommMonoid.{u6} M₃ _inst_9) _inst_12 _inst_13 _inst_14 _inst_15 g₂ f)) +but is expected to have type + forall {R : Type.{u6}} [_inst_1 : Ring.{u6} R] {R₂ : Type.{u5}} [_inst_2 : Ring.{u5} R₂] {R₃ : Type.{u4}} [_inst_3 : Ring.{u4} R₃] {M : Type.{u1}} [_inst_4 : TopologicalSpace.{u1} M] [_inst_5 : AddCommGroup.{u1} M] {M₂ : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M₂] [_inst_7 : AddCommGroup.{u2} M₂] {M₃ : Type.{u3}} [_inst_8 : TopologicalSpace.{u3} M₃] [_inst_9 : AddCommGroup.{u3} M₃] [_inst_12 : Module.{u6, u1} R M (Ring.toSemiring.{u6} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5)] [_inst_13 : Module.{u5, u2} R₂ M₂ (Ring.toSemiring.{u5} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_7)] [_inst_14 : Module.{u4, u3} R₃ M₃ (Ring.toSemiring.{u4} R₃ _inst_3) (AddCommGroup.toAddCommMonoid.{u3} M₃ _inst_9)] {σ₁₂ : RingHom.{u6, u5} R R₂ (NonAssocRing.toNonAssocSemiring.{u6} R (Ring.toNonAssocRing.{u6} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2))} {σ₂₃ : RingHom.{u5, u4} R₂ R₃ (NonAssocRing.toNonAssocSemiring.{u5} R₂ (Ring.toNonAssocRing.{u5} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u4} R₃ (Ring.toNonAssocRing.{u4} R₃ _inst_3))} {σ₁₃ : RingHom.{u6, u4} R R₃ (NonAssocRing.toNonAssocSemiring.{u6} R 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inaccurate. Consider using '#align continuous_linear_map.sub_comp ContinuousLinearMap.sub_compₓ'. -/ @[simp] theorem sub_comp [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [TopologicalAddGroup M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : (g₁ - g₂).comp f = g₁.comp f - g₂.comp f := @@ -1504,6 +2425,12 @@ instance [TopologicalAddGroup M] : Ring (M →L[R] M) := mul := (· * ·) one := 1 } +/- warning: continuous_linear_map.smul_right_one_pow -> ContinuousLinearMap.smulRight_one_pow is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_15 : TopologicalSpace.{u1} R] [_inst_16 : TopologicalRing.{u1} R _inst_15 (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))] (c : R) (n : Nat), Eq.{succ u1} (ContinuousLinearMap.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R _inst_15 (AddCommGroup.toAddCommMonoid.{u1} 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Consider using '#align continuous_linear_map.smul_right_one_pow ContinuousLinearMap.smulRight_one_powₓ'. -/ theorem smulRight_one_pow [TopologicalSpace R] [TopologicalRing R] (c : R) (n : ℕ) : smulRight (1 : R →L[R] R) c ^ n = smulRight (1 : R →L[R] R) (c ^ n) := by @@ -1517,6 +2444,12 @@ section variable {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] +/- warning: continuous_linear_map.proj_ker_of_right_inverse -> ContinuousLinearMap.projKerOfRightInverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_15 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u3} M _inst_4 (AddCommGroup.toAddGroup.{u3} M _inst_5)] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u1} R _inst_1) σ₂₁ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ 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(Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13)) f₁))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_15 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u3} M _inst_4 (AddCommGroup.toAddGroup.{u3} M _inst_5)] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u1} R _inst_1) σ₂₁ M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) M _inst_4 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Consider using '#align continuous_linear_map.proj_ker_of_right_inverse ContinuousLinearMap.projKerOfRightInverseₓ'. -/ /-- Given a right inverse `f₂ : M₂ →L[R] M` to `f₁ : M →L[R] M₂`, `proj_ker_of_right_inverse f₁ f₂ h` is the projection `M →L[R] f₁.ker` along `f₂.range`. -/ def projKerOfRightInverse [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) @@ -1524,6 +2457,12 @@ def projKerOfRightInverse [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂ (id R M - f₂.comp f₁).codRestrict (LinearMap.ker f₁) fun x => by simp [h (f₁ x)] #align continuous_linear_map.proj_ker_of_right_inverse ContinuousLinearMap.projKerOfRightInverse +/- warning: continuous_linear_map.coe_proj_ker_of_right_inverse_apply -> ContinuousLinearMap.coe_projKerOfRightInverse_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_15 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u3} M _inst_4 (AddCommGroup.toAddGroup.{u3} M _inst_5)] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ 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[_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] [_inst_12 : Module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u1} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] {σ₁₂ : RingHom.{u3, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u3} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))} [_inst_15 : RingHomInvPair.{u3, u2} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u4} M _inst_4 (AddCommGroup.toAddGroup.{u4} M _inst_5)] (f₁ : ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R 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σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_12 (Submodule.module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12 (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) M (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) 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M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) _inst_4 (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_4) (ContinuousSemilinearMapClass.toContinuousMapClass.{u4, u3, u3, u4, u4} (ContinuousLinearMap.{u3, u3, u4, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 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_inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_12 (Submodule.module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12 (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 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Consider using '#align continuous_linear_map.coe_proj_ker_of_right_inverse_apply ContinuousLinearMap.coe_projKerOfRightInverse_applyₓ'. -/ @[simp] theorem coe_projKerOfRightInverse_apply [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : M) : @@ -1531,6 +2470,12 @@ theorem coe_projKerOfRightInverse_apply [TopologicalAddGroup M] (f₁ : M →SL[ rfl #align continuous_linear_map.coe_proj_ker_of_right_inverse_apply ContinuousLinearMap.coe_projKerOfRightInverse_apply +/- warning: continuous_linear_map.proj_ker_of_right_inverse_apply_idem -> ContinuousLinearMap.projKerOfRightInverse_apply_idem is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_15 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u3} M _inst_4 (AddCommGroup.toAddGroup.{u3} M _inst_5)] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} 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M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_12 _inst_13)) f₁)))))) x)) x +but is expected to have type + forall {R : Type.{u3}} [_inst_1 : Ring.{u3} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u4}} [_inst_4 : TopologicalSpace.{u4} M] [_inst_5 : AddCommGroup.{u4} M] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] [_inst_12 : Module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5)] [_inst_13 : Module.{u2, u1} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] {σ₁₂ : RingHom.{u3, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1)) 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_inst_12 _inst_13)) f₁))) _x) (ContinuousMapClass.toFunLike.{u4, u4, u4} (ContinuousLinearMap.{u3, u3, u4, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) 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(AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 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(AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) 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M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))))) (ContinuousLinearMap.projKerOfRightInverse.{u3, u2, u4, u1} R _inst_1 R₂ _inst_2 M _inst_4 _inst_5 M₂ _inst_6 _inst_7 _inst_12 _inst_13 σ₁₂ σ₂₁ _inst_15 _inst_16 f₁ f₂ h) (Subtype.val.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Set.{u4} M) (Set.instMembershipSet.{u4} M) x (SetLike.coe.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R 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_inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) x)) x +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.proj_ker_of_right_inverse_apply_idem ContinuousLinearMap.projKerOfRightInverse_apply_idemₓ'. -/ @[simp] theorem projKerOfRightInverse_apply_idem [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (x : LinearMap.ker f₁) : @@ -1538,6 +2483,12 @@ theorem projKerOfRightInverse_apply_idem [TopologicalAddGroup M] (f₁ : M →SL Subtype.ext_iff_val.2 <| by simp #align continuous_linear_map.proj_ker_of_right_inverse_apply_idem ContinuousLinearMap.projKerOfRightInverse_apply_idem +/- warning: continuous_linear_map.proj_ker_of_right_inverse_comp_inv -> ContinuousLinearMap.projKerOfRightInverse_comp_inv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M] [_inst_5 : AddCommGroup.{u3} M] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_12 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_5)] [_inst_13 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_15 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_16 : TopologicalAddGroup.{u3} M _inst_4 (AddCommGroup.toAddGroup.{u3} M _inst_5)] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M _inst_5) M₂ _inst_6 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(AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 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(ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) _x) (ContinuousMapClass.toFunLike.{u4, u4, u4} (ContinuousLinearMap.{u3, u3, u4, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_4) 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_inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_12 (Submodule.module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12 (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) M (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R 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_inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_4) (ContinuousSemilinearMapClass.toContinuousMapClass.{u4, u3, u3, u4, u4} (ContinuousLinearMap.{u3, u3, u4, u4} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R 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_inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) (instTopologicalSpaceSubtype.{u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_4) (Submodule.addCommMonoid.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12 (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)) _inst_12 (Submodule.module.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12 (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 σ₁₂ (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u4 u1, u3, u2, u4, u1} (ContinuousLinearMap.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13) R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁))) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) (Subtype.{succ u4} M (fun (x : M) => Membership.mem.{u4, u4} M (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) (SetLike.instMembership.{u4, u4} (Submodule.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12) M (Submodule.setLike.{u3, u4} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) _inst_12)) x (LinearMap.ker.{u3, u2, u4, u1, max u4 u1} R R₂ M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) 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R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u2, u4, u1} R R₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ M _inst_4 (AddCommGroup.toAddCommMonoid.{u4} M _inst_5) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_12 _inst_13)) f₁)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.proj_ker_of_right_inverse_comp_inv ContinuousLinearMap.projKerOfRightInverse_comp_invₓ'. -/ @[simp] theorem projKerOfRightInverse_comp_inv [TopologicalAddGroup M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : Function.RightInverse f₂ f₁) (y : M₂) : @@ -1553,6 +2504,12 @@ section DivisionMonoid variable {R M : Type _} +/- warning: continuous_linear_map.is_open_map_of_ne_zero -> ContinuousLinearMap.isOpenMap_of_ne_zero is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} R] [_inst_2 : DivisionRing.{u1} R] [_inst_3 : ContinuousSub.{u1} R _inst_1 (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (DivisionRing.toRing.{u1} R _inst_2))))))] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : TopologicalSpace.{u2} M] [_inst_6 : ContinuousAdd.{u2} M _inst_5 (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4)))))] [_inst_7 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_8 : ContinuousSMul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_7)))) _inst_1 _inst_5] (f : ContinuousLinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) R _inst_1 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (DivisionRing.toRing.{u1} R _inst_2))))) _inst_7 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(Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) R _inst_1 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (DivisionRing.toRing.{u1} R _inst_2))))) _inst_7 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))))))) -> (IsOpenMap.{u2, u1} M R _inst_5 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (ContinuousLinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) R _inst_1 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R 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(Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) R _inst_1 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (DivisionRing.toRing.{u1} R _inst_2))))) _inst_7 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (DivisionRing.toRing.{u1} R _inst_2)))) f)) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} R] [_inst_2 : DivisionRing.{u2} R] [_inst_3 : ContinuousSub.{u2} R _inst_1 (Ring.toSub.{u2} R (DivisionRing.toRing.{u2} R _inst_2))] [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : TopologicalSpace.{u1} M] [_inst_6 : ContinuousAdd.{u1} M _inst_5 (AddZeroClass.toAdd.{u1} M (AddMonoid.toAddZeroClass.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_4)))))] [_inst_7 : Module.{u2, u1} R M (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_8 : ContinuousSMul.{u2, u1} R M (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_7)))) _inst_1 _inst_5] (f : ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))), (Ne.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) f (OfNat.ofNat.{max u2 u1} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) 0 (Zero.toOfNat0.{max u2 u1} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) (ContinuousLinearMap.zero.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2))))))) -> (IsOpenMap.{u1, u2} M R _inst_5 _inst_1 (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => R) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) M R _inst_5 _inst_1 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u2, u2, u1, u2} (ContinuousLinearMap.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))) R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2))) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u1, u2} R R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2)))) M _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (DivisionRing.toRing.{u2} R _inst_2))))) _inst_7 (Semiring.toModule.{u2} R (DivisionSemiring.toSemiring.{u2} R (DivisionRing.toDivisionSemiring.{u2} R _inst_2)))))) f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.is_open_map_of_ne_zero ContinuousLinearMap.isOpenMap_of_ne_zeroₓ'. -/ /-- A nonzero continuous linear functional is open. -/ protected theorem isOpenMap_of_ne_zero [TopologicalSpace R] [DivisionRing R] [ContinuousSub R] [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] @@ -1579,6 +2536,12 @@ variable {R R₂ R₃ S S₃ : Type _} [Semiring R] [Semiring R₂] [Semiring R include σ₁₃ +/- warning: continuous_linear_map.smul_comp -> ContinuousLinearMap.smul_comp is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} {S₃ : Type.{u4}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] [_inst_5 : Monoid.{u4} S₃] {M : Type.{u5}} [_inst_6 : TopologicalSpace.{u5} M] [_inst_7 : AddCommMonoid.{u5} M] [_inst_8 : Module.{u1, u5} R M _inst_1 _inst_7] {M₂ : Type.{u6}} [_inst_9 : TopologicalSpace.{u6} M₂] [_inst_10 : AddCommMonoid.{u6} M₂] [_inst_11 : Module.{u2, u6} R₂ M₂ _inst_2 _inst_10] {M₃ : Type.{u7}} [_inst_12 : TopologicalSpace.{u7} M₃] [_inst_13 : AddCommMonoid.{u7} M₃] [_inst_14 : Module.{u3, u7} R₃ M₃ _inst_3 _inst_13] [_inst_21 : DistribMulAction.{u4, u7} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13)] [_inst_22 : SMulCommClass.{u3, u4, u7} R₃ S₃ M₃ (SMulZeroClass.toHasSmul.{u3, u7} R₃ M₃ (AddZeroClass.toHasZero.{u7} M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u7} R₃ M₃ (MulZeroClass.toHasZero.{u3} R₃ (MulZeroOneClass.toMulZeroClass.{u3} R₃ (MonoidWithZero.toMulZeroOneClass.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)))) (AddZeroClass.toHasZero.{u7} M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u7} R₃ M₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3) (AddZeroClass.toHasZero.{u7} M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u7} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toHasSmul.{u4, u7} S₃ M₃ (AddZeroClass.toHasZero.{u7} M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13))) (DistribSMul.toSmulZeroClass.{u4, u7} S₃ M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13)) (DistribMulAction.toDistribSMul.{u4, u7} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13) _inst_21)))] [_inst_23 : ContinuousConstSMul.{u4, u7} S₃ M₃ _inst_12 (SMulZeroClass.toHasSmul.{u4, u7} S₃ M₃ (AddZeroClass.toHasZero.{u7} M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13))) (DistribSMul.toSmulZeroClass.{u4, u7} S₃ M₃ (AddMonoid.toAddZeroClass.{u7} M₃ (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13)) (DistribMulAction.toDistribSMul.{u4, u7} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u7} M₃ _inst_13) _inst_21)))] {σ₁₂ : RingHom.{u1, u2} R R₂ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R R₃ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} [_inst_27 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (c : S₃) (h : ContinuousLinearMap.{u2, u3, u6, u7} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (f : ContinuousLinearMap.{u1, u2, u5, u6} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11), Eq.{max (succ u5) (succ u7)} (ContinuousLinearMap.{u1, u3, u5, u7} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.comp.{u1, u2, u3, u5, u6, u7} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 (SMul.smul.{u4, max u6 u7} S₃ (ContinuousLinearMap.{u2, u3, u6, u7} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (MulAction.toHasSmul.{u4, max u6 u7} S₃ (ContinuousLinearMap.{u2, u3, u6, u7} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) _inst_5 (ContinuousLinearMap.mulAction.{u2, u3, u6, u7, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14 S₃ _inst_5 _inst_21 _inst_22 _inst_23)) c h) f) (SMul.smul.{u4, max u5 u7} S₃ (ContinuousLinearMap.{u1, u3, u5, u7} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MulAction.toHasSmul.{u4, max u5 u7} S₃ (ContinuousLinearMap.{u1, u3, u5, u7} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) _inst_5 (ContinuousLinearMap.mulAction.{u1, u3, u5, u7, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 S₃ _inst_5 _inst_21 _inst_22 _inst_23)) c (ContinuousLinearMap.comp.{u1, u2, u3, u5, u6, u7} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h f)) +but is expected to have type + forall {R : Type.{u3}} {R₂ : Type.{u7}} {R₃ : Type.{u6}} {S₃ : Type.{u1}} [_inst_1 : Semiring.{u3} R] [_inst_2 : Semiring.{u7} R₂] [_inst_3 : Semiring.{u6} R₃] [_inst_5 : Monoid.{u1} S₃] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u3, u2} R M _inst_1 _inst_7] {M₂ : Type.{u5}} [_inst_9 : TopologicalSpace.{u5} M₂] [_inst_10 : AddCommMonoid.{u5} M₂] [_inst_11 : Module.{u7, u5} R₂ M₂ _inst_2 _inst_10] {M₃ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₃] [_inst_13 : AddCommMonoid.{u4} M₃] [_inst_14 : Module.{u6, u4} R₃ M₃ _inst_3 _inst_13] [_inst_21 : DistribMulAction.{u1, u4} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)] [_inst_22 : SMulCommClass.{u6, u1, u4} R₃ S₃ M₃ (SMulZeroClass.toSMul.{u6, u4} R₃ M₃ (AddMonoid.toZero.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u6, u4} R₃ M₃ (MonoidWithZero.toZero.{u6} R₃ (Semiring.toMonoidWithZero.{u6} R₃ _inst_3)) (AddMonoid.toZero.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u6, u4} R₃ M₃ (Semiring.toMonoidWithZero.{u6} R₃ _inst_3) (AddMonoid.toZero.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (Module.toMulActionWithZero.{u6, u4} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toSMul.{u1, u4} S₃ M₃ (AddMonoid.toZero.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (DistribSMul.toSMulZeroClass.{u1, u4} S₃ M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (DistribMulAction.toDistribSMul.{u1, u4} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13) _inst_21)))] [_inst_23 : ContinuousConstSMul.{u1, u4} S₃ M₃ _inst_12 (SMulZeroClass.toSMul.{u1, u4} S₃ M₃ (AddMonoid.toZero.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (DistribSMul.toSMulZeroClass.{u1, u4} S₃ M₃ (AddMonoid.toAddZeroClass.{u4} M₃ (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13)) (DistribMulAction.toDistribSMul.{u1, u4} S₃ M₃ _inst_5 (AddCommMonoid.toAddMonoid.{u4} M₃ _inst_13) _inst_21)))] {σ₁₂ : RingHom.{u3, u7} R R₂ (Semiring.toNonAssocSemiring.{u3} R _inst_1) (Semiring.toNonAssocSemiring.{u7} R₂ _inst_2)} {σ₂₃ : RingHom.{u7, u6} R₂ R₃ (Semiring.toNonAssocSemiring.{u7} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u6} R₃ _inst_3)} {σ₁₃ : RingHom.{u3, u6} R R₃ (Semiring.toNonAssocSemiring.{u3} R _inst_1) (Semiring.toNonAssocSemiring.{u6} R₃ _inst_3)} [_inst_27 : RingHomCompTriple.{u3, u7, u6} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] (c : S₃) (h : ContinuousLinearMap.{u7, u6, u5, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (f : ContinuousLinearMap.{u3, u7, u2, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11), Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u3, u6, u2, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.comp.{u3, u7, u6, u2, u5, u4} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 (HSMul.hSMul.{u1, max u4 u5, max u4 u5} S₃ (ContinuousLinearMap.{u7, u6, u5, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (ContinuousLinearMap.{u7, u6, u5, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (instHSMul.{u1, max u4 u5} S₃ (ContinuousLinearMap.{u7, u6, u5, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (MulAction.toSMul.{u1, max u4 u5} S₃ (ContinuousLinearMap.{u7, u6, u5, u4} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) _inst_5 (ContinuousLinearMap.mulAction.{u7, u6, u5, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14 S₃ _inst_5 _inst_21 _inst_22 _inst_23))) c h) f) (HSMul.hSMul.{u1, max u4 u2, max u4 u2} S₃ (ContinuousLinearMap.{u3, u6, u2, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.{u3, u6, u2, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (instHSMul.{u1, max u4 u2} S₃ (ContinuousLinearMap.{u3, u6, u2, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MulAction.toSMul.{u1, max u4 u2} S₃ (ContinuousLinearMap.{u3, u6, u2, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) _inst_5 (ContinuousLinearMap.mulAction.{u3, u6, u2, u4, u1} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 S₃ _inst_5 _inst_21 _inst_22 _inst_23))) c (ContinuousLinearMap.comp.{u3, u7, u6, u2, u5, u4} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_comp ContinuousLinearMap.smul_compₓ'. -/ @[simp] theorem smul_comp (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) : (c • h).comp f = c • h.comp f := @@ -1591,6 +2554,12 @@ variable [DistribMulAction S₃ M₂] [ContinuousConstSMul S₃ M₂] [SMulCommC variable [DistribMulAction S N₂] [ContinuousConstSMul S N₂] [SMulCommClass R S N₂] +/- warning: continuous_linear_map.comp_smul -> ContinuousLinearMap.comp_smul is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_4 : Monoid.{u2} S] {M : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : AddCommMonoid.{u3} M] [_inst_8 : Module.{u1, u3} R M _inst_1 _inst_7] {N₂ : Type.{u4}} [_inst_15 : TopologicalSpace.{u4} N₂] [_inst_16 : AddCommMonoid.{u4} N₂] [_inst_17 : Module.{u1, u4} R N₂ _inst_1 _inst_16] {N₃ : Type.{u5}} [_inst_18 : TopologicalSpace.{u5} N₃] [_inst_19 : AddCommMonoid.{u5} N₃] [_inst_20 : Module.{u1, u5} R N₃ _inst_1 _inst_19] [_inst_24 : DistribMulAction.{u2, u5} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)] [_inst_25 : SMulCommClass.{u1, u2, u5} R S N₃ (SMulZeroClass.toHasSmul.{u1, u5} R N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u1, u5} R N₃ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u1, u5} R N₃ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (Module.toMulActionWithZero.{u1, u5} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toHasSmul.{u2, u5} S N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (DistribSMul.toSmulZeroClass.{u2, u5} S N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u2, u5} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) _inst_24)))] [_inst_26 : ContinuousConstSMul.{u2, u5} S N₃ _inst_18 (SMulZeroClass.toHasSmul.{u2, u5} S N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (DistribSMul.toSmulZeroClass.{u2, u5} S N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u2, u5} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) _inst_24)))] [_inst_31 : DistribMulAction.{u2, u4} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)] [_inst_32 : ContinuousConstSMul.{u2, u4} S N₂ _inst_15 (SMulZeroClass.toHasSmul.{u2, u4} S N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (DistribSMul.toSmulZeroClass.{u2, u4} S N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u2, u4} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) _inst_31)))] [_inst_33 : SMulCommClass.{u1, u2, u4} R S N₂ (SMulZeroClass.toHasSmul.{u1, u4} R N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (SMulWithZero.toSmulZeroClass.{u1, u4} R N₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (MulActionWithZero.toSMulWithZero.{u1, u4} R N₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (Module.toMulActionWithZero.{u1, u4} R N₂ _inst_1 _inst_16 _inst_17)))) (SMulZeroClass.toHasSmul.{u2, u4} S N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (DistribSMul.toSmulZeroClass.{u2, u4} S N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u2, u4} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) _inst_31)))] [_inst_34 : LinearMap.CompatibleSMul.{u4, u5, u2, u1} N₂ N₃ _inst_16 _inst_19 S R _inst_1 (SMulZeroClass.toHasSmul.{u2, u4} S N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (DistribSMul.toSmulZeroClass.{u2, u4} S N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u2, u4} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) _inst_31))) _inst_17 (SMulZeroClass.toHasSmul.{u2, u5} S N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (DistribSMul.toSmulZeroClass.{u2, u5} S N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u2, u5} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) _inst_24))) _inst_20] (hₗ : ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_17 _inst_20) (c : S) (fₗ : ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17), Eq.{max (succ u3) (succ u5)} (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.comp.{u1, u1, u1, u3, u4, u5} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_8 _inst_17 _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) hₗ (SMul.smul.{u2, max u3 u4} S (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (MulAction.toHasSmul.{u2, max u3 u4} S (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) _inst_4 (ContinuousLinearMap.mulAction.{u1, u1, u3, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 S _inst_4 _inst_31 _inst_33 _inst_32)) c fₗ)) (SMul.smul.{u2, max u3 u5} S (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (MulAction.toHasSmul.{u2, max u3 u5} S (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) _inst_4 (ContinuousLinearMap.mulAction.{u1, u1, u3, u5, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 S _inst_4 _inst_24 _inst_25 _inst_26)) c (ContinuousLinearMap.comp.{u1, u1, u1, u3, u4, u5} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_8 _inst_17 _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) hₗ fₗ)) +but is expected to have type + forall {R : Type.{u2}} {S : Type.{u3}} [_inst_1 : Semiring.{u2} R] [_inst_4 : Monoid.{u3} S] {M : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M] [_inst_7 : AddCommMonoid.{u1} M] [_inst_8 : Module.{u2, u1} R M _inst_1 _inst_7] {N₂ : Type.{u5}} [_inst_15 : TopologicalSpace.{u5} N₂] [_inst_16 : AddCommMonoid.{u5} N₂] [_inst_17 : Module.{u2, u5} R N₂ _inst_1 _inst_16] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u2, u4} R N₃ _inst_1 _inst_19] [_inst_24 : DistribMulAction.{u3, u4} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)] [_inst_25 : SMulCommClass.{u2, u3, u4} R S N₃ (SMulZeroClass.toSMul.{u2, u4} R N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u2, u4} R N₃ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u2, u4} R N₃ (Semiring.toMonoidWithZero.{u2} R _inst_1) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (Module.toMulActionWithZero.{u2, u4} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toSMul.{u3, u4} S N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribSMul.toSMulZeroClass.{u3, u4} S N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u3, u4} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19) _inst_24)))] [_inst_26 : ContinuousConstSMul.{u3, u4} S N₃ _inst_18 (SMulZeroClass.toSMul.{u3, u4} S N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribSMul.toSMulZeroClass.{u3, u4} S N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u3, u4} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19) _inst_24)))] [_inst_31 : DistribMulAction.{u3, u5} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)] [_inst_32 : ContinuousConstSMul.{u3, u5} S N₂ _inst_15 (SMulZeroClass.toSMul.{u3, u5} S N₂ (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribSMul.toSMulZeroClass.{u3, u5} S N₂ (AddMonoid.toAddZeroClass.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u3, u5} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16) _inst_31)))] [_inst_33 : SMulCommClass.{u2, u3, u5} R S N₂ (SMulZeroClass.toSMul.{u2, u5} R N₂ (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (SMulWithZero.toSMulZeroClass.{u2, u5} R N₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (MulActionWithZero.toSMulWithZero.{u2, u5} R N₂ (Semiring.toMonoidWithZero.{u2} R _inst_1) (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (Module.toMulActionWithZero.{u2, u5} R N₂ _inst_1 _inst_16 _inst_17)))) (SMulZeroClass.toSMul.{u3, u5} S N₂ (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribSMul.toSMulZeroClass.{u3, u5} S N₂ (AddMonoid.toAddZeroClass.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u3, u5} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16) _inst_31)))] [_inst_34 : LinearMap.CompatibleSMul.{u5, u4, u3, u2} N₂ N₃ _inst_16 _inst_19 S R _inst_1 (SMulZeroClass.toSMul.{u3, u5} S N₂ (AddMonoid.toZero.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribSMul.toSMulZeroClass.{u3, u5} S N₂ (AddMonoid.toAddZeroClass.{u5} N₂ (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16)) (DistribMulAction.toDistribSMul.{u3, u5} S N₂ _inst_4 (AddCommMonoid.toAddMonoid.{u5} N₂ _inst_16) _inst_31))) _inst_17 (SMulZeroClass.toSMul.{u3, u4} S N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribSMul.toSMulZeroClass.{u3, u4} S N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (DistribMulAction.toDistribSMul.{u3, u4} S N₃ _inst_4 (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19) _inst_24))) _inst_20] (hₗ : ContinuousLinearMap.{u2, u2, u5, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_17 _inst_20) (c : S) (fₗ : ContinuousLinearMap.{u2, u2, u1, u5} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17), Eq.{max (succ u1) (succ u4)} (ContinuousLinearMap.{u2, u2, u1, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.comp.{u2, u2, u2, u1, u5, u4} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_8 _inst_17 _inst_20 (RingHomCompTriple.ids.{u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) hₗ (HSMul.hSMul.{u3, max u5 u1, max u5 u1} S (ContinuousLinearMap.{u2, u2, u1, u5} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u2, u2, u1, u5} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (instHSMul.{u3, max u5 u1} S (ContinuousLinearMap.{u2, u2, u1, u5} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (MulAction.toSMul.{u3, max u5 u1} S (ContinuousLinearMap.{u2, u2, u1, u5} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) _inst_4 (ContinuousLinearMap.mulAction.{u2, u2, u1, u5, u3} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 S _inst_4 _inst_31 _inst_33 _inst_32))) c fₗ)) (HSMul.hSMul.{u3, max u4 u1, max u4 u1} S (ContinuousLinearMap.{u2, u2, u1, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.{u2, u2, u1, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (instHSMul.{u3, max u4 u1} S (ContinuousLinearMap.{u2, u2, u1, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (MulAction.toSMul.{u3, max u4 u1} S (ContinuousLinearMap.{u2, u2, u1, u4} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) _inst_4 (ContinuousLinearMap.mulAction.{u2, u2, u1, u4, u3} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 S _inst_4 _inst_24 _inst_25 _inst_26))) c (ContinuousLinearMap.comp.{u2, u2, u2, u1, u5, u4} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_8 _inst_17 _inst_20 (RingHomCompTriple.ids.{u2, u2} R R _inst_1 _inst_1 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) hₗ fₗ)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_smul ContinuousLinearMap.comp_smulₓ'. -/ @[simp] theorem comp_smul [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂) : hₗ.comp (c • fₗ) = c • hₗ.comp fₗ := @@ -1601,6 +2570,12 @@ theorem comp_smul [LinearMap.CompatibleSMul N₂ N₃ S R] (hₗ : N₂ →L[R] include σ₁₃ +/- warning: continuous_linear_map.comp_smulₛₗ -> ContinuousLinearMap.comp_smulₛₗ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {M : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M] [_inst_7 : AddCommMonoid.{u4} M] [_inst_8 : Module.{u1, u4} R M _inst_1 _inst_7] {M₂ : Type.{u5}} [_inst_9 : TopologicalSpace.{u5} M₂] [_inst_10 : AddCommMonoid.{u5} M₂] [_inst_11 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_10] {M₃ : Type.{u6}} [_inst_12 : TopologicalSpace.{u6} M₃] [_inst_13 : AddCommMonoid.{u6} M₃] [_inst_14 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_13] {σ₁₂ : RingHom.{u1, u2} R R₂ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₁₃ : RingHom.{u1, u3} R R₃ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} [_inst_27 : RingHomCompTriple.{u1, u2, u3} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_34 : SMulCommClass.{u2, u2, u5} R₂ R₂ M₂ (SMulZeroClass.toHasSmul.{u2, u5} R₂ M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (SMulWithZero.toSmulZeroClass.{u2, u5} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (MulActionWithZero.toSMulWithZero.{u2, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (Module.toMulActionWithZero.{u2, u5} R₂ M₂ _inst_2 _inst_10 _inst_11)))) (SMulZeroClass.toHasSmul.{u2, u5} R₂ M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (SMulWithZero.toSmulZeroClass.{u2, u5} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (MulActionWithZero.toSMulWithZero.{u2, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (Module.toMulActionWithZero.{u2, u5} R₂ M₂ _inst_2 _inst_10 _inst_11))))] [_inst_35 : SMulCommClass.{u3, u3, u6} R₃ R₃ M₃ (SMulZeroClass.toHasSmul.{u3, u6} R₃ M₃ (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u6} R₃ M₃ (MulZeroClass.toHasZero.{u3} R₃ (MulZeroOneClass.toMulZeroClass.{u3} R₃ (MonoidWithZero.toMulZeroOneClass.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)))) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u6} R₃ M₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u6} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toHasSmul.{u3, u6} R₃ M₃ (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u6} R₃ M₃ (MulZeroClass.toHasZero.{u3} R₃ (MulZeroOneClass.toMulZeroClass.{u3} R₃ (MonoidWithZero.toMulZeroOneClass.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)))) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u6} R₃ M₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u6} R₃ M₃ _inst_3 _inst_13 _inst_14))))] [_inst_36 : ContinuousConstSMul.{u2, u5} R₂ M₂ _inst_9 (SMulZeroClass.toHasSmul.{u2, u5} R₂ M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (SMulWithZero.toSmulZeroClass.{u2, u5} R₂ M₂ (MulZeroClass.toHasZero.{u2} R₂ (MulZeroOneClass.toMulZeroClass.{u2} R₂ (MonoidWithZero.toMulZeroOneClass.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (MulActionWithZero.toSMulWithZero.{u2, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10))) (Module.toMulActionWithZero.{u2, u5} R₂ M₂ _inst_2 _inst_10 _inst_11))))] [_inst_37 : ContinuousConstSMul.{u3, u6} R₃ M₃ _inst_12 (SMulZeroClass.toHasSmul.{u3, u6} R₃ M₃ (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u6} R₃ M₃ (MulZeroClass.toHasZero.{u3} R₃ (MulZeroOneClass.toMulZeroClass.{u3} R₃ (MonoidWithZero.toMulZeroOneClass.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)))) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u6} R₃ M₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3) (AddZeroClass.toHasZero.{u6} M₃ (AddMonoid.toAddZeroClass.{u6} M₃ (AddCommMonoid.toAddMonoid.{u6} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u6} R₃ M₃ _inst_3 _inst_13 _inst_14))))] (h : ContinuousLinearMap.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (c : R₂) (f : ContinuousLinearMap.{u1, u2, u4, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h (SMul.smul.{u2, max u4 u5} R₂ (ContinuousLinearMap.{u1, u2, u4, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (MulAction.toHasSmul.{u2, max u4 u5} R₂ (ContinuousLinearMap.{u1, u2, u4, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (MonoidWithZero.toMonoid.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)) (ContinuousLinearMap.mulAction.{u1, u2, u4, u5, u2} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11 R₂ (MonoidWithZero.toMonoid.{u2} R₂ (Semiring.toMonoidWithZero.{u2} R₂ _inst_2)) (Module.toDistribMulAction.{u2, u5} R₂ M₂ _inst_2 _inst_10 _inst_11) _inst_34 _inst_36)) c f)) (SMul.smul.{u3, max u4 u6} R₃ (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MulAction.toHasSmul.{u3, max u4 u6} R₃ (ContinuousLinearMap.{u1, u3, u4, u6} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MonoidWithZero.toMonoid.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)) (ContinuousLinearMap.mulAction.{u1, u3, u4, u6, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 R₃ (MonoidWithZero.toMonoid.{u3} R₃ (Semiring.toMonoidWithZero.{u3} R₃ _inst_3)) (Module.toDistribMulAction.{u3, u6} R₃ M₃ _inst_3 _inst_13 _inst_14) _inst_35 _inst_37)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)) (fun (_x : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)) => R₂ -> R₃) (RingHom.hasCoeToFun.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)) σ₂₃ c) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h f)) +but is expected to have type + forall {R : Type.{u2}} {R₂ : Type.{u6}} {R₃ : Type.{u4}} [_inst_1 : Semiring.{u2} R] [_inst_2 : Semiring.{u6} R₂] [_inst_3 : Semiring.{u4} R₃] {M : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M] [_inst_7 : AddCommMonoid.{u1} M] [_inst_8 : Module.{u2, u1} R M _inst_1 _inst_7] {M₂ : Type.{u5}} [_inst_9 : TopologicalSpace.{u5} M₂] [_inst_10 : AddCommMonoid.{u5} M₂] [_inst_11 : Module.{u6, u5} R₂ M₂ _inst_2 _inst_10] {M₃ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₃] [_inst_13 : AddCommMonoid.{u3} M₃] [_inst_14 : Module.{u4, u3} R₃ M₃ _inst_3 _inst_13] {σ₁₂ : RingHom.{u2, u6} R R₂ (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)} {σ₂₃ : RingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} {σ₁₃ : RingHom.{u2, u4} R R₃ (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)} [_inst_27 : RingHomCompTriple.{u2, u6, u4} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_34 : SMulCommClass.{u6, u6, u5} R₂ R₂ M₂ (SMulZeroClass.toSMul.{u6, u5} R₂ M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (SMulWithZero.toSMulZeroClass.{u6, u5} R₂ M₂ (MonoidWithZero.toZero.{u6} R₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (MulActionWithZero.toSMulWithZero.{u6, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (Module.toMulActionWithZero.{u6, u5} R₂ M₂ _inst_2 _inst_10 _inst_11)))) (SMulZeroClass.toSMul.{u6, u5} R₂ M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (SMulWithZero.toSMulZeroClass.{u6, u5} R₂ M₂ (MonoidWithZero.toZero.{u6} R₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (MulActionWithZero.toSMulWithZero.{u6, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (Module.toMulActionWithZero.{u6, u5} R₂ M₂ _inst_2 _inst_10 _inst_11))))] [_inst_35 : SMulCommClass.{u4, u4, u3} R₃ R₃ M₃ (SMulZeroClass.toSMul.{u4, u3} R₃ M₃ (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u4, u3} R₃ M₃ (MonoidWithZero.toZero.{u4} R₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3)) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u4, u3} R₃ M₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (Module.toMulActionWithZero.{u4, u3} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toSMul.{u4, u3} R₃ M₃ (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u4, u3} R₃ M₃ (MonoidWithZero.toZero.{u4} R₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3)) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u4, u3} R₃ M₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (Module.toMulActionWithZero.{u4, u3} R₃ M₃ _inst_3 _inst_13 _inst_14))))] [_inst_36 : ContinuousConstSMul.{u6, u5} R₂ M₂ _inst_9 (SMulZeroClass.toSMul.{u6, u5} R₂ M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (SMulWithZero.toSMulZeroClass.{u6, u5} R₂ M₂ (MonoidWithZero.toZero.{u6} R₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (MulActionWithZero.toSMulWithZero.{u6, u5} R₂ M₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_10)) (Module.toMulActionWithZero.{u6, u5} R₂ M₂ _inst_2 _inst_10 _inst_11))))] [_inst_37 : ContinuousConstSMul.{u4, u3} R₃ M₃ _inst_12 (SMulZeroClass.toSMul.{u4, u3} R₃ M₃ (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u4, u3} R₃ M₃ (MonoidWithZero.toZero.{u4} R₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3)) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u4, u3} R₃ M₃ (Semiring.toMonoidWithZero.{u4} R₃ _inst_3) (AddMonoid.toZero.{u3} M₃ (AddCommMonoid.toAddMonoid.{u3} M₃ _inst_13)) (Module.toMulActionWithZero.{u4, u3} R₃ M₃ _inst_3 _inst_13 _inst_14))))] (h : ContinuousLinearMap.{u6, u4, u5, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_11 _inst_14) (c : R₂) (f : ContinuousLinearMap.{u2, u6, u1, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11), Eq.{max (succ u1) (succ u3)} (ContinuousLinearMap.{u2, u4, u1, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.comp.{u2, u6, u4, u1, u5, u3} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h (HSMul.hSMul.{u6, max u5 u1, max u5 u1} R₂ (ContinuousLinearMap.{u2, u6, u1, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (ContinuousLinearMap.{u2, u6, u1, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (instHSMul.{u6, max u5 u1} R₂ (ContinuousLinearMap.{u2, u6, u1, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (MulAction.toSMul.{u6, max u5 u1} R₂ (ContinuousLinearMap.{u2, u6, u1, u5} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11) (MonoidWithZero.toMonoid.{u6} R₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2)) (ContinuousLinearMap.mulAction.{u2, u6, u1, u5, u6} R R₂ _inst_1 _inst_2 σ₁₂ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 _inst_8 _inst_11 R₂ (MonoidWithZero.toMonoid.{u6} R₂ (Semiring.toMonoidWithZero.{u6} R₂ _inst_2)) (Module.toDistribMulAction.{u6, u5} R₂ M₂ _inst_2 _inst_10 _inst_11) _inst_34 _inst_36))) c f)) (HSMul.hSMul.{u4, max u3 u1, max u3 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (ContinuousLinearMap.{u2, u4, u1, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.{u2, u4, u1, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (instHSMul.{u4, max u3 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (ContinuousLinearMap.{u2, u4, u1, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MulAction.toSMul.{u4, max u3 u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (ContinuousLinearMap.{u2, u4, u1, u3} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (MonoidWithZero.toMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (Semiring.toMonoidWithZero.{u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) _inst_3)) (ContinuousLinearMap.mulAction.{u2, u4, u1, u3, u4} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (MonoidWithZero.toMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) (Semiring.toMonoidWithZero.{u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) _inst_3)) (Module.toDistribMulAction.{u4, u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) c) M₃ _inst_3 _inst_13 _inst_14) _inst_35 _inst_37))) (FunLike.coe.{max (succ u6) (succ u4), succ u6, succ u4} (RingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)) R₂ (fun (_x : R₂) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₂) => R₃) _x) (MulHomClass.toFunLike.{max u6 u4, u6, u4} (RingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)) R₂ R₃ (NonUnitalNonAssocSemiring.toMul.{u6} R₂ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R₂ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2))) (NonUnitalNonAssocSemiring.toMul.{u4} R₃ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R₃ (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3))) (NonUnitalRingHomClass.toMulHomClass.{max u6 u4, u6, u4} (RingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)) R₂ R₃ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R₂ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R₃ (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)) (RingHomClass.toNonUnitalRingHomClass.{max u6 u4, u6, u4} (RingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3)) R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3) (RingHom.instRingHomClassRingHom.{u6, u4} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₃ _inst_3))))) σ₂₃ c) (ContinuousLinearMap.comp.{u2, u6, u4, u1, u5, u3} R R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M _inst_6 _inst_7 M₂ _inst_9 _inst_10 M₃ _inst_12 _inst_13 _inst_8 _inst_11 _inst_14 _inst_27 h f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.comp_smulₛₗ ContinuousLinearMap.comp_smulₛₗₓ'. -/ @[simp] theorem comp_smulₛₗ [SMulCommClass R₂ R₂ M₂] [SMulCommClass R₃ R₃ M₃] [ContinuousConstSMul R₂ M₂] [ContinuousConstSMul R₃ M₃] (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) : @@ -1632,6 +2607,12 @@ variable {R R₂ R₃ S S₃ : Type _} [Semiring R] [Semiring R₂] [Semiring R {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] (c : S) (h : M₂ →SL[σ₂₃] M₃) (f g : M →SL[σ₁₂] M₂) (x y z : M) +/- warning: continuous_linear_map.prod_equiv -> ContinuousLinearMap.prodEquiv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u1, u2} R M _inst_1 _inst_7] {N₂ : Type.{u3}} [_inst_15 : TopologicalSpace.{u3} N₂] [_inst_16 : AddCommMonoid.{u3} N₂] [_inst_17 : Module.{u1, u3} R N₂ _inst_1 _inst_16] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 _inst_19], Equiv.{max (succ (max u2 u3)) (succ (max u2 u4)), max (succ u2) (succ (max u3 u4))} (Prod.{max u2 u3, max u2 u4} (ContinuousLinearMap.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20)) (ContinuousLinearMap.{u1, u1, u2, max u3 u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u3, u4} N₂ N₃) (Prod.topologicalSpace.{u3, u4} N₂ N₃ _inst_15 _inst_18) (Prod.addCommMonoid.{u3, u4} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u3, u4} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20)) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u1, u2} R M _inst_1 _inst_7] {N₂ : Type.{u3}} [_inst_15 : TopologicalSpace.{u3} N₂] [_inst_16 : AddCommMonoid.{u3} N₂] [_inst_17 : Module.{u1, u3} R N₂ _inst_1 _inst_16] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 _inst_19], Equiv.{max (succ (max u4 u2)) (succ (max u3 u2)), max (succ (max u4 u3)) (succ u2)} (Prod.{max u3 u2, max u4 u2} (ContinuousLinearMap.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u2, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20)) (ContinuousLinearMap.{u1, u1, u2, max u4 u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u3, u4} N₂ N₃) (instTopologicalSpaceProd.{u3, u4} N₂ N₃ _inst_15 _inst_18) (Prod.instAddCommMonoidSum.{u3, u4} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u3, u4} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prod_equiv ContinuousLinearMap.prodEquivₓ'. -/ /-- `continuous_linear_map.prod` as an `equiv`. -/ @[simps apply] def prodEquiv : (M →L[R] N₂) × (M →L[R] N₃) ≃ (M →L[R] N₂ × N₃) @@ -1642,6 +2623,12 @@ def prodEquiv : (M →L[R] N₂) × (M →L[R] N₃) ≃ (M →L[R] N₂ × N₃ right_inv f := by ext <;> rfl #align continuous_linear_map.prod_equiv ContinuousLinearMap.prodEquiv +/- warning: continuous_linear_map.prod_ext_iff -> ContinuousLinearMap.prod_ext_iff is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u1, u2} R M _inst_1 _inst_7] {N₂ : Type.{u3}} [_inst_15 : TopologicalSpace.{u3} N₂] [_inst_16 : AddCommMonoid.{u3} N₂] [_inst_17 : Module.{u1, u3} R N₂ _inst_1 _inst_16] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 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inaccurate. Consider using '#align continuous_linear_map.prod_ext_iff ContinuousLinearMap.prod_ext_iffₓ'. -/ theorem prod_ext_iff {f g : M × N₂ →L[R] N₃} : f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) := by @@ -1649,6 +2636,12 @@ theorem prod_ext_iff {f g : M × N₂ →L[R] N₃} : rfl #align continuous_linear_map.prod_ext_iff ContinuousLinearMap.prod_ext_iff +/- warning: continuous_linear_map.prod_ext -> ContinuousLinearMap.prod_ext is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u1, u2} R M _inst_1 _inst_7] {N₂ : Type.{u3}} [_inst_15 : TopologicalSpace.{u3} N₂] [_inst_16 : AddCommMonoid.{u3} N₂] [_inst_17 : Module.{u1, u3} R N₂ _inst_1 _inst_16] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 _inst_19] {f : 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_inst_1)) M _inst_6 _inst_7 (Prod.{u2, u3} M N₂) (Prod.topologicalSpace.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.addCommMonoid.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_8 (Prod.module.{u1, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) g (ContinuousLinearMap.inl.{u1, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17))) -> (Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_17 _inst_20) (ContinuousLinearMap.comp.{u1, u1, u1, u3, max u2 u3, u4} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) N₂ _inst_15 _inst_16 (Prod.{u2, u3} M N₂) (Prod.topologicalSpace.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.addCommMonoid.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_17 (Prod.module.{u1, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) f (ContinuousLinearMap.inr.{u1, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17)) (ContinuousLinearMap.comp.{u1, u1, u1, u3, max u2 u3, u4} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) N₂ _inst_15 _inst_16 (Prod.{u2, u3} M N₂) (Prod.topologicalSpace.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.addCommMonoid.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_17 (Prod.module.{u1, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.right_ids.{u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) g (ContinuousLinearMap.inr.{u1, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17))) -> (Eq.{max (succ (max u2 u3)) (succ u4)} (ContinuousLinearMap.{u1, u1, max u2 u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (Prod.{u2, u3} M N₂) (Prod.topologicalSpace.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.addCommMonoid.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 (Prod.module.{u1, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20) f g) +but is expected to have type + forall {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] {M : Type.{u2}} [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : AddCommMonoid.{u2} M] [_inst_8 : Module.{u4, u2} R M _inst_1 _inst_7] {N₂ : Type.{u3}} [_inst_15 : TopologicalSpace.{u3} N₂] [_inst_16 : AddCommMonoid.{u3} N₂] [_inst_17 : Module.{u4, u3} R N₂ _inst_1 _inst_16] {N₃ : Type.{u1}} [_inst_18 : TopologicalSpace.{u1} N₃] [_inst_19 : AddCommMonoid.{u1} N₃] [_inst_20 : Module.{u4, u1} R N₃ _inst_1 _inst_19] {f : ContinuousLinearMap.{u4, u4, max u3 u2, u1} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20} {g : ContinuousLinearMap.{u4, u4, max u3 u2, u1} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20}, (Eq.{max (succ u2) (succ u1)} 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u2, max u2 u3, u1} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) M _inst_6 _inst_7 (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_8 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.ids.{u4, u4} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1))) g (ContinuousLinearMap.inl.{u4, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17))) -> (Eq.{max (succ u3) (succ u1)} (ContinuousLinearMap.{u4, u4, u3, u1} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_17 _inst_20) (ContinuousLinearMap.comp.{u4, u4, u4, u3, max u2 u3, u1} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) N₂ _inst_15 _inst_16 (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_17 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.ids.{u4, u4} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1))) f (ContinuousLinearMap.inr.{u4, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17)) (ContinuousLinearMap.comp.{u4, u4, u4, u3, max u2 u3, u1} R R R _inst_1 _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) N₂ _inst_15 _inst_16 (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 _inst_17 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20 (RingHomCompTriple.ids.{u4, u4} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1))) g (ContinuousLinearMap.inr.{u4, u2, u3} R _inst_1 M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17))) -> (Eq.{max (max (succ u2) (succ u3)) (succ u1)} (ContinuousLinearMap.{u4, u4, max u3 u2, u1} R R _inst_1 _inst_1 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R _inst_1)) (Prod.{u2, u3} M N₂) (instTopologicalSpaceProd.{u2, u3} M N₂ _inst_6 _inst_15) (Prod.instAddCommMonoidSum.{u2, u3} M N₂ _inst_7 _inst_16) N₃ _inst_18 _inst_19 (Prod.module.{u4, u2, u3} R M N₂ _inst_1 _inst_7 _inst_16 _inst_8 _inst_17) _inst_20) f g) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prod_ext ContinuousLinearMap.prod_extₓ'. -/ @[ext] theorem prod_ext {f g : M × N₂ →L[R] N₃} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _)) (hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g := @@ -1667,6 +2660,12 @@ instance [Module S₃ᵐᵒᵖ M₃] [IsCentralScalar S₃ M₃] : IsCentralScal variable (S) [ContinuousAdd N₃] +/- warning: continuous_linear_map.prodₗ -> ContinuousLinearMap.prodₗ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : Semiring.{u1} R] [_inst_4 : Semiring.{u2} S] {M : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : AddCommMonoid.{u3} M] [_inst_8 : Module.{u1, u3} R M _inst_1 _inst_7] {N₂ : Type.{u4}} [_inst_15 : TopologicalSpace.{u4} N₂] [_inst_16 : AddCommMonoid.{u4} N₂] [_inst_17 : Module.{u1, u4} R N₂ _inst_1 _inst_16] {N₃ : Type.{u5}} [_inst_18 : TopologicalSpace.{u5} N₃] [_inst_19 : AddCommMonoid.{u5} N₃] [_inst_20 : Module.{u1, u5} R N₃ _inst_1 _inst_19] [_inst_24 : Module.{u2, u4} S N₂ _inst_4 _inst_16] [_inst_25 : ContinuousConstSMul.{u2, u4} S N₂ _inst_15 (SMulZeroClass.toHasSmul.{u2, u4} S N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (SMulWithZero.toSmulZeroClass.{u2, u4} S N₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₂ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (Module.toMulActionWithZero.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))))] [_inst_26 : SMulCommClass.{u1, u2, u4} R S N₂ (SMulZeroClass.toHasSmul.{u1, u4} R N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (SMulWithZero.toSmulZeroClass.{u1, u4} R N₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (MulActionWithZero.toSMulWithZero.{u1, u4} R N₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (Module.toMulActionWithZero.{u1, u4} R N₂ _inst_1 _inst_16 _inst_17)))) (SMulZeroClass.toHasSmul.{u2, u4} S N₂ (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (SMulWithZero.toSmulZeroClass.{u2, u4} S N₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₂ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) (Module.toMulActionWithZero.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))))] [_inst_27 : Module.{u2, u5} S N₃ _inst_4 _inst_19] [_inst_28 : SMulCommClass.{u1, u2, u5} R S N₃ (SMulZeroClass.toHasSmul.{u1, u5} R N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u1, u5} R N₃ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u1, u5} R N₃ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (Module.toMulActionWithZero.{u1, u5} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toHasSmul.{u2, u5} S N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u2, u5} S N₃ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u2, u5} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (Module.toMulActionWithZero.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_29 : ContinuousConstSMul.{u2, u5} S N₃ _inst_18 (SMulZeroClass.toHasSmul.{u2, u5} S N₃ (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u2, u5} S N₃ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u2, u5} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) (Module.toMulActionWithZero.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_33 : ContinuousAdd.{u4} N₂ _inst_15 (AddZeroClass.toHasAdd.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)))] [_inst_34 : ContinuousAdd.{u5} N₃ _inst_18 (AddZeroClass.toHasAdd.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)))], LinearEquiv.{u2, u2, max (max u3 u4) u3 u5, max u3 u4 u5} S S _inst_4 _inst_4 (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (RingHomInvPair.ids.{u2} S _inst_4) (RingHomInvPair.ids.{u2} S _inst_4) (Prod.{max u3 u4, max u3 u5} (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20)) (ContinuousLinearMap.{u1, u1, u3, max u4 u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u4, u5} N₂ N₃) (Prod.topologicalSpace.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.addCommMonoid.{u4, u5} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20)) (Prod.addCommMonoid.{max u3 u4, max u3 u5} (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 _inst_33) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34)) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, max u4 u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u4, u5} N₂ N₃) (Prod.topologicalSpace.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.addCommMonoid.{u4, u5} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20) (ContinuousLinearMap.prodₗ._proof_1.{u5, u4} N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_33 _inst_34)) (Prod.module.{u2, max u3 u4, max u3 u5} S (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) _inst_4 (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 _inst_33) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34) (ContinuousLinearMap.module.{u1, u1, u2, u3, u4} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₂ _inst_15 _inst_16 _inst_17 _inst_24 _inst_26 _inst_25 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_33) (ContinuousLinearMap.module.{u1, u1, u2, u3, u5} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₃ _inst_18 _inst_19 _inst_20 _inst_27 _inst_28 _inst_29 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_34)) (ContinuousLinearMap.module.{u1, u1, u2, u3, max u4 u5} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 (Prod.{u4, u5} N₂ N₃) (Prod.topologicalSpace.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.addCommMonoid.{u4, u5} N₂ N₃ _inst_16 _inst_19) (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20) (Prod.module.{u2, u4, u5} S N₂ N₃ _inst_4 _inst_16 _inst_19 _inst_24 _inst_27) (ContinuousLinearMap.prodₗ._proof_2.{u1, u5, u2, u4} R S _inst_1 _inst_4 N₂ _inst_16 _inst_17 N₃ _inst_19 _inst_20 _inst_24 _inst_26 _inst_27 _inst_28) (ContinuousLinearMap.prodₗ._proof_3.{u5, u2, u4} S _inst_4 N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_24 _inst_25 _inst_27 _inst_29) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (ContinuousLinearMap.prodₗ._proof_4.{u5, u4} N₂ _inst_15 _inst_16 N₃ _inst_18 _inst_19 _inst_33 _inst_34)) +but is expected to have type + forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : Semiring.{u1} R] [_inst_4 : Semiring.{u2} S] {M : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : AddCommMonoid.{u3} M] [_inst_8 : Module.{u1, u3} R M _inst_1 _inst_7] {N₂ : Type.{u4}} [_inst_15 : TopologicalSpace.{u4} N₂] [_inst_16 : AddCommMonoid.{u4} N₂] [_inst_17 : Module.{u1, u4} R N₂ _inst_1 _inst_16] {N₃ : Type.{u5}} [_inst_18 : TopologicalSpace.{u5} N₃] [_inst_19 : AddCommMonoid.{u5} N₃] [_inst_20 : Module.{u1, u5} R N₃ _inst_1 _inst_19] [_inst_24 : Module.{u2, u4} S N₂ _inst_4 _inst_16] [_inst_25 : ContinuousConstSMul.{u2, u4} S N₂ _inst_15 (SMulZeroClass.toSMul.{u2, u4} S N₂ (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (SMulWithZero.toSMulZeroClass.{u2, u4} S N₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₂ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (Module.toMulActionWithZero.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))))] [_inst_26 : SMulCommClass.{u1, u2, u4} R S N₂ (SMulZeroClass.toSMul.{u1, u4} R N₂ (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (SMulWithZero.toSMulZeroClass.{u1, u4} R N₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (MulActionWithZero.toSMulWithZero.{u1, u4} R N₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (Module.toMulActionWithZero.{u1, u4} R N₂ _inst_1 _inst_16 _inst_17)))) (SMulZeroClass.toSMul.{u2, u4} S N₂ (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (SMulWithZero.toSMulZeroClass.{u2, u4} S N₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₂ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)) (Module.toMulActionWithZero.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))))] [_inst_27 : Module.{u2, u5} S N₃ _inst_4 _inst_19] [_inst_28 : SMulCommClass.{u1, u2, u5} R S N₃ (SMulZeroClass.toSMul.{u1, u5} R N₃ (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u1, u5} R N₃ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u1, u5} R N₃ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (Module.toMulActionWithZero.{u1, u5} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toSMul.{u2, u5} S N₃ (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u2, u5} S N₃ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u2, u5} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (Module.toMulActionWithZero.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_29 : ContinuousConstSMul.{u2, u5} S N₃ _inst_18 (SMulZeroClass.toSMul.{u2, u5} S N₃ (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u2, u5} S N₃ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u2, u5} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)) (Module.toMulActionWithZero.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_33 : ContinuousAdd.{u4} N₂ _inst_15 (AddZeroClass.toAdd.{u4} N₂ (AddMonoid.toAddZeroClass.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16)))] [_inst_34 : ContinuousAdd.{u5} N₃ _inst_18 (AddZeroClass.toAdd.{u5} N₃ (AddMonoid.toAddZeroClass.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19)))], LinearEquiv.{u2, u2, max (max u5 u3) u4 u3, max (max u5 u4) u3} S S _inst_4 _inst_4 (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (RingHomInvPair.ids.{u2} S _inst_4) (RingHomInvPair.ids.{u2} S _inst_4) (Prod.{max u4 u3, max u5 u3} (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20)) (ContinuousLinearMap.{u1, u1, u3, max u5 u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u4, u5} N₂ N₃) (instTopologicalSpaceProd.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.instAddCommMonoidSum.{u4, u5} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20)) (Prod.instAddCommMonoidSum.{max u3 u4, max u3 u5} (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 _inst_33) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34)) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, max u4 u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 (Prod.{u4, u5} N₂ N₃) (instTopologicalSpaceProd.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.instAddCommMonoidSum.{u4, u5} N₂ N₃ _inst_16 _inst_19) _inst_8 (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20) (instContinuousAddSumInstTopologicalSpaceSumInstAddSum.{u4, u5} N₂ N₃ _inst_15 (AddSemigroup.toAdd.{u4} N₂ (AddMonoid.toAddSemigroup.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) _inst_33 _inst_18 (AddSemigroup.toAdd.{u5} N₃ (AddMonoid.toAddSemigroup.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) _inst_34)) (Prod.module.{u2, max u3 u4, max u3 u5} S (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17) (ContinuousLinearMap.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) _inst_4 (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₂ _inst_15 _inst_16 _inst_8 _inst_17 _inst_33) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34) (ContinuousLinearMap.module.{u1, u1, u2, u3, u4} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₂ _inst_15 _inst_16 _inst_17 _inst_24 _inst_26 _inst_25 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_33) (ContinuousLinearMap.module.{u1, u1, u2, u3, u5} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₃ _inst_18 _inst_19 _inst_20 _inst_27 _inst_28 _inst_29 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_34)) (ContinuousLinearMap.module.{u1, u1, u2, u3, max u4 u5} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 (Prod.{u4, u5} N₂ N₃) (instTopologicalSpaceProd.{u4, u5} N₂ N₃ _inst_15 _inst_18) (Prod.instAddCommMonoidSum.{u4, u5} N₂ N₃ _inst_16 _inst_19) (Prod.module.{u1, u4, u5} R N₂ N₃ _inst_1 _inst_16 _inst_19 _inst_17 _inst_20) (Prod.module.{u2, u4, u5} S N₂ N₃ _inst_4 _inst_16 _inst_19 _inst_24 _inst_27) (Prod.smulCommClass.{u1, u2, u4, u5} R S N₂ N₃ (MulAction.toSMul.{u1, u4} R N₂ (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (DistribMulAction.toMulAction.{u1, u4} R N₂ (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) (Module.toDistribMulAction.{u1, u4} R N₂ _inst_1 _inst_16 _inst_17))) (MulAction.toSMul.{u1, u5} R N₃ (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (DistribMulAction.toMulAction.{u1, u5} R N₃ (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) (Module.toDistribMulAction.{u1, u5} R N₃ _inst_1 _inst_19 _inst_20))) (MulAction.toSMul.{u2, u4} S N₂ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (DistribMulAction.toMulAction.{u2, u4} S N₂ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) (Module.toDistribMulAction.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))) (MulAction.toSMul.{u2, u5} S N₃ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (DistribMulAction.toMulAction.{u2, u5} S N₃ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) (Module.toDistribMulAction.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))) _inst_26 _inst_28) (instContinuousConstSMulProdInstTopologicalSpaceProdSmul.{u2, u4, u5} S N₂ N₃ _inst_15 (MulAction.toSMul.{u2, u4} S N₂ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (DistribMulAction.toMulAction.{u2, u4} S N₂ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16) (Module.toDistribMulAction.{u2, u4} S N₂ _inst_4 _inst_16 _inst_24))) _inst_25 _inst_18 (MulAction.toSMul.{u2, u5} S N₃ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (DistribMulAction.toMulAction.{u2, u5} S N₃ (MonoidWithZero.toMonoid.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19) (Module.toDistribMulAction.{u2, u5} S N₃ _inst_4 _inst_19 _inst_27))) _inst_29) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (instContinuousAddSumInstTopologicalSpaceSumInstAddSum.{u4, u5} N₂ N₃ _inst_15 (AddSemigroup.toAdd.{u4} N₂ (AddMonoid.toAddSemigroup.{u4} N₂ (AddCommMonoid.toAddMonoid.{u4} N₂ _inst_16))) _inst_33 _inst_18 (AddSemigroup.toAdd.{u5} N₃ (AddMonoid.toAddSemigroup.{u5} N₃ (AddCommMonoid.toAddMonoid.{u5} N₃ _inst_19))) _inst_34)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.prodₗ ContinuousLinearMap.prodₗₓ'. -/ /-- `continuous_linear_map.prod` as a `linear_equiv`. -/ @[simps apply] def prodₗ : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ₗ[S] M →L[R] N₂ × N₃ := @@ -1675,25 +2674,37 @@ def prodₗ : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ₗ[S] M →L[R] N₂ × map_smul' := fun c f => rfl } #align continuous_linear_map.prodₗ ContinuousLinearMap.prodₗ +/- warning: continuous_linear_map.coe_lm -> ContinuousLinearMap.coeLM is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : Semiring.{u1} R] [_inst_4 : Semiring.{u2} S] {M : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : AddCommMonoid.{u3} M] [_inst_8 : Module.{u1, u3} R M _inst_1 _inst_7] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 _inst_19] [_inst_27 : Module.{u2, u4} S N₃ _inst_4 _inst_19] [_inst_28 : SMulCommClass.{u1, u2, u4} R S N₃ (SMulZeroClass.toHasSmul.{u1, u4} R N₃ (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u1, u4} R N₃ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u1, u4} R N₃ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (Module.toMulActionWithZero.{u1, u4} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toHasSmul.{u2, u4} S N₃ (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u2, u4} S N₃ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (Module.toMulActionWithZero.{u2, u4} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_29 : ContinuousConstSMul.{u2, u4} S N₃ _inst_18 (SMulZeroClass.toHasSmul.{u2, u4} S N₃ (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (SMulWithZero.toSmulZeroClass.{u2, u4} S N₃ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)))) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddZeroClass.toHasZero.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19))) (Module.toMulActionWithZero.{u2, u4} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_34 : ContinuousAdd.{u4} N₃ _inst_18 (AddZeroClass.toHasAdd.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)))], LinearMap.{u2, u2, max u3 u4, max u3 u4} S S _inst_4 _inst_4 (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (LinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M N₃ _inst_7 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34) (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R M N₃ _inst_1 _inst_1 _inst_7 _inst_19 _inst_8 _inst_20 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (ContinuousLinearMap.module.{u1, u1, u2, u3, u4} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₃ _inst_18 _inst_19 _inst_20 _inst_27 _inst_28 _inst_29 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_34) (LinearMap.module.{u1, u1, u2, u3, u4} R R S M N₃ _inst_1 _inst_1 _inst_7 _inst_19 _inst_8 _inst_20 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_4 _inst_27 _inst_28) +but is expected to have type + forall {R : Type.{u1}} (S : Type.{u2}) [_inst_1 : Semiring.{u1} R] [_inst_4 : Semiring.{u2} S] {M : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : AddCommMonoid.{u3} M] [_inst_8 : Module.{u1, u3} R M _inst_1 _inst_7] {N₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} N₃] [_inst_19 : AddCommMonoid.{u4} N₃] [_inst_20 : Module.{u1, u4} R N₃ _inst_1 _inst_19] [_inst_27 : Module.{u2, u4} S N₃ _inst_4 _inst_19] [_inst_28 : SMulCommClass.{u1, u2, u4} R S N₃ (SMulZeroClass.toSMul.{u1, u4} R N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u1, u4} R N₃ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u1, u4} R N₃ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (Module.toMulActionWithZero.{u1, u4} R N₃ _inst_1 _inst_19 _inst_20)))) (SMulZeroClass.toSMul.{u2, u4} S N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u2, u4} S N₃ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (Module.toMulActionWithZero.{u2, u4} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_29 : ContinuousConstSMul.{u2, u4} S N₃ _inst_18 (SMulZeroClass.toSMul.{u2, u4} S N₃ (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (SMulWithZero.toSMulZeroClass.{u2, u4} S N₃ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_4)) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (MulActionWithZero.toSMulWithZero.{u2, u4} S N₃ (Semiring.toMonoidWithZero.{u2} S _inst_4) (AddMonoid.toZero.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)) (Module.toMulActionWithZero.{u2, u4} S N₃ _inst_4 _inst_19 _inst_27))))] [_inst_34 : ContinuousAdd.{u4} N₃ _inst_18 (AddZeroClass.toAdd.{u4} N₃ (AddMonoid.toAddZeroClass.{u4} N₃ (AddCommMonoid.toAddMonoid.{u4} N₃ _inst_19)))], LinearMap.{u2, u2, max u4 u3, max u4 u3} S S _inst_4 _inst_4 (RingHom.id.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_4)) (ContinuousLinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20) (LinearMap.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M N₃ _inst_7 _inst_19 _inst_8 _inst_20) (ContinuousLinearMap.addCommMonoid.{u1, u1, u3, u4} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_6 _inst_7 N₃ _inst_18 _inst_19 _inst_8 _inst_20 _inst_34) (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R M N₃ _inst_1 _inst_1 _inst_7 _inst_19 _inst_8 _inst_20 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (ContinuousLinearMap.module.{u1, u1, u2, u3, u4} R R S _inst_1 _inst_1 _inst_4 M _inst_6 _inst_7 _inst_8 N₃ _inst_18 _inst_19 _inst_20 _inst_27 _inst_28 _inst_29 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_34) (LinearMap.instModuleLinearMapAddCommMonoid.{u1, u1, u2, u3, u4} R R S M N₃ _inst_1 _inst_1 _inst_7 _inst_19 _inst_8 _inst_20 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_4 _inst_27 _inst_28) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_lm ContinuousLinearMap.coeLMₓ'. -/ /-- The coercion from `M →L[R] M₂` to `M →ₗ[R] M₂`, as a linear map. -/ @[simps] -def coeLm : (M →L[R] N₃) →ₗ[S] M →ₗ[R] N₃ +def coeLM : (M →L[R] N₃) →ₗ[S] M →ₗ[R] N₃ where toFun := coe map_add' f g := coe_add f g map_smul' c f := coe_smul c f -#align continuous_linear_map.coe_lm ContinuousLinearMap.coeLm +#align continuous_linear_map.coe_lm ContinuousLinearMap.coeLM variable {S} (σ₁₃) +/- warning: continuous_linear_map.coe_lmₛₗ -> ContinuousLinearMap.coeLMₛₗ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {R₃ : Type.{u2}} {S₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R] [_inst_3 : Semiring.{u2} R₃] [_inst_5 : Semiring.{u3} S₃] {M : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M] [_inst_7 : AddCommMonoid.{u4} M] [_inst_8 : Module.{u1, u4} R M _inst_1 _inst_7] {M₃ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₃] [_inst_13 : AddCommMonoid.{u5} M₃] [_inst_14 : Module.{u2, u5} R₃ M₃ _inst_3 _inst_13] [_inst_21 : Module.{u3, u5} S₃ M₃ _inst_5 _inst_13] [_inst_22 : SMulCommClass.{u2, u3, u5} R₃ S₃ M₃ (SMulZeroClass.toHasSmul.{u2, u5} R₃ M₃ (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u2, u5} R₃ M₃ (MulZeroClass.toHasZero.{u2} R₃ (MulZeroOneClass.toMulZeroClass.{u2} R₃ (MonoidWithZero.toMulZeroOneClass.{u2} R₃ (Semiring.toMonoidWithZero.{u2} R₃ _inst_3)))) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u2, u5} R₃ M₃ (Semiring.toMonoidWithZero.{u2} R₃ _inst_3) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (Module.toMulActionWithZero.{u2, u5} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toHasSmul.{u3, u5} S₃ M₃ (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u5} S₃ M₃ (MulZeroClass.toHasZero.{u3} S₃ (MulZeroOneClass.toMulZeroClass.{u3} S₃ (MonoidWithZero.toMulZeroOneClass.{u3} S₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5)))) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u5} S₃ M₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u5} S₃ M₃ _inst_5 _inst_13 _inst_21))))] [_inst_23 : ContinuousConstSMul.{u3, u5} S₃ M₃ _inst_12 (SMulZeroClass.toHasSmul.{u3, u5} S₃ M₃ (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (SMulWithZero.toSmulZeroClass.{u3, u5} S₃ M₃ (MulZeroClass.toHasZero.{u3} S₃ (MulZeroOneClass.toMulZeroClass.{u3} S₃ (MonoidWithZero.toMulZeroOneClass.{u3} S₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5)))) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (MulActionWithZero.toSMulWithZero.{u3, u5} S₃ M₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5) (AddZeroClass.toHasZero.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13))) (Module.toMulActionWithZero.{u3, u5} S₃ M₃ _inst_5 _inst_13 _inst_21))))] (σ₁₃ : RingHom.{u1, u2} R R₃ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)) [_inst_32 : ContinuousAdd.{u5} M₃ _inst_12 (AddZeroClass.toHasAdd.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)))], LinearMap.{u3, u3, max u4 u5, max u4 u5} S₃ S₃ _inst_5 _inst_5 (RingHom.id.{u3} S₃ (Semiring.toNonAssocSemiring.{u3} S₃ _inst_5)) (ContinuousLinearMap.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (LinearMap.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M M₃ _inst_7 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.addCommMonoid.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 _inst_32) (LinearMap.addCommMonoid.{u1, u2, u4, u5} R R₃ M M₃ _inst_1 _inst_3 _inst_7 _inst_13 _inst_8 _inst_14 σ₁₃) (ContinuousLinearMap.module.{u1, u2, u3, u4, u5} R R₃ S₃ _inst_1 _inst_3 _inst_5 M _inst_6 _inst_7 _inst_8 M₃ _inst_12 _inst_13 _inst_14 _inst_21 _inst_22 _inst_23 σ₁₃ _inst_32) (LinearMap.module.{u1, u2, u3, u4, u5} R R₃ S₃ M M₃ _inst_1 _inst_3 _inst_7 _inst_13 _inst_8 _inst_14 σ₁₃ _inst_5 _inst_21 _inst_22) +but is expected to have type + forall {R : Type.{u1}} {R₃ : Type.{u2}} {S₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R] [_inst_3 : Semiring.{u2} R₃] [_inst_5 : Semiring.{u3} S₃] {M : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M] [_inst_7 : AddCommMonoid.{u4} M] [_inst_8 : Module.{u1, u4} R M _inst_1 _inst_7] {M₃ : Type.{u5}} [_inst_12 : TopologicalSpace.{u5} M₃] [_inst_13 : AddCommMonoid.{u5} M₃] [_inst_14 : Module.{u2, u5} R₃ M₃ _inst_3 _inst_13] [_inst_21 : Module.{u3, u5} S₃ M₃ _inst_5 _inst_13] [_inst_22 : SMulCommClass.{u2, u3, u5} R₃ S₃ M₃ (SMulZeroClass.toSMul.{u2, u5} R₃ M₃ (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u2, u5} R₃ M₃ (MonoidWithZero.toZero.{u2} R₃ (Semiring.toMonoidWithZero.{u2} R₃ _inst_3)) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u2, u5} R₃ M₃ (Semiring.toMonoidWithZero.{u2} R₃ _inst_3) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (Module.toMulActionWithZero.{u2, u5} R₃ M₃ _inst_3 _inst_13 _inst_14)))) (SMulZeroClass.toSMul.{u3, u5} S₃ M₃ (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u3, u5} S₃ M₃ (MonoidWithZero.toZero.{u3} S₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5)) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u3, u5} S₃ M₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (Module.toMulActionWithZero.{u3, u5} S₃ M₃ _inst_5 _inst_13 _inst_21))))] [_inst_23 : ContinuousConstSMul.{u3, u5} S₃ M₃ _inst_12 (SMulZeroClass.toSMul.{u3, u5} S₃ M₃ (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (SMulWithZero.toSMulZeroClass.{u3, u5} S₃ M₃ (MonoidWithZero.toZero.{u3} S₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5)) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (MulActionWithZero.toSMulWithZero.{u3, u5} S₃ M₃ (Semiring.toMonoidWithZero.{u3} S₃ _inst_5) (AddMonoid.toZero.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)) (Module.toMulActionWithZero.{u3, u5} S₃ M₃ _inst_5 _inst_13 _inst_21))))] (σ₁₃ : RingHom.{u1, u2} R R₃ (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)) [_inst_32 : ContinuousAdd.{u5} M₃ _inst_12 (AddZeroClass.toAdd.{u5} M₃ (AddMonoid.toAddZeroClass.{u5} M₃ (AddCommMonoid.toAddMonoid.{u5} M₃ _inst_13)))], LinearMap.{u3, u3, max u5 u4, max u5 u4} S₃ S₃ _inst_5 _inst_5 (RingHom.id.{u3} S₃ (Semiring.toNonAssocSemiring.{u3} S₃ _inst_5)) (ContinuousLinearMap.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14) (LinearMap.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M M₃ _inst_7 _inst_13 _inst_8 _inst_14) (ContinuousLinearMap.addCommMonoid.{u1, u2, u4, u5} R R₃ _inst_1 _inst_3 σ₁₃ M _inst_6 _inst_7 M₃ _inst_12 _inst_13 _inst_8 _inst_14 _inst_32) (LinearMap.addCommMonoid.{u1, u2, u4, u5} R R₃ M M₃ _inst_1 _inst_3 _inst_7 _inst_13 _inst_8 _inst_14 σ₁₃) (ContinuousLinearMap.module.{u1, u2, u3, u4, u5} R R₃ S₃ _inst_1 _inst_3 _inst_5 M _inst_6 _inst_7 _inst_8 M₃ _inst_12 _inst_13 _inst_14 _inst_21 _inst_22 _inst_23 σ₁₃ _inst_32) (LinearMap.instModuleLinearMapAddCommMonoid.{u1, u2, u3, u4, u5} R R₃ S₃ M M₃ _inst_1 _inst_3 _inst_7 _inst_13 _inst_8 _inst_14 σ₁₃ _inst_5 _inst_21 _inst_22) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_lmₛₗ ContinuousLinearMap.coeLMₛₗₓ'. -/ /-- The coercion from `M →SL[σ] M₂` to `M →ₛₗ[σ] M₂`, as a linear map. -/ @[simps] -def coeLmₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] M →ₛₗ[σ₁₃] M₃ +def coeLMₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] M →ₛₗ[σ₁₃] M₃ where toFun := coe map_add' f g := coe_add f g map_smul' c f := coe_smul c f -#align continuous_linear_map.coe_lmₛₗ ContinuousLinearMap.coeLmₛₗ +#align continuous_linear_map.coe_lmₛₗ ContinuousLinearMap.coeLMₛₗ variable {σ₁₃} @@ -1707,6 +2718,12 @@ variable {R S T M M₂ : Type _} [Semiring R] [Semiring S] [Semiring T] [Module [ContinuousAdd M₂] [Module T M₂] [ContinuousConstSMul T M₂] [SMulCommClass R T M₂] [SMulCommClass S T M₂] +/- warning: continuous_linear_map.smul_rightₗ -> ContinuousLinearMap.smulRightₗ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} {M : Type.{u4}} {M₂ : Type.{u5}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Semiring.{u2} S] [_inst_3 : Semiring.{u3} T] [_inst_4 : Module.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))] [_inst_5 : AddCommMonoid.{u5} M₂] [_inst_6 : Module.{u1, u5} R M₂ _inst_1 _inst_5] [_inst_7 : Module.{u2, u5} S M₂ _inst_2 _inst_5] [_inst_8 : IsScalarTower.{u1, u2, u5} R S M₂ (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_4)))) (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toHasSmul.{u1, u5} R M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u5} R M₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6))))] [_inst_9 : TopologicalSpace.{u2} S] [_inst_10 : TopologicalSpace.{u5} M₂] [_inst_11 : ContinuousSMul.{u2, u5} S M₂ (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) _inst_9 _inst_10] [_inst_12 : TopologicalSpace.{u4} M] [_inst_13 : AddCommMonoid.{u4} M] [_inst_14 : Module.{u1, u4} R M _inst_1 _inst_13] [_inst_15 : ContinuousAdd.{u5} M₂ _inst_10 (AddZeroClass.toHasAdd.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)))] [_inst_16 : Module.{u3, u5} T M₂ _inst_3 _inst_5] [_inst_17 : ContinuousConstSMul.{u3, u5} T M₂ _inst_10 (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_18 : SMulCommClass.{u1, u3, u5} R T M₂ (SMulZeroClass.toHasSmul.{u1, u5} R M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u5} R M₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6)))) (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_19 : SMulCommClass.{u2, u3, u5} S T M₂ (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))], (ContinuousLinearMap.{u1, u1, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 S _inst_9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_14 _inst_4) -> (LinearMap.{u3, u3, u5, max u4 u5} T T _inst_3 _inst_3 (RingHom.id.{u3} T (Semiring.toNonAssocSemiring.{u3} T _inst_3)) M₂ (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_5 (ContinuousLinearMap.addCommMonoid.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u1, u1, u3, u4, u5} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_15)) +but is expected to have type + forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} {M : Type.{u4}} {M₂ : Type.{u5}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Semiring.{u2} S] [_inst_3 : Semiring.{u3} T] [_inst_4 : Module.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))] [_inst_5 : AddCommMonoid.{u5} M₂] [_inst_6 : Module.{u1, u5} R M₂ _inst_1 _inst_5] [_inst_7 : Module.{u2, u5} S M₂ _inst_2 _inst_5] [_inst_8 : IsScalarTower.{u1, u2, u5} R S M₂ (SMulZeroClass.toSMul.{u1, u2} R S (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R S (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R _inst_1) (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (Module.toMulActionWithZero.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_4)))) (SMulZeroClass.toSMul.{u2, u5} S M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u5} S M₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toSMul.{u1, u5} R M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u5} R M₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6))))] [_inst_9 : TopologicalSpace.{u2} S] [_inst_10 : TopologicalSpace.{u5} M₂] [_inst_11 : ContinuousSMul.{u2, u5} S M₂ (SMulZeroClass.toSMul.{u2, u5} S M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u5} S M₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) _inst_9 _inst_10] [_inst_12 : TopologicalSpace.{u4} M] [_inst_13 : AddCommMonoid.{u4} M] [_inst_14 : Module.{u1, u4} R M _inst_1 _inst_13] [_inst_15 : ContinuousAdd.{u5} M₂ _inst_10 (AddZeroClass.toAdd.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)))] [_inst_16 : Module.{u3, u5} T M₂ _inst_3 _inst_5] [_inst_17 : ContinuousConstSMul.{u3, u5} T M₂ _inst_10 (SMulZeroClass.toSMul.{u3, u5} T M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u5} T M₂ (MonoidWithZero.toZero.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_18 : SMulCommClass.{u1, u3, u5} R T M₂ (SMulZeroClass.toSMul.{u1, u5} R M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u5} R M₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6)))) (SMulZeroClass.toSMul.{u3, u5} T M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u5} T M₂ (MonoidWithZero.toZero.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_19 : SMulCommClass.{u2, u3, u5} S T M₂ (SMulZeroClass.toSMul.{u2, u5} S M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u2, u5} S M₂ (MonoidWithZero.toZero.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toSMul.{u3, u5} T M₂ (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u5} T M₂ (MonoidWithZero.toZero.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddMonoid.toZero.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))], (ContinuousLinearMap.{u1, u1, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 S _inst_9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_14 _inst_4) -> (LinearMap.{u3, u3, u5, max u5 u4} T T _inst_3 _inst_3 (RingHom.id.{u3} T (Semiring.toNonAssocSemiring.{u3} T _inst_3)) M₂ (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_5 (ContinuousLinearMap.addCommMonoid.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u1, u1, u3, u4, u5} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_15)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.smul_rightₗ ContinuousLinearMap.smulRightₗₓ'. -/ /-- Given `c : E →L[𝕜] 𝕜`, `c.smul_rightₗ` is the linear map from `F` to `E →L[𝕜] F` sending `f` to `λ e, c e • f`. See also `continuous_linear_map.smul_rightL`. -/ def smulRightₗ (c : M →L[R] S) : M₂ →ₗ[T] M →L[R] M₂ @@ -1721,6 +2738,12 @@ def smulRightₗ (c : M →L[R] S) : M₂ →ₗ[T] M →L[R] M₂ apply smul_comm #align continuous_linear_map.smul_rightₗ ContinuousLinearMap.smulRightₗ +/- warning: continuous_linear_map.coe_smul_rightₗ -> ContinuousLinearMap.coe_smulRightₗ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} {M : Type.{u4}} {M₂ : Type.{u5}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Semiring.{u2} S] [_inst_3 : Semiring.{u3} T] [_inst_4 : Module.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))] [_inst_5 : AddCommMonoid.{u5} M₂] [_inst_6 : Module.{u1, u5} R M₂ _inst_1 _inst_5] [_inst_7 : Module.{u2, u5} S M₂ _inst_2 _inst_5] [_inst_8 : IsScalarTower.{u1, u2, u5} R S M₂ (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_4)))) (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toHasSmul.{u1, u5} R M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u5} R M₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6))))] [_inst_9 : TopologicalSpace.{u2} S] [_inst_10 : TopologicalSpace.{u5} M₂] [_inst_11 : ContinuousSMul.{u2, u5} S M₂ (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) _inst_9 _inst_10] [_inst_12 : TopologicalSpace.{u4} M] [_inst_13 : AddCommMonoid.{u4} M] [_inst_14 : Module.{u1, u4} R M _inst_1 _inst_13] [_inst_15 : ContinuousAdd.{u5} M₂ _inst_10 (AddZeroClass.toHasAdd.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5)))] [_inst_16 : Module.{u3, u5} T M₂ _inst_3 _inst_5] [_inst_17 : ContinuousConstSMul.{u3, u5} T M₂ _inst_10 (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_18 : SMulCommClass.{u1, u3, u5} R T M₂ (SMulZeroClass.toHasSmul.{u1, u5} R M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u1, u5} R M₂ (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u1, u5} R M₂ (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u1, u5} R M₂ _inst_1 _inst_5 _inst_6)))) (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_19 : SMulCommClass.{u2, u3, u5} S T M₂ (SMulZeroClass.toHasSmul.{u2, u5} S M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u2, u5} S M₂ (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u2, u5} S M₂ (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u2, u5} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toHasSmul.{u3, u5} T M₂ (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (SMulWithZero.toSmulZeroClass.{u3, u5} T M₂ (MulZeroClass.toHasZero.{u3} T (MulZeroOneClass.toMulZeroClass.{u3} T (MonoidWithZero.toMulZeroOneClass.{u3} T (Semiring.toMonoidWithZero.{u3} T _inst_3)))) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (MulActionWithZero.toSMulWithZero.{u3, u5} T M₂ (Semiring.toMonoidWithZero.{u3} T _inst_3) (AddZeroClass.toHasZero.{u5} M₂ (AddMonoid.toAddZeroClass.{u5} M₂ (AddCommMonoid.toAddMonoid.{u5} M₂ _inst_5))) (Module.toMulActionWithZero.{u3, u5} T M₂ _inst_3 _inst_5 _inst_16))))] (c : ContinuousLinearMap.{u1, u1, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 S _inst_9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) _inst_14 _inst_4), Eq.{max (succ u5) (succ (max u4 u5))} (M₂ -> (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6)) (coeFn.{max (succ u5) (succ (max u4 u5)), max (succ u5) (succ (max u4 u5))} (LinearMap.{u3, u3, u5, max u4 u5} T T _inst_3 _inst_3 (RingHom.id.{u3} T (Semiring.toNonAssocSemiring.{u3} T _inst_3)) M₂ (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_5 (ContinuousLinearMap.addCommMonoid.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u1, u1, u3, u4, u5} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_15)) (fun (_x : LinearMap.{u3, u3, u5, max u4 u5} T T _inst_3 _inst_3 (RingHom.id.{u3} T (Semiring.toNonAssocSemiring.{u3} T _inst_3)) M₂ (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_5 (ContinuousLinearMap.addCommMonoid.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u1, u1, u3, u4, u5} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_15)) => M₂ -> (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6)) (LinearMap.hasCoeToFun.{u3, u3, u5, max u4 u5} T T M₂ (ContinuousLinearMap.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_3 _inst_3 _inst_5 (ContinuousLinearMap.addCommMonoid.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u1, u1, u3, u4, u5} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) _inst_15) (RingHom.id.{u3} T (Semiring.toNonAssocSemiring.{u3} T _inst_3))) (ContinuousLinearMap.smulRightₗ.{u1, u2, u3, u4, u5} R S T M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13 _inst_14 _inst_15 _inst_16 _inst_17 _inst_18 _inst_19 c)) (ContinuousLinearMap.smulRight.{u4, u5, u1, u2} M _inst_12 _inst_13 M₂ _inst_10 _inst_5 R S _inst_1 _inst_2 _inst_14 _inst_6 _inst_4 _inst_7 _inst_8 _inst_9 _inst_11 c) +but is expected to have type + forall {R : Type.{u5}} {S : Type.{u3}} {T : Type.{u1}} {M : Type.{u4}} {M₂ : Type.{u2}} [_inst_1 : Semiring.{u5} R] [_inst_2 : Semiring.{u3} S] [_inst_3 : Semiring.{u1} T] [_inst_4 : Module.{u5, u3} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2)))] [_inst_5 : AddCommMonoid.{u2} M₂] [_inst_6 : Module.{u5, u2} R M₂ _inst_1 _inst_5] [_inst_7 : Module.{u3, u2} S M₂ _inst_2 _inst_5] [_inst_8 : IsScalarTower.{u5, u3, u2} R S M₂ (SMulZeroClass.toSMul.{u5, u3} R S (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (SMulWithZero.toSMulZeroClass.{u5, u3} R S (MonoidWithZero.toZero.{u5} R (Semiring.toMonoidWithZero.{u5} R _inst_1)) (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (MulActionWithZero.toSMulWithZero.{u5, u3} R S (Semiring.toMonoidWithZero.{u5} R _inst_1) (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (Module.toMulActionWithZero.{u5, u3} R S _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2))) _inst_4)))) (SMulZeroClass.toSMul.{u3, u2} S M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u2} S M₂ (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u2} S M₂ (Semiring.toMonoidWithZero.{u3} S _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u2} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toSMul.{u5, u2} R M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u5, u2} R M₂ (MonoidWithZero.toZero.{u5} R (Semiring.toMonoidWithZero.{u5} R _inst_1)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u5, u2} R M₂ (Semiring.toMonoidWithZero.{u5} R _inst_1) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u5, u2} R M₂ _inst_1 _inst_5 _inst_6))))] [_inst_9 : TopologicalSpace.{u3} S] [_inst_10 : TopologicalSpace.{u2} M₂] [_inst_11 : ContinuousSMul.{u3, u2} S M₂ (SMulZeroClass.toSMul.{u3, u2} S M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u2} S M₂ (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u2} S M₂ (Semiring.toMonoidWithZero.{u3} S _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u2} S M₂ _inst_2 _inst_5 _inst_7)))) _inst_9 _inst_10] [_inst_12 : TopologicalSpace.{u4} M] [_inst_13 : AddCommMonoid.{u4} M] [_inst_14 : Module.{u5, u4} R M _inst_1 _inst_13] [_inst_15 : ContinuousAdd.{u2} M₂ _inst_10 (AddZeroClass.toAdd.{u2} M₂ (AddMonoid.toAddZeroClass.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)))] [_inst_16 : Module.{u1, u2} T M₂ _inst_3 _inst_5] [_inst_17 : ContinuousConstSMul.{u1, u2} T M₂ _inst_10 (SMulZeroClass.toSMul.{u1, u2} T M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u2} T M₂ (MonoidWithZero.toZero.{u1} T (Semiring.toMonoidWithZero.{u1} T _inst_3)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u2} T M₂ (Semiring.toMonoidWithZero.{u1} T _inst_3) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u1, u2} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_18 : SMulCommClass.{u5, u1, u2} R T M₂ (SMulZeroClass.toSMul.{u5, u2} R M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u5, u2} R M₂ (MonoidWithZero.toZero.{u5} R (Semiring.toMonoidWithZero.{u5} R _inst_1)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u5, u2} R M₂ (Semiring.toMonoidWithZero.{u5} R _inst_1) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u5, u2} R M₂ _inst_1 _inst_5 _inst_6)))) (SMulZeroClass.toSMul.{u1, u2} T M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u2} T M₂ (MonoidWithZero.toZero.{u1} T (Semiring.toMonoidWithZero.{u1} T _inst_3)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u2} T M₂ (Semiring.toMonoidWithZero.{u1} T _inst_3) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u1, u2} T M₂ _inst_3 _inst_5 _inst_16))))] [_inst_19 : SMulCommClass.{u3, u1, u2} S T M₂ (SMulZeroClass.toSMul.{u3, u2} S M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u3, u2} S M₂ (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u3, u2} S M₂ (Semiring.toMonoidWithZero.{u3} S _inst_2) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u3, u2} S M₂ _inst_2 _inst_5 _inst_7)))) (SMulZeroClass.toSMul.{u1, u2} T M₂ (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (SMulWithZero.toSMulZeroClass.{u1, u2} T M₂ (MonoidWithZero.toZero.{u1} T (Semiring.toMonoidWithZero.{u1} T _inst_3)) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (MulActionWithZero.toSMulWithZero.{u1, u2} T M₂ (Semiring.toMonoidWithZero.{u1} T _inst_3) (AddMonoid.toZero.{u2} M₂ (AddCommMonoid.toAddMonoid.{u2} M₂ _inst_5)) (Module.toMulActionWithZero.{u1, u2} T M₂ _inst_3 _inst_5 _inst_16))))] (c : ContinuousLinearMap.{u5, u5, u4, u3} R R _inst_1 _inst_1 (RingHom.id.{u5} R (Semiring.toNonAssocSemiring.{u5} R _inst_1)) M _inst_12 _inst_13 S _inst_9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2))) _inst_14 _inst_4), Eq.{max (succ u4) (succ u2)} (forall (ᾰ : M₂), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₂) => ContinuousLinearMap.{u5, u5, u4, 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M₂) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M₂) => ContinuousLinearMap.{u5, u5, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u5} R (Semiring.toNonAssocSemiring.{u5} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, max u4 u2} T T M₂ (ContinuousLinearMap.{u5, u5, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u5} R (Semiring.toNonAssocSemiring.{u5} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6) _inst_3 _inst_3 _inst_5 (ContinuousLinearMap.addCommMonoid.{u5, u5, u4, u2} R R _inst_1 _inst_1 (RingHom.id.{u5} R (Semiring.toNonAssocSemiring.{u5} R _inst_1)) M _inst_12 _inst_13 M₂ _inst_10 _inst_5 _inst_14 _inst_6 _inst_15) _inst_16 (ContinuousLinearMap.module.{u5, u5, u1, u4, u2} R R T _inst_1 _inst_1 _inst_3 M _inst_12 _inst_13 _inst_14 M₂ _inst_10 _inst_5 _inst_6 _inst_16 _inst_18 _inst_17 (RingHom.id.{u5} R (Semiring.toNonAssocSemiring.{u5} R _inst_1)) _inst_15) (RingHom.id.{u1} T (Semiring.toNonAssocSemiring.{u1} T _inst_3))) (ContinuousLinearMap.smulRightₗ.{u5, u3, u1, u4, u2} R S T M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13 _inst_14 _inst_15 _inst_16 _inst_17 _inst_18 _inst_19 c)) (ContinuousLinearMap.smulRight.{u4, u2, u5, u3} M _inst_12 _inst_13 M₂ _inst_10 _inst_5 R S _inst_1 _inst_2 _inst_14 _inst_6 _inst_4 _inst_7 _inst_8 _inst_9 _inst_11 c) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_smul_rightₗ ContinuousLinearMap.coe_smulRightₗₓ'. -/ @[simp] theorem coe_smulRightₗ (c : M →L[R] S) : ⇑(smulRightₗ c : M₂ →ₗ[T] M →L[R] M₂) = c.smul_right := rfl @@ -1747,26 +2770,46 @@ variable {A M M₂ : Type _} [Ring A] [AddCommGroup M] [AddCommGroup M₂] [Modu [TopologicalSpace M] [TopologicalSpace M₂] (R : Type _) [Ring R] [Module R M] [Module R M₂] [LinearMap.CompatibleSMul M M₂ R A] +#print ContinuousLinearMap.restrictScalars /- /-- If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous `R`-linear map. We assume `linear_map.compatible_smul M M₂ R A` to match assumptions of `linear_map.map_smul_of_tower`. -/ def restrictScalars (f : M →L[A] M₂) : M →L[R] M₂ := ⟨(f : M →ₗ[A] M₂).restrictScalars R, f.Continuous⟩ #align continuous_linear_map.restrict_scalars ContinuousLinearMap.restrictScalars +-/ variable {R} +/- warning: continuous_linear_map.coe_restrict_scalars -> ContinuousLinearMap.coe_restrictScalars is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R 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[_inst_1 : Ring.{u4} A] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u2} M₂] [_inst_4 : Module.{u4, u3} A M (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_5 : Module.{u4, u2} A M₂ (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : TopologicalSpace.{u2} M₂] {R : Type.{u1}} [_inst_8 : Ring.{u1} R] [_inst_9 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_10 : Module.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u3, u2, u1, u4} M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) R A (Ring.toSemiring.{u4} A _inst_1) (SMulZeroClass.toSMul.{u1, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M 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R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_10)))) _inst_5] (f : ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u3) (succ u2)} (LinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.toLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 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Consider using '#align continuous_linear_map.coe_restrict_scalars ContinuousLinearMap.coe_restrictScalarsₓ'. -/ @[simp, norm_cast] theorem coe_restrictScalars (f : M →L[A] M₂) : (f.restrictScalars R : M →ₗ[R] M₂) = (f : M →ₗ[A] M₂).restrictScalars R := rfl #align continuous_linear_map.coe_restrict_scalars ContinuousLinearMap.coe_restrictScalars +/- warning: continuous_linear_map.coe_restrict_scalars' -> ContinuousLinearMap.coe_restrictScalars' is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] (f : ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u2) (succ u3)} (M -> M₂) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (fun (_x : ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) => M -> M₂) (ContinuousLinearMap.toFun.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) (coeFn.{max (succ u2) 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(AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) f) +but is expected to have type + forall {A : Type.{u4}} {M : Type.{u3}} {M₂ : Type.{u2}} [_inst_1 : Ring.{u4} A] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u2} M₂] [_inst_4 : Module.{u4, u3} A M (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_5 : Module.{u4, u2} A M₂ (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : TopologicalSpace.{u2} M₂] {R : Type.{u1}} [_inst_8 : Ring.{u1} R] [_inst_9 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_10 : Module.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u3, u2, u1, u4} M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) R A (Ring.toSemiring.{u4} A _inst_1) (SMulZeroClass.toSMul.{u1, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u3} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (Module.toMulActionWithZero.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u1, u2} R M₂ (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M₂ (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_10)))) _inst_5] (f : ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u3) (succ u2)} (forall (ᾰ : M), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) ᾰ) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) M M₂ _inst_6 _inst_7 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u1, u1, u3, u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10))) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u2, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) M M₂ _inst_6 _inst_7 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u2, u4, u4, u3, u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5))) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.coe_restrict_scalars' ContinuousLinearMap.coe_restrictScalars'ₓ'. -/ @[simp] -theorem coe_restrict_scalars' (f : M →L[A] M₂) : ⇑(f.restrictScalars R) = f := +theorem coe_restrictScalars' (f : M →L[A] M₂) : ⇑(f.restrictScalars R) = f := rfl -#align continuous_linear_map.coe_restrict_scalars' ContinuousLinearMap.coe_restrict_scalars' - +#align continuous_linear_map.coe_restrict_scalars' ContinuousLinearMap.coe_restrictScalars' + +/- warning: continuous_linear_map.restrict_scalars_zero -> ContinuousLinearMap.restrictScalars_zero is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5], Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (OfNat.ofNat.{max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) 0 (OfNat.mk.{max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) 0 (Zero.zero.{max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.zero.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5))))) (OfNat.ofNat.{max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) 0 (OfNat.mk.{max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) 0 (Zero.zero.{max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.zero.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10)))) +but is expected to have type + forall {A : Type.{u1}} {M : Type.{u4}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u4} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u4} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u4} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u2}} [_inst_8 : Ring.{u2} R] [_inst_9 : Module.{u2, u4} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2)] [_inst_10 : Module.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u4, u3, u2, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toSMul.{u2, u4} R M (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u4} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u4} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (Module.toMulActionWithZero.{u2, u4} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u2, u3} R M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M₂ (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5], Eq.{max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u4, u3, u2} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (OfNat.ofNat.{max u4 u3} (ContinuousLinearMap.{u1, u1, u4, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (NonAssocRing.toNonAssocSemiring.{u1} A (Ring.toNonAssocRing.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) 0 (Zero.toOfNat0.{max u4 u3} (ContinuousLinearMap.{u1, u1, u4, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (NonAssocRing.toNonAssocSemiring.{u1} A (Ring.toNonAssocRing.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.zero.{u1, u1, u4, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (NonAssocRing.toNonAssocSemiring.{u1} A (Ring.toNonAssocRing.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5)))) (OfNat.ofNat.{max u4 u3} (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) 0 (Zero.toOfNat0.{max u4 u3} (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.zero.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.restrict_scalars_zero ContinuousLinearMap.restrictScalars_zeroₓ'. -/ @[simp] theorem restrictScalars_zero : (0 : M →L[A] M₂).restrictScalars R = 0 := rfl @@ -1776,12 +2819,24 @@ section variable [TopologicalAddGroup M₂] +/- warning: continuous_linear_map.restrict_scalars_add -> ContinuousLinearMap.restrictScalars_add is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] [_inst_12 : TopologicalAddGroup.{u3} M₂ _inst_7 (AddCommGroup.toAddGroup.{u3} M₂ _inst_3)] (f : ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (g : ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (instHAdd.{max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.add.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5 (TopologicalAddGroup.to_continuousAdd.{u3} M₂ _inst_7 (AddCommGroup.toAddGroup.{u3} M₂ _inst_3) _inst_12))) f g)) (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (instHAdd.{max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.add.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10 (TopologicalAddGroup.to_continuousAdd.{u3} M₂ _inst_7 (AddCommGroup.toAddGroup.{u3} M₂ _inst_3) _inst_12))) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 g)) +but is expected to have type + forall {A : Type.{u4}} {M : Type.{u3}} {M₂ : Type.{u2}} [_inst_1 : Ring.{u4} A] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u2} M₂] [_inst_4 : Module.{u4, u3} A M (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_5 : Module.{u4, u2} A M₂ (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : TopologicalSpace.{u2} M₂] {R : Type.{u1}} [_inst_8 : Ring.{u1} R] [_inst_9 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_10 : Module.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u3, u2, u1, u4} M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) R A (Ring.toSemiring.{u4} A _inst_1) (SMulZeroClass.toSMul.{u1, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u3} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (Module.toMulActionWithZero.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u1, u2} R M₂ (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M₂ (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_10)))) _inst_5] [_inst_12 : TopologicalAddGroup.{u2} M₂ _inst_7 (AddCommGroup.toAddGroup.{u2} M₂ _inst_3)] (f : ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (g : ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u3) (succ u2)} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (HAdd.hAdd.{max u3 u2, max u3 u2, max u3 u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (instHAdd.{max u3 u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.add.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5 (TopologicalAddGroup.toContinuousAdd.{u2} M₂ _inst_7 (AddCommGroup.toAddGroup.{u2} M₂ _inst_3) _inst_12))) f g)) (HAdd.hAdd.{max u3 u2, max u3 u2, max u3 u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (instHAdd.{max u3 u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.add.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10 (TopologicalAddGroup.toContinuousAdd.{u2} M₂ _inst_7 (AddCommGroup.toAddGroup.{u2} M₂ _inst_3) _inst_12))) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 g)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.restrict_scalars_add ContinuousLinearMap.restrictScalars_addₓ'. -/ @[simp] theorem restrictScalars_add (f g : M →L[A] M₂) : (f + g).restrictScalars R = f.restrictScalars R + g.restrictScalars R := rfl #align continuous_linear_map.restrict_scalars_add ContinuousLinearMap.restrictScalars_add +/- warning: continuous_linear_map.restrict_scalars_neg -> ContinuousLinearMap.restrictScalars_neg is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] [_inst_12 : TopologicalAddGroup.{u3} M₂ _inst_7 (AddCommGroup.toAddGroup.{u3} M₂ _inst_3)] (f : ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (Neg.neg.{max u2 u3} (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.neg.{u1, u1, u2, u3} A _inst_1 A _inst_1 M _inst_6 _inst_2 M₂ _inst_7 _inst_3 _inst_4 _inst_5 (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) _inst_12) f)) (Neg.neg.{max u2 u3} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.neg.{u4, u4, u2, u3} R _inst_8 R _inst_8 M _inst_6 _inst_2 M₂ _inst_7 _inst_3 _inst_9 _inst_10 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) _inst_12) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) +but is expected to have type + forall {A : Type.{u4}} {M : Type.{u3}} {M₂ : Type.{u2}} [_inst_1 : Ring.{u4} A] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u2} M₂] [_inst_4 : Module.{u4, u3} A M (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_5 : Module.{u4, u2} A M₂ (Ring.toSemiring.{u4} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u3} M] [_inst_7 : TopologicalSpace.{u2} M₂] {R : Type.{u1}} [_inst_8 : Ring.{u1} R] [_inst_9 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_10 : Module.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u3, u2, u1, u4} M M₂ (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) R A (Ring.toSemiring.{u4} A _inst_1) (SMulZeroClass.toSMul.{u1, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u3} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_2))))) (Module.toMulActionWithZero.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u1, u2} R M₂ (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M₂ (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8))) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M₂ (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_8)) (NegZeroClass.toZero.{u2} M₂ (SubNegZeroMonoid.toNegZeroClass.{u2} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u2} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u2} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u2} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u2} R M₂ (Ring.toSemiring.{u1} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_10)))) _inst_5] [_inst_12 : TopologicalAddGroup.{u2} M₂ _inst_7 (AddCommGroup.toAddGroup.{u2} M₂ _inst_3)] (f : ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u3) (succ u2)} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (Neg.neg.{max u3 u2} (ContinuousLinearMap.{u4, u4, u3, u2} A A (Ring.toSemiring.{u4} A _inst_1) (Ring.toSemiring.{u4} A _inst_1) (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.neg.{u4, u4, u3, u2} A _inst_1 A _inst_1 M _inst_6 _inst_2 M₂ _inst_7 _inst_3 _inst_4 _inst_5 (RingHom.id.{u4} A (NonAssocRing.toNonAssocSemiring.{u4} A (Ring.toNonAssocRing.{u4} A _inst_1))) _inst_12) f)) (Neg.neg.{max u3 u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_8) (Ring.toSemiring.{u1} R _inst_8) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.neg.{u1, u1, u3, u2} R _inst_8 R _inst_8 M _inst_6 _inst_2 M₂ _inst_7 _inst_3 _inst_9 _inst_10 (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_8))) _inst_12) (ContinuousLinearMap.restrictScalars.{u4, u3, u2, u1} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.restrict_scalars_neg ContinuousLinearMap.restrictScalars_negₓ'. -/ @[simp] theorem restrictScalars_neg (f : M →L[A] M₂) : (-f).restrictScalars R = -f.restrictScalars R := rfl @@ -1792,6 +2847,12 @@ end variable {S : Type _} [Ring S] [Module S M₂] [ContinuousConstSMul S M₂] [SMulCommClass A S M₂] [SMulCommClass R S M₂] +/- warning: continuous_linear_map.restrict_scalars_smul -> ContinuousLinearMap.restrictScalars_smul is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] {S : Type.{u5}} [_inst_12 : Ring.{u5} S] [_inst_13 : Module.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_14 : ContinuousConstSMul.{u5, u3} S M₂ _inst_7 (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_15 : SMulCommClass.{u1, u5, u3} A S M₂ (SMulZeroClass.toHasSmul.{u1, u3} A M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u1, u3} A M₂ (MulZeroClass.toHasZero.{u1} A (MulZeroOneClass.toMulZeroClass.{u1} A (MonoidWithZero.toMulZeroOneClass.{u1} A (Semiring.toMonoidWithZero.{u1} A (Ring.toSemiring.{u1} A _inst_1))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M₂ (Semiring.toMonoidWithZero.{u1} A (Ring.toSemiring.{u1} A _inst_1)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5)))) (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_16 : SMulCommClass.{u4, u5, u3} R S M₂ (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] (c : S) (f : ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u2) (succ u3)} (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (SMul.smul.{u5, max u2 u3} S (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (MulAction.toHasSmul.{u5, max u2 u3} S (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (Ring.toMonoid.{u5} S _inst_12) (ContinuousLinearMap.mulAction.{u1, u1, u2, u3, u5} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5 S (Ring.toMonoid.{u5} S _inst_12) (Module.toDistribMulAction.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13) _inst_15 _inst_14)) c f)) (SMul.smul.{u5, max u2 u3} S (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (MulAction.toHasSmul.{u5, max u2 u3} S (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (Ring.toMonoid.{u5} S _inst_12) (ContinuousLinearMap.mulAction.{u4, u4, u2, u3, u5} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10 S (Ring.toMonoid.{u5} S _inst_12) (Module.toDistribMulAction.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13) _inst_16 _inst_14)) c (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) +but is expected to have type + forall {A : Type.{u5}} {M : Type.{u4}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u5} A] [_inst_2 : AddCommGroup.{u4} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u5, u4} A M (Ring.toSemiring.{u5} A _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2)] [_inst_5 : Module.{u5, u3} A M₂ (Ring.toSemiring.{u5} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u4} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u2}} [_inst_8 : Ring.{u2} R] [_inst_9 : Module.{u2, u4} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2)] [_inst_10 : Module.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u4, u3, u2, u5} M M₂ (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u5} A _inst_1) (SMulZeroClass.toSMul.{u2, u4} R M (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u4} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u4} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u4} M (SubNegZeroMonoid.toNegZeroClass.{u4} M (SubtractionMonoid.toSubNegZeroMonoid.{u4} M (SubtractionCommMonoid.toSubtractionMonoid.{u4} M (AddCommGroup.toDivisionAddCommMonoid.{u4} M _inst_2))))) (Module.toMulActionWithZero.{u2, u4} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u2, u3} R M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M₂ (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] {S : Type.{u1}} [_inst_12 : Ring.{u1} S] [_inst_13 : Module.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_14 : ContinuousConstSMul.{u1, u3} S M₂ _inst_7 (SMulZeroClass.toSMul.{u1, u3} S M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_15 : SMulCommClass.{u5, u1, u3} A S M₂ (SMulZeroClass.toSMul.{u5, u3} A M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u5, u3} A M₂ (MonoidWithZero.toZero.{u5} A (Semiring.toMonoidWithZero.{u5} A (Ring.toSemiring.{u5} A _inst_1))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u5, u3} A M₂ (Semiring.toMonoidWithZero.{u5} A (Ring.toSemiring.{u5} A _inst_1)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u5, u3} A M₂ (Ring.toSemiring.{u5} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5)))) (SMulZeroClass.toSMul.{u1, u3} S M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_16 : SMulCommClass.{u2, u1, u3} R S M₂ (SMulZeroClass.toSMul.{u2, u3} R M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M₂ (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u2, u3} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) (SMulZeroClass.toSMul.{u1, u3} S M₂ (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u3} M₂ (SubNegZeroMonoid.toNegZeroClass.{u3} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u3} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u3} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u3} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] (c : S) (f : ContinuousLinearMap.{u5, u5, u4, u3} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5), Eq.{max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.restrictScalars.{u5, u4, u3, u2} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 (HSMul.hSMul.{u1, max u3 u4, max u3 u4} S (ContinuousLinearMap.{u5, u5, u4, u3} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u5, u5, u4, u3} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (instHSMul.{u1, max u3 u4} S (ContinuousLinearMap.{u5, u5, u4, u3} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (MulAction.toSMul.{u1, max u3 u4} S (ContinuousLinearMap.{u5, u5, u4, u3} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (MonoidWithZero.toMonoid.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (ContinuousLinearMap.mulAction.{u5, u5, u4, u3, u1} A A (Ring.toSemiring.{u5} A _inst_1) (Ring.toSemiring.{u5} A _inst_1) (RingHom.id.{u5} A (NonAssocRing.toNonAssocSemiring.{u5} A (Ring.toNonAssocRing.{u5} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5 S (MonoidWithZero.toMonoid.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (Module.toDistribMulAction.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13) _inst_15 _inst_14))) c f)) (HSMul.hSMul.{u1, max u3 u4, max u3 u4} S (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (instHSMul.{u1, max u3 u4} S (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (MulAction.toSMul.{u1, max u3 u4} S (ContinuousLinearMap.{u2, u2, u4, u3} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (MonoidWithZero.toMonoid.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (ContinuousLinearMap.mulAction.{u2, u2, u4, u3, u1} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10 S (MonoidWithZero.toMonoid.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (Module.toDistribMulAction.{u1, u3} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13) _inst_16 _inst_14))) c (ContinuousLinearMap.restrictScalars.{u5, u4, u3, u2} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 f)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.restrict_scalars_smul ContinuousLinearMap.restrictScalars_smulₓ'. -/ @[simp] theorem restrictScalars_smul (c : S) (f : M →L[A] M₂) : (c • f).restrictScalars R = c • f.restrictScalars R := @@ -1800,6 +2861,7 @@ theorem restrictScalars_smul (c : S) (f : M →L[A] M₂) : variable (A M M₂ R S) [TopologicalAddGroup M₂] +#print ContinuousLinearMap.restrictScalarsₗ /- /-- `continuous_linear_map.restrict_scalars` as a `linear_map`. See also `continuous_linear_map.restrict_scalarsL`. -/ def restrictScalarsₗ : (M →L[A] M₂) →ₗ[S] M →L[R] M₂ @@ -1808,9 +2870,16 @@ def restrictScalarsₗ : (M →L[A] M₂) →ₗ[S] M →L[R] M₂ map_add' := restrictScalars_add map_smul' := restrictScalars_smul #align continuous_linear_map.restrict_scalarsₗ ContinuousLinearMap.restrictScalarsₗ +-/ variable {A M M₂ R S} +/- warning: continuous_linear_map.coe_restrict_scalarsₗ -> ContinuousLinearMap.coe_restrictScalarsₗ is a dubious translation: +lean 3 declaration is + forall {A : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : Ring.{u1} A] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} M₂] [_inst_4 : Module.{u1, u2} A M (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_5 : Module.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u2} M] [_inst_7 : TopologicalSpace.{u3} M₂] {R : Type.{u4}} [_inst_8 : Ring.{u4} R] [_inst_9 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_10 : Module.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u2, u3, u4, u1} M M₂ (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) R A (Ring.toSemiring.{u1} A _inst_1) (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u4, u2} R M (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) _inst_5] {S : Type.{u5}} [_inst_12 : Ring.{u5} S] [_inst_13 : Module.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)] [_inst_14 : ContinuousConstSMul.{u5, u3} S M₂ _inst_7 (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_15 : SMulCommClass.{u1, u5, u3} A S M₂ (SMulZeroClass.toHasSmul.{u1, u3} A M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u1, u3} A M₂ (MulZeroClass.toHasZero.{u1} A (MulZeroOneClass.toMulZeroClass.{u1} A (MonoidWithZero.toMulZeroOneClass.{u1} A (Semiring.toMonoidWithZero.{u1} A (Ring.toSemiring.{u1} A _inst_1))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M₂ (Semiring.toMonoidWithZero.{u1} A (Ring.toSemiring.{u1} A _inst_1)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u1, u3} A M₂ (Ring.toSemiring.{u1} A _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5)))) (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_16 : SMulCommClass.{u4, u5, u3} R S M₂ (SMulZeroClass.toHasSmul.{u4, u3} R M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u4, u3} R M₂ (MulZeroClass.toHasZero.{u4} R (MulZeroOneClass.toMulZeroClass.{u4} R (MonoidWithZero.toMulZeroOneClass.{u4} R (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u4, u3} R M₂ (Semiring.toMonoidWithZero.{u4} R (Ring.toSemiring.{u4} R _inst_8)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u4, u3} R M₂ (Ring.toSemiring.{u4} R _inst_8) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10)))) (SMulZeroClass.toHasSmul.{u5, u3} S M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (SMulWithZero.toSmulZeroClass.{u5, u3} S M₂ (MulZeroClass.toHasZero.{u5} S (MulZeroOneClass.toMulZeroClass.{u5} S (MonoidWithZero.toMulZeroOneClass.{u5} S (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12))))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (MulActionWithZero.toSMulWithZero.{u5, u3} S M₂ (Semiring.toMonoidWithZero.{u5} S (Ring.toSemiring.{u5} S _inst_12)) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3)))) (Module.toMulActionWithZero.{u5, u3} S M₂ (Ring.toSemiring.{u5} S _inst_12) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_13))))] [_inst_17 : TopologicalAddGroup.{u3} M₂ _inst_7 (AddCommGroup.toAddGroup.{u3} M₂ _inst_3)], Eq.{succ (max u2 u3)} ((ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) -> (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10)) (coeFn.{succ (max u2 u3), succ (max u2 u3)} (LinearMap.{u5, u5, max u2 u3, max u2 u3} S S (Ring.toSemiring.{u5} S _inst_12) (Ring.toSemiring.{u5} S _inst_12) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (Ring.toSemiring.{u5} S _inst_12))) (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) 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_inst_1) (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u5} S _inst_12) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_4 M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_13 _inst_15 _inst_14 (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) (ContinuousLinearMap.restrictScalarsₗ._proof_3.{u3} M₂ _inst_3 _inst_7 _inst_17)) (ContinuousLinearMap.module.{u4, u4, u5, u2, u3} R R S (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u5} S _inst_12) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9 M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10 _inst_13 _inst_16 _inst_14 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) (ContinuousLinearMap.restrictScalarsₗ._proof_4.{u3} M₂ _inst_3 _inst_7 _inst_17))) => (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) -> (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10)) (LinearMap.hasCoeToFun.{u5, u5, max u2 u3, max u2 u3} S S (ContinuousLinearMap.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5) (ContinuousLinearMap.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10) (Ring.toSemiring.{u5} S _inst_12) (Ring.toSemiring.{u5} S _inst_12) (ContinuousLinearMap.addCommMonoid.{u1, u1, u2, u3} A A (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_4 _inst_5 (ContinuousLinearMap.restrictScalarsₗ._proof_1.{u3} M₂ _inst_3 _inst_7 _inst_17)) (ContinuousLinearMap.addCommMonoid.{u4, u4, u2, u3} R R (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_9 _inst_10 (ContinuousLinearMap.restrictScalarsₗ._proof_2.{u3} M₂ _inst_3 _inst_7 _inst_17)) (ContinuousLinearMap.module.{u1, u1, u5, u2, u3} A A S (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u1} A _inst_1) (Ring.toSemiring.{u5} S _inst_12) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_4 M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_5 _inst_13 _inst_15 _inst_14 (RingHom.id.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_1))) (ContinuousLinearMap.restrictScalarsₗ._proof_3.{u3} M₂ _inst_3 _inst_7 _inst_17)) (ContinuousLinearMap.module.{u4, u4, u5, u2, u3} R R S (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u4} R _inst_8) (Ring.toSemiring.{u5} S _inst_12) M _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_9 M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_3) _inst_10 _inst_13 _inst_16 _inst_14 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (Ring.toSemiring.{u4} R _inst_8))) (ContinuousLinearMap.restrictScalarsₗ._proof_4.{u3} M₂ _inst_3 _inst_7 _inst_17)) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (Ring.toSemiring.{u5} S _inst_12)))) (ContinuousLinearMap.restrictScalarsₗ.{u1, u2, u3, u4, u5} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11 S _inst_12 _inst_13 _inst_14 _inst_15 _inst_16 _inst_17)) (ContinuousLinearMap.restrictScalars.{u1, u2, u3, u4} A M M₂ _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 R _inst_8 _inst_9 _inst_10 _inst_11) +but is expected to have type + forall {A : Type.{u3}} {M : Type.{u5}} {M₂ : Type.{u4}} [_inst_1 : Ring.{u3} A] [_inst_2 : AddCommGroup.{u5} M] [_inst_3 : AddCommGroup.{u4} M₂] [_inst_4 : Module.{u3, u5} A M (Ring.toSemiring.{u3} A _inst_1) (AddCommGroup.toAddCommMonoid.{u5} M _inst_2)] [_inst_5 : Module.{u3, u4} A M₂ (Ring.toSemiring.{u3} A _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3)] [_inst_6 : TopologicalSpace.{u5} M] [_inst_7 : TopologicalSpace.{u4} M₂] {R : Type.{u2}} [_inst_8 : Ring.{u2} R] [_inst_9 : Module.{u2, u5} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u5} M _inst_2)] [_inst_10 : Module.{u2, u4} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3)] [_inst_11 : LinearMap.CompatibleSMul.{u5, u4, u2, u3} M M₂ (AddCommGroup.toAddCommMonoid.{u5} M _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) R A (Ring.toSemiring.{u3} A _inst_1) (SMulZeroClass.toSMul.{u2, u5} R M (NegZeroClass.toZero.{u5} M (SubNegZeroMonoid.toNegZeroClass.{u5} M (SubtractionMonoid.toSubNegZeroMonoid.{u5} M (SubtractionCommMonoid.toSubtractionMonoid.{u5} M (AddCommGroup.toDivisionAddCommMonoid.{u5} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u2, u5} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u5} M (SubNegZeroMonoid.toNegZeroClass.{u5} M (SubtractionMonoid.toSubNegZeroMonoid.{u5} M (SubtractionCommMonoid.toSubtractionMonoid.{u5} M (AddCommGroup.toDivisionAddCommMonoid.{u5} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u2, u5} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u5} M (SubNegZeroMonoid.toNegZeroClass.{u5} M (SubtractionMonoid.toSubNegZeroMonoid.{u5} M (SubtractionCommMonoid.toSubtractionMonoid.{u5} M (AddCommGroup.toDivisionAddCommMonoid.{u5} M _inst_2))))) (Module.toMulActionWithZero.{u2, u5} R M (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u5} M _inst_2) _inst_9)))) _inst_4 (SMulZeroClass.toSMul.{u2, u4} R M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u2, u4} R M₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u2, u4} R M₂ (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u2, u4} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_10)))) _inst_5] {S : Type.{u1}} [_inst_12 : Ring.{u1} S] [_inst_13 : Module.{u1, u4} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3)] [_inst_14 : ContinuousConstSMul.{u1, u4} S M₂ _inst_7 (SMulZeroClass.toSMul.{u1, u4} S M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u4} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u4} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u4} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_13))))] [_inst_15 : SMulCommClass.{u3, u1, u4} A S M₂ (SMulZeroClass.toSMul.{u3, u4} A M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u3, u4} A M₂ (MonoidWithZero.toZero.{u3} A (Semiring.toMonoidWithZero.{u3} A (Ring.toSemiring.{u3} A _inst_1))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u3, u4} A M₂ (Semiring.toMonoidWithZero.{u3} A (Ring.toSemiring.{u3} A _inst_1)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u3, u4} A M₂ (Ring.toSemiring.{u3} A _inst_1) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_5)))) (SMulZeroClass.toSMul.{u1, u4} S M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u4} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u4} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u4} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_13))))] [_inst_16 : SMulCommClass.{u2, u1, u4} R S M₂ (SMulZeroClass.toSMul.{u2, u4} R M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u2, u4} R M₂ (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u2, u4} R M₂ (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_8)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u2, u4} R M₂ (Ring.toSemiring.{u2} R _inst_8) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_10)))) (SMulZeroClass.toSMul.{u1, u4} S M₂ (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (SMulWithZero.toSMulZeroClass.{u1, u4} S M₂ (MonoidWithZero.toZero.{u1} S (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12))) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (MulActionWithZero.toSMulWithZero.{u1, u4} S M₂ (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S _inst_12)) (NegZeroClass.toZero.{u4} M₂ (SubNegZeroMonoid.toNegZeroClass.{u4} M₂ (SubtractionMonoid.toSubNegZeroMonoid.{u4} M₂ (SubtractionCommMonoid.toSubtractionMonoid.{u4} M₂ (AddCommGroup.toDivisionAddCommMonoid.{u4} M₂ _inst_3))))) (Module.toMulActionWithZero.{u1, u4} S M₂ (Ring.toSemiring.{u1} S _inst_12) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_13))))] [_inst_17 : TopologicalAddGroup.{u4} M₂ _inst_7 (AddCommGroup.toAddGroup.{u4} M₂ _inst_3)], Eq.{max (succ u5) (succ u4)} (forall (ᾰ : ContinuousLinearMap.{u3, u3, u5, u4} A A (Ring.toSemiring.{u3} A _inst_1) (Ring.toSemiring.{u3} A _inst_1) (RingHom.id.{u3} A (NonAssocRing.toNonAssocSemiring.{u3} A (Ring.toNonAssocRing.{u3} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_4 _inst_5), (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : ContinuousLinearMap.{u3, u3, u5, u4} A A (Ring.toSemiring.{u3} A _inst_1) (Ring.toSemiring.{u3} A _inst_1) (RingHom.id.{u3} A (NonAssocRing.toNonAssocSemiring.{u3} A (Ring.toNonAssocRing.{u3} A _inst_1))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_4 _inst_5) => ContinuousLinearMap.{u2, u2, u5, u4} R R (Ring.toSemiring.{u2} R _inst_8) (Ring.toSemiring.{u2} R _inst_8) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_8))) M _inst_6 (AddCommGroup.toAddCommMonoid.{u5} M _inst_2) M₂ _inst_7 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_3) _inst_9 _inst_10) ᾰ) (FunLike.coe.{max (succ u5) (succ 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Consider using '#align continuous_linear_map.coe_restrict_scalarsₗ ContinuousLinearMap.coe_restrictScalarsₗₓ'. -/ @[simp] theorem coe_restrictScalarsₗ : ⇑(restrictScalarsₗ A M M₂ R S) = restrictScalars R := rfl @@ -1836,10 +2905,12 @@ variable {R₁ : Type _} {R₂ : Type _} {R₃ : Type _} [Semiring R₁] [Semiri include σ₂₁ +#print ContinuousLinearEquiv.toContinuousLinearMap /- /-- A continuous linear equivalence induces a continuous linear map. -/ def toContinuousLinearMap (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁₂] M₂ := { e.toLinearEquiv.toLinearMap with cont := e.continuous_toFun } #align continuous_linear_equiv.to_continuous_linear_map ContinuousLinearEquiv.toContinuousLinearMap +-/ /-- Coerce continuous linear equivs to continuous linear maps. -/ instance : Coe (M₁ ≃SL[σ₁₂] M₂) (M₁ →SL[σ₁₂] M₂) := @@ -1867,82 +2938,181 @@ instance : ContinuousSemilinearEquivClass (M₁ ≃SL[σ₁₂] M₂) σ₁₂ M instance : CoeFun (M₁ ≃SL[σ₁₂] M₂) fun _ => M₁ → M₂ := ⟨fun f => f⟩ +/- warning: continuous_linear_equiv.coe_def_rev clashes with [anonymous] -> [anonymous] +warning: continuous_linear_equiv.coe_def_rev -> [anonymous] is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u_1}} {R₂ : Type.{u_2}} [_inst_1 : Semiring.{u_1} R₁] [_inst_2 : Semiring.{u_2} R₂] {σ₁₂ : RingHom.{u_1, u_2} R₁ R₂ (Semiring.toNonAssocSemiring.{u_1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u_2} R₂ _inst_2)} {σ₂₁ : RingHom.{u_2, u_1} R₂ R₁ (Semiring.toNonAssocSemiring.{u_2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u_1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u_1, u_2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u_2, u_1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u_4}} [_inst_12 : TopologicalSpace.{u_4} M₁] [_inst_13 : AddCommMonoid.{u_4} M₁] {M₂ : Type.{u_6}} [_inst_16 : TopologicalSpace.{u_6} M₂] [_inst_17 : AddCommMonoid.{u_6} M₂] [_inst_22 : Module.{u_1, u_4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u_2, u_6} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u_4) (succ u_6)} (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.toContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) ((fun (a : Sort.{max (succ u_4) (succ u_6)}) (b : Sort.{max (succ u_4) (succ u_6)}) [self : HasLiftT.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} a b] => self.0) (ContinuousLinearEquiv.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (HasLiftT.mk.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearEquiv.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (CoeTCₓ.coe.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearEquiv.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (coeBase.{max (succ u_4) (succ u_6), max (succ u_4) (succ u_6)} (ContinuousLinearEquiv.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u_1, u_2, u_4, u_6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +but is expected to have type + forall {R₁ : Type.{u}} {R₂ : Type.{v}}, (Nat -> R₁ -> R₂) -> Nat -> (List.{u} R₁) -> (List.{v} R₂) +Case conversion may be inaccurate. 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_inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e b) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (b : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) b) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) b) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e b) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_apply ContinuousLinearEquiv.coe_applyₓ'. -/ theorem coe_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : (e : M₁ →SL[σ₁₂] M₂) b = e b := rfl #align continuous_linear_equiv.coe_apply ContinuousLinearEquiv.coe_apply +/- warning: continuous_linear_equiv.coe_to_linear_equiv -> ContinuousLinearEquiv.coe_toLinearEquiv is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (f : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) (fun (_x : LinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (LinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_13 _inst_17 _inst_22 _inst_24 σ₁₂ σ₂₁ _inst_4 _inst_5) (ContinuousLinearEquiv.toLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f)) (coeFn.{max (succ u3) (succ 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_inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (f : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (LinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M₁) => M₂) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (LinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) M₁ M₂ (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (LinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) M₁ M₂ (AddEquivClass.toEquivLike.{max u2 u1, u2, u1} (LinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) M₁ M₂ (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))) (AddZeroClass.toAdd.{u1} M₂ (AddMonoid.toAddZeroClass.{u1} M₂ (AddCommMonoid.toAddMonoid.{u1} M₂ _inst_17))) (SemilinearEquivClass.toAddEquivClass.{max u2 u1, u4, u3, u2, u1} (LinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24 (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_13 _inst_17 _inst_22 _inst_24 σ₁₂ σ₂₁ _inst_4 _inst_5))))) (ContinuousLinearEquiv.toLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_to_linear_equiv ContinuousLinearEquiv.coe_toLinearEquivₓ'. -/ @[simp] theorem coe_toLinearEquiv (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.toLinearEquiv = f := rfl #align continuous_linear_equiv.coe_to_linear_equiv ContinuousLinearEquiv.coe_toLinearEquiv +/- warning: continuous_linear_equiv.coe_coe -> ContinuousLinearEquiv.coe_coe is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => 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+but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (forall (ᾰ : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) ᾰ) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_coe ContinuousLinearEquiv.coe_coeₓ'. -/ @[simp, norm_cast] theorem coe_coe (e : M₁ ≃SL[σ₁₂] M₂) : ⇑(e : M₁ →SL[σ₁₂] M₂) = e := rfl #align continuous_linear_equiv.coe_coe ContinuousLinearEquiv.coe_coe +/- warning: continuous_linear_equiv.to_linear_equiv_injective -> ContinuousLinearEquiv.toLinearEquiv_injective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17], Function.Injective.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (LinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.toLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {σ₂₁ : RingHom.{u1, u2} R₂ R₁ (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] [_inst_22 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_17], Function.Injective.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (LinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ M₂ _inst_13 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.toLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.to_linear_equiv_injective ContinuousLinearEquiv.toLinearEquiv_injectiveₓ'. -/ theorem toLinearEquiv_injective : Function.Injective (toLinearEquiv : (M₁ ≃SL[σ₁₂] M₂) → M₁ ≃ₛₗ[σ₁₂] M₂) | ⟨e, _, _⟩, ⟨e', _, _⟩, rfl => rfl #align continuous_linear_equiv.to_linear_equiv_injective ContinuousLinearEquiv.toLinearEquiv_injective +/- warning: continuous_linear_equiv.ext -> ContinuousLinearEquiv.ext is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] {f : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24} {g : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24}, (Eq.{max (succ u3) (succ u4)} ((fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) f) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : 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M₂ _inst_16 _inst_17 _inst_22 _inst_24) f g) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] {f : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24} {g : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24}, (Eq.{max (succ u2) (succ u1)} (forall (a : M₁), (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) f) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) g)) -> (Eq.{max (succ u2) (succ u1)} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) f g) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.ext ContinuousLinearEquiv.extₓ'. -/ @[ext] theorem ext {f g : M₁ ≃SL[σ₁₂] M₂} (h : (f : M₁ → M₂) = g) : f = g := toLinearEquiv_injective <| LinearEquiv.ext <| congr_fun h #align continuous_linear_equiv.ext ContinuousLinearEquiv.ext +/- warning: continuous_linear_equiv.coe_injective -> ContinuousLinearEquiv.coe_injective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17], Function.Injective.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))))) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {σ₂₁ : RingHom.{u1, u2} R₂ R₁ (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] [_inst_22 : Module.{u2, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u1, u3} R₂ M₂ _inst_2 _inst_17], Function.Injective.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.toContinuousLinearMap.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_injective ContinuousLinearEquiv.coe_injectiveₓ'. -/ theorem coe_injective : Function.Injective (coe : (M₁ ≃SL[σ₁₂] M₂) → M₁ →SL[σ₁₂] M₂) := fun e e' h => ext <| funext <| ContinuousLinearMap.ext_iff.1 h #align continuous_linear_equiv.coe_injective ContinuousLinearEquiv.coe_injective +/- warning: continuous_linear_equiv.coe_inj -> ContinuousLinearEquiv.coe_inj is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] {e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24} {e' : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24}, Iff (Eq.{max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max (succ u3) (succ u4)}) [self : HasLiftT.{max (succ u3) (succ u4), max (succ u3) (succ u4)} a b] => self.0) (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 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_inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24} {e' : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24}, Iff (Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e')) (Eq.{max (succ u2) (succ u1)} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e e') +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_inj ContinuousLinearEquiv.coe_injₓ'. -/ @[simp, norm_cast] theorem coe_inj {e e' : M₁ ≃SL[σ₁₂] M₂} : (e : M₁ →SL[σ₁₂] M₂) = e' ↔ e = e' := coe_injective.eq_iff #align continuous_linear_equiv.coe_inj ContinuousLinearEquiv.coe_inj +#print ContinuousLinearEquiv.toHomeomorph /- /-- A continuous linear equivalence induces a homeomorphism. -/ def toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : M₁ ≃ₜ M₂ := { e with toEquiv := e.toLinearEquiv.toEquiv } #align continuous_linear_equiv.to_homeomorph ContinuousLinearEquiv.toHomeomorph +-/ +/- warning: continuous_linear_equiv.coe_to_homeomorph -> ContinuousLinearEquiv.coe_toHomeomorph is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u3) (succ u4)} (M₁ -> M₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (Homeomorph.{u3, u4} M₁ M₂ _inst_12 _inst_16) (fun (_x : Homeomorph.{u3, u4} M₁ M₂ _inst_12 _inst_16) => M₁ -> M₂) (Homeomorph.hasCoeToFun.{u3, u4} M₁ M₂ _inst_12 _inst_16) (ContinuousLinearEquiv.toHomeomorph.{u1, u2, u3, u4} R₁ R₂ _inst_1 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_inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (M₁ -> M₂) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} M₁ M₂ _inst_12 _inst_16) M₁ (fun (_x : M₁) => M₂) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} M₁ M₂ _inst_12 _inst_16) M₁ M₂ (EquivLike.toEmbeddingLike.{max (succ u2) (succ u1), succ u2, succ u1} (Homeomorph.{u2, u1} M₁ M₂ _inst_12 _inst_16) M₁ M₂ 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Consider using '#align continuous_linear_equiv.coe_to_homeomorph ContinuousLinearEquiv.coe_toHomeomorphₓ'. -/ @[simp] theorem coe_toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toHomeomorph = e := rfl #align continuous_linear_equiv.coe_to_homeomorph ContinuousLinearEquiv.coe_toHomeomorph +/- warning: continuous_linear_equiv.image_closure -> ContinuousLinearEquiv.image_closure is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u3} M₁), Eq.{succ u4} (Set.{u4} M₂) (Set.image.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (closure.{u3} M₁ _inst_12 s)) (closure.{u4} M₂ 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_inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u2} M₁), Eq.{succ u1} (Set.{u1} M₂) (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ 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(closure.{u1} M₂ _inst_16 (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.image_closure ContinuousLinearEquiv.image_closureₓ'. -/ theorem image_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e '' closure s = closure (e '' s) := e.toHomeomorph.image_closure s #align continuous_linear_equiv.image_closure ContinuousLinearEquiv.image_closure +/- warning: continuous_linear_equiv.preimage_closure -> ContinuousLinearEquiv.preimage_closure is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u4} M₂), Eq.{succ u3} (Set.{u3} M₁) (Set.preimage.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (closure.{u4} M₂ _inst_16 s)) (closure.{u3} M₁ _inst_12 (Set.preimage.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u1} M₂), Eq.{succ u2} (Set.{u2} M₁) (Set.preimage.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) (closure.{u1} M₂ _inst_16 s)) (closure.{u2} M₁ _inst_12 (Set.preimage.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.preimage_closure ContinuousLinearEquiv.preimage_closureₓ'. -/ theorem preimage_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e ⁻¹' closure s = closure (e ⁻¹' s) := e.toHomeomorph.preimage_closure s #align continuous_linear_equiv.preimage_closure ContinuousLinearEquiv.preimage_closure +/- warning: continuous_linear_equiv.is_closed_image -> ContinuousLinearEquiv.isClosed_image is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u3} M₁}, Iff (IsClosed.{u4} M₂ _inst_16 (Set.image.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s)) (IsClosed.{u3} M₁ _inst_12 s) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u2} M₁}, Iff (IsClosed.{u1} M₂ _inst_16 (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s)) (IsClosed.{u2} M₁ _inst_12 s) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.is_closed_image ContinuousLinearEquiv.isClosed_imageₓ'. -/ @[simp] theorem isClosed_image (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : IsClosed (e '' s) ↔ IsClosed s := e.toHomeomorph.isClosed_image #align continuous_linear_equiv.is_closed_image ContinuousLinearEquiv.isClosed_image +/- warning: continuous_linear_equiv.map_nhds_eq -> ContinuousLinearEquiv.map_nhds_eq is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u4} (Filter.{u4} M₂) (Filter.map.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (nhds.{u3} M₁ _inst_12 x)) (nhds.{u4} M₂ _inst_16 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e x)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u1} (Filter.{u1} M₂) (Filter.map.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) (nhds.{u2} M₁ _inst_12 x)) (nhds.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_nhds_eq ContinuousLinearEquiv.map_nhds_eqₓ'. -/ theorem map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e (𝓝 x) = 𝓝 (e x) := e.toHomeomorph.map_nhds_eq x #align continuous_linear_equiv.map_nhds_eq ContinuousLinearEquiv.map_nhds_eq +/- warning: continuous_linear_equiv.map_zero -> ContinuousLinearEquiv.map_zero is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e (OfNat.ofNat.{u3} M₁ 0 (OfNat.mk.{u3} M₁ 0 (Zero.zero.{u3} M₁ (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_13))))))) (OfNat.ofNat.{u4} M₂ 0 (OfNat.mk.{u4} M₂ 0 (Zero.zero.{u4} M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_17)))))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) (AddMonoid.toZero.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) (AddCommMonoid.toAddMonoid.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) _inst_17)))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_zero ContinuousLinearEquiv.map_zeroₓ'. -/ -- Make some straightforward lemmas available to `simp`. @[simp] theorem map_zero (e : M₁ ≃SL[σ₁₂] M₂) : e (0 : M₁) = 0 := (e : M₁ →SL[σ₁₂] M₂).map_zero #align continuous_linear_equiv.map_zero ContinuousLinearEquiv.map_zero +/- warning: continuous_linear_equiv.map_add -> ContinuousLinearEquiv.map_add is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁) (y : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e (HAdd.hAdd.{u3, u3, u3} M₁ M₁ M₁ (instHAdd.{u3} M₁ (AddZeroClass.toHasAdd.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_13)))) x y)) (HAdd.hAdd.{u4, u4, u4} M₂ M₂ M₂ (instHAdd.{u4} M₂ (AddZeroClass.toHasAdd.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_17)))) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e x) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 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TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁) (y : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (HAdd.hAdd.{u2, u2, u2} M₁ M₁ M₁ (instHAdd.{u2} M₁ (AddZeroClass.toAdd.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)))) x y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 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_inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e y)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_add ContinuousLinearEquiv.map_addₓ'. -/ @[simp] theorem map_add (e : M₁ ≃SL[σ₁₂] M₂) (x y : M₁) : e (x + y) = e x + e y := (e : M₁ →SL[σ₁₂] M₂).map_add x y #align continuous_linear_equiv.map_add ContinuousLinearEquiv.map_add +/- warning: continuous_linear_equiv.map_smulₛₗ -> ContinuousLinearEquiv.map_smulₛₗ is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (c : R₁) (x : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e (SMul.smul.{u1, u3} R₁ M₁ (SMulZeroClass.toHasSmul.{u1, u3} R₁ M₁ (AddZeroClass.toHasZero.{u3} M₁ 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Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (c : R₁) (x : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (HSMul.hSMul.{u4, u2, u2} R₁ M₁ M₁ (instHSMul.{u4, u2} R₁ M₁ (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_13 _inst_22))))) c x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e (HSMul.hSMul.{u4, u2, u2} R₁ M₁ M₁ (instHSMul.{u4, u2} R₁ M₁ (SMulZeroClass.toSMul.{u4, u2} R₁ M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (SMulWithZero.toSMulZeroClass.{u4, u2} R₁ M₁ (MonoidWithZero.toZero.{u4} R₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1)) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (MulActionWithZero.toSMulWithZero.{u4, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u4} R₁ _inst_1) (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13)) (Module.toMulActionWithZero.{u4, u2} R₁ M₁ _inst_1 _inst_13 _inst_22))))) c x)) (HSMul.hSMul.{u3, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₁) => R₂) c) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (instHSMul.{u3, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₁) => R₂) c) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (SMulZeroClass.toSMul.{u3, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R₁) => R₂) c) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_17)) (SMulWithZero.toSMulZeroClass.{u3, u1} 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_inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ R₂ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (RingHom.instRingHomClassRingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2))))) σ₁₂ c) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_smulₛₗ ContinuousLinearEquiv.map_smulₛₗₓ'. -/ @[simp] theorem map_smulₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e (c • x) = σ₁₂ c • e x := (e : M₁ →SL[σ₁₂] M₂).map_smulₛₗ c x @@ -1950,6 +3120,12 @@ theorem map_smulₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e omit σ₂₁ +/- warning: continuous_linear_equiv.map_smul -> ContinuousLinearEquiv.map_smul is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_17] (e : ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) (c : R₁) (x : M₁), Eq.{succ u3} M₂ (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) (fun (_x : ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) e (SMul.smul.{u1, u2} R₁ M₁ (SMulZeroClass.toHasSmul.{u1, u2} R₁ M₁ (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))) (SMulWithZero.toSmulZeroClass.{u1, u2} R₁ M₁ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))) (MulActionWithZero.toSMulWithZero.{u1, u2} R₁ M₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u2} M₁ (AddMonoid.toAddZeroClass.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))) (Module.toMulActionWithZero.{u1, u2} R₁ M₁ _inst_1 _inst_13 _inst_22)))) c x)) (SMul.smul.{u1, u3} R₁ M₂ (SMulZeroClass.toHasSmul.{u1, u3} R₁ M₂ (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_17))) (SMulWithZero.toSmulZeroClass.{u1, u3} R₁ M₂ (MulZeroClass.toHasZero.{u1} R₁ (MulZeroOneClass.toMulZeroClass.{u1} R₁ (MonoidWithZero.toMulZeroOneClass.{u1} R₁ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1)))) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_17))) (MulActionWithZero.toSMulWithZero.{u1, u3} R₁ M₂ (Semiring.toMonoidWithZero.{u1} R₁ _inst_1) (AddZeroClass.toHasZero.{u3} M₂ (AddMonoid.toAddZeroClass.{u3} M₂ (AddCommMonoid.toAddMonoid.{u3} M₂ _inst_17))) (Module.toMulActionWithZero.{u1, u3} R₁ M₂ _inst_1 _inst_17 _inst_26)))) c (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ 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Type.{u3}} [_inst_1 : Semiring.{u3} R₁] {M₁ : Type.{u1}} [_inst_12 : TopologicalSpace.{u1} M₁] [_inst_13 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u2}} [_inst_16 : TopologicalSpace.{u2} M₂] [_inst_17 : AddCommMonoid.{u2} M₂] [_inst_22 : Module.{u3, u1} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u3, u2} R₁ M₂ _inst_1 _inst_17] (e : ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) (c : R₁) (x : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (HSMul.hSMul.{u3, u1, u1} R₁ M₁ M₁ (instHSMul.{u3, u1} R₁ M₁ (SMulZeroClass.toSMul.{u3, u1} R₁ M₁ (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₁ M₁ (MonoidWithZero.toZero.{u3} R₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₁ M₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (Module.toMulActionWithZero.{u3, u1} R₁ M₁ _inst_1 _inst_13 _inst_22))))) c x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u1 u2, u3, u3, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26)))) e (HSMul.hSMul.{u3, u1, u1} R₁ M₁ M₁ (instHSMul.{u3, u1} R₁ M₁ (SMulZeroClass.toSMul.{u3, u1} R₁ M₁ (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (SMulWithZero.toSMulZeroClass.{u3, u1} R₁ M₁ (MonoidWithZero.toZero.{u3} R₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1)) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (MulActionWithZero.toSMulWithZero.{u3, u1} R₁ M₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1) (AddMonoid.toZero.{u1} M₁ (AddCommMonoid.toAddMonoid.{u1} M₁ _inst_13)) (Module.toMulActionWithZero.{u3, u1} R₁ M₁ _inst_1 _inst_13 _inst_22))))) c x)) (HSMul.hSMul.{u3, u2, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (instHSMul.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (SMulZeroClass.toSMul.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_17)) (SMulWithZero.toSMulZeroClass.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (MonoidWithZero.toZero.{u3} R₁ (Semiring.toMonoidWithZero.{u3} R₁ _inst_1)) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_17)) (MulActionWithZero.toSMulWithZero.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (Semiring.toMonoidWithZero.{u3} R₁ _inst_1) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_17)) (Module.toMulActionWithZero.{u3, u2} R₁ ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_1 _inst_17 _inst_26))))) c (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u1 u2, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u1 u2, u3, u3, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u1 u2, u3, u3, u1, u2} (ContinuousLinearEquiv.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u3, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHom.id.{u3} R₁ (Semiring.toNonAssocSemiring.{u3} R₁ _inst_1)) (RingHomInvPair.ids.{u3} R₁ _inst_1) (RingHomInvPair.ids.{u3} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26)))) e x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_smul ContinuousLinearEquiv.map_smulₓ'. -/ @[simp] theorem map_smul [Module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M₁) : e (c • x) = c • e x := (e : M₁ →L[R₁] M₂).map_smul c x @@ -1957,6 +3133,12 @@ theorem map_smul [Module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M include σ₂₁ +/- warning: continuous_linear_equiv.map_eq_zero_iff -> ContinuousLinearEquiv.map_eq_zero_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₁}, Iff (Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e x) (OfNat.ofNat.{u4} M₂ 0 (OfNat.mk.{u4} M₂ 0 (Zero.zero.{u4} M₂ (AddZeroClass.toHasZero.{u4} M₂ (AddMonoid.toAddZeroClass.{u4} M₂ (AddCommMonoid.toAddMonoid.{u4} M₂ _inst_17))))))) (Eq.{succ u3} M₁ x (OfNat.ofNat.{u3} M₁ 0 (OfNat.mk.{u3} M₁ 0 (Zero.zero.{u3} M₁ (AddZeroClass.toHasZero.{u3} M₁ (AddMonoid.toAddZeroClass.{u3} M₁ (AddCommMonoid.toAddMonoid.{u3} M₁ _inst_13))))))) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₁}, Iff (Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddMonoid.toZero.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (AddCommMonoid.toAddMonoid.{u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) _inst_17))))) (Eq.{succ u2} M₁ x (OfNat.ofNat.{u2} M₁ 0 (Zero.toOfNat0.{u2} M₁ (AddMonoid.toZero.{u2} M₁ (AddCommMonoid.toAddMonoid.{u2} M₁ _inst_13))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_eq_zero_iff ContinuousLinearEquiv.map_eq_zero_iffₓ'. -/ @[simp] theorem map_eq_zero_iff (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ x = 0 := e.toLinearEquiv.map_eq_zero_iff @@ -1965,29 +3147,65 @@ theorem map_eq_zero_iff (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ attribute [continuity] ContinuousLinearEquiv.continuous_toFun ContinuousLinearEquiv.continuous_invFun +/- warning: continuous_linear_equiv.continuous -> ContinuousLinearEquiv.continuous is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Continuous.{u3, u4} M₁ M₂ _inst_12 _inst_16 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Continuous.{u2, u1} M₁ M₂ _inst_12 _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.continuous ContinuousLinearEquiv.continuousₓ'. -/ @[continuity] protected theorem continuous (e : M₁ ≃SL[σ₁₂] M₂) : Continuous (e : M₁ → M₂) := e.continuous_toFun #align continuous_linear_equiv.continuous ContinuousLinearEquiv.continuous +/- warning: continuous_linear_equiv.continuous_on -> ContinuousLinearEquiv.continuousOn is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u3} M₁}, ContinuousOn.{u3, u4} M₁ M₂ _inst_12 _inst_16 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u2} M₁}, ContinuousOn.{u2, u1} M₁ M₂ _inst_12 _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.continuous_on ContinuousLinearEquiv.continuousOnₓ'. -/ protected theorem continuousOn (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : ContinuousOn (e : M₁ → M₂) s := e.Continuous.ContinuousOn #align continuous_linear_equiv.continuous_on ContinuousLinearEquiv.continuousOn +/- warning: continuous_linear_equiv.continuous_at -> ContinuousLinearEquiv.continuousAt is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₁}, ContinuousAt.{u3, u4} M₁ M₂ _inst_12 _inst_16 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) x +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₁}, ContinuousAt.{u2, u1} M₁ M₂ _inst_12 _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) x +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.continuous_at ContinuousLinearEquiv.continuousAtₓ'. -/ protected theorem continuousAt (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : ContinuousAt (e : M₁ → M₂) x := e.Continuous.ContinuousAt #align continuous_linear_equiv.continuous_at ContinuousLinearEquiv.continuousAt +/- warning: continuous_linear_equiv.continuous_within_at -> ContinuousLinearEquiv.continuousWithinAt is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u3} M₁} {x : M₁}, ContinuousWithinAt.{u3, u4} M₁ M₂ _inst_12 _inst_16 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s x +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {s : Set.{u2} M₁} {x : M₁}, ContinuousWithinAt.{u2, u1} M₁ M₂ _inst_12 _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s x +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.continuous_within_at ContinuousLinearEquiv.continuousWithinAtₓ'. -/ protected theorem continuousWithinAt (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} {x : M₁} : ContinuousWithinAt (e : M₁ → M₂) s x := e.Continuous.ContinuousWithinAt #align continuous_linear_equiv.continuous_within_at ContinuousLinearEquiv.continuousWithinAt +/- warning: continuous_linear_equiv.comp_continuous_on_iff -> ContinuousLinearEquiv.comp_continuousOn_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] {α : Type.{u5}} [_inst_26 : TopologicalSpace.{u5} α] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {f : α -> M₁} {s : Set.{u5} α}, Iff (ContinuousOn.{u5, u4} α M₂ _inst_26 _inst_16 (Function.comp.{succ u5, succ u3, succ u4} α M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) f) s) (ContinuousOn.{u5, u3} α M₁ _inst_26 _inst_12 f s) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] {α : Type.{u5}} [_inst_26 : TopologicalSpace.{u5} α] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {f : α -> M₁} {s : Set.{u5} α}, Iff (ContinuousOn.{u5, u1} α M₂ _inst_26 _inst_16 (Function.comp.{succ u5, succ u2, succ u1} α M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) f) s) (ContinuousOn.{u5, u2} α M₁ _inst_26 _inst_12 f s) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.comp_continuous_on_iff ContinuousLinearEquiv.comp_continuousOn_iffₓ'. -/ theorem comp_continuousOn_iff {α : Type _} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} {s : Set α} : ContinuousOn (e ∘ f) s ↔ ContinuousOn f s := e.toHomeomorph.comp_continuousOn_iff _ _ #align continuous_linear_equiv.comp_continuous_on_iff ContinuousLinearEquiv.comp_continuousOn_iff +/- warning: continuous_linear_equiv.comp_continuous_iff -> ContinuousLinearEquiv.comp_continuous_iff is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] {α : Type.{u5}} [_inst_26 : TopologicalSpace.{u5} α] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {f : α -> M₁}, Iff (Continuous.{u5, u4} α M₂ _inst_26 _inst_16 (Function.comp.{succ u5, succ u3, succ u4} α M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) f)) (Continuous.{u5, u3} α M₁ _inst_26 _inst_12 f) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] {α : Type.{u5}} [_inst_26 : TopologicalSpace.{u5} α] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {f : α -> M₁}, Iff (Continuous.{u5, u1} α M₂ _inst_26 _inst_16 (Function.comp.{succ u5, succ u2, succ u1} α M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) f)) (Continuous.{u5, u2} α M₁ _inst_26 _inst_12 f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.comp_continuous_iff ContinuousLinearEquiv.comp_continuous_iffₓ'. -/ theorem comp_continuous_iff {α : Type _} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} : Continuous (e ∘ f) ↔ Continuous f := e.toHomeomorph.comp_continuous_iff @@ -1995,6 +3213,12 @@ theorem comp_continuous_iff {α : Type _} [TopologicalSpace α] (e : M₁ ≃SL[ omit σ₂₁ +/- warning: continuous_linear_equiv.ext₁ -> ContinuousLinearEquiv.ext₁ is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : TopologicalSpace.{u1} R₁] {f : ContinuousLinearEquiv.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) R₁ _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₁ _inst_12 _inst_13 (Semiring.toModule.{u1} R₁ _inst_1) _inst_22} {g : ContinuousLinearEquiv.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) R₁ _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1))) M₁ _inst_12 _inst_13 (Semiring.toModule.{u1} R₁ _inst_1) _inst_22}, (Eq.{succ u2} M₁ (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (ContinuousLinearEquiv.{u1, u1, u1, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ 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_inst_13 (Semiring.toModule.{u1} R₁ _inst_1) _inst_22) f g) +but is expected to have type + forall {R₁ : Type.{u2}} [_inst_1 : Semiring.{u2} R₁] {M₁ : Type.{u1}} [_inst_12 : TopologicalSpace.{u1} M₁] [_inst_13 : AddCommMonoid.{u1} M₁] [_inst_22 : Module.{u2, u1} R₁ M₁ _inst_1 _inst_13] [_inst_26 : TopologicalSpace.{u2} R₁] {f : ContinuousLinearEquiv.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHomInvPair.ids.{u2} R₁ _inst_1) (RingHomInvPair.ids.{u2} R₁ _inst_1) R₁ _inst_26 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R₁ (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1))) M₁ _inst_12 _inst_13 (Semiring.toModule.{u2} R₁ _inst_1) _inst_22} {g : ContinuousLinearEquiv.{u2, u2, u2, u1} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u2} R₁ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)) (RingHom.id.{u2} R₁ 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Consider using '#align continuous_linear_equiv.ext₁ ContinuousLinearEquiv.ext₁ₓ'. -/ /-- An extensionality lemma for `R ≃L[R] M`. -/ theorem ext₁ [TopologicalSpace R₁] {f g : R₁ ≃L[R₁] M₁} (h : f 1 = g 1) : f = g := ext <| funext fun x => mul_one x ▸ by rw [← smul_eq_mul, map_smul, h, map_smul] @@ -2004,6 +3228,7 @@ section variable (R₁ M₁) +#print ContinuousLinearEquiv.refl /- /-- The identity map as a continuous linear equivalence. -/ @[refl] protected def refl : M₁ ≃L[R₁] M₁ := @@ -2011,19 +3236,25 @@ protected def refl : M₁ ≃L[R₁] M₁ := continuous_toFun := continuous_id continuous_invFun := continuous_id } #align continuous_linear_equiv.refl ContinuousLinearEquiv.refl +-/ end +#print ContinuousLinearEquiv.coe_refl /- @[simp, norm_cast] theorem coe_refl : ↑(ContinuousLinearEquiv.refl R₁ M₁) = ContinuousLinearMap.id R₁ M₁ := rfl #align continuous_linear_equiv.coe_refl ContinuousLinearEquiv.coe_refl +-/ +#print ContinuousLinearEquiv.coe_refl' /- @[simp, norm_cast] theorem coe_refl' : ⇑(ContinuousLinearEquiv.refl R₁ M₁) = id := rfl #align continuous_linear_equiv.coe_refl' ContinuousLinearEquiv.coe_refl' +-/ +#print ContinuousLinearEquiv.symm /- /-- The inverse of a continuous linear equivalence as a continuous linear equivalence-/ @[symm] protected def symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ := @@ -2031,9 +3262,16 @@ protected def symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ : continuous_toFun := e.continuous_invFun continuous_invFun := e.continuous_toFun } #align continuous_linear_equiv.symm ContinuousLinearEquiv.symm +-/ include σ₂₁ +/- warning: continuous_linear_equiv.symm_to_linear_equiv -> ContinuousLinearEquiv.symm_toLinearEquiv is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u4) (succ u3)} (LinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ M₁ _inst_17 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.toLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (LinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_13 _inst_17 _inst_22 _inst_24 σ₁₂ σ₂₁ _inst_4 _inst_5 (ContinuousLinearEquiv.toLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (LinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ M₁ _inst_17 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.toLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (LinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ M₁ M₂ _inst_1 _inst_2 _inst_13 _inst_17 _inst_22 _inst_24 σ₁₂ σ₂₁ _inst_4 _inst_5 (ContinuousLinearEquiv.toLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_to_linear_equiv ContinuousLinearEquiv.symm_toLinearEquivₓ'. -/ @[simp] theorem symm_toLinearEquiv (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toLinearEquiv = e.toLinearEquiv.symm := by @@ -2041,25 +3279,41 @@ theorem symm_toLinearEquiv (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toLinearEqui rfl #align continuous_linear_equiv.symm_to_linear_equiv ContinuousLinearEquiv.symm_toLinearEquiv +/- warning: continuous_linear_equiv.symm_to_homeomorph -> ContinuousLinearEquiv.symm_toHomeomorph is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u4) (succ u3)} (Homeomorph.{u4, u3} M₂ M₁ _inst_16 _inst_12) (Homeomorph.symm.{u3, u4} M₁ M₂ _inst_12 _inst_16 (ContinuousLinearEquiv.toHomeomorph.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (ContinuousLinearEquiv.toHomeomorph.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (Homeomorph.{u1, u2} M₂ M₁ _inst_16 _inst_12) (Homeomorph.symm.{u2, u1} M₁ M₂ _inst_12 _inst_16 (ContinuousLinearEquiv.toHomeomorph.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (ContinuousLinearEquiv.toHomeomorph.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_to_homeomorph ContinuousLinearEquiv.symm_toHomeomorphₓ'. -/ @[simp] theorem symm_toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : e.toHomeomorph.symm = e.symm.toHomeomorph := rfl #align continuous_linear_equiv.symm_to_homeomorph ContinuousLinearEquiv.symm_toHomeomorph +#print ContinuousLinearEquiv.Simps.apply /- /-- See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections. -/ def Simps.apply (h : M₁ ≃SL[σ₁₂] M₂) : M₁ → M₂ := h #align continuous_linear_equiv.simps.apply ContinuousLinearEquiv.Simps.apply +-/ +#print ContinuousLinearEquiv.Simps.symm_apply /- /-- See Note [custom simps projection] -/ -def Simps.symmApply (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁ := +def Simps.symm_apply (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁ := h.symm -#align continuous_linear_equiv.simps.symm_apply ContinuousLinearEquiv.Simps.symmApply +#align continuous_linear_equiv.simps.symm_apply ContinuousLinearEquiv.Simps.symm_apply +-/ initialize_simps_projections ContinuousLinearEquiv (to_linear_equiv_to_fun → apply, to_linear_equiv_inv_fun → symm_apply) +/- warning: continuous_linear_equiv.symm_map_nhds_eq -> ContinuousLinearEquiv.symm_map_nhds_eq is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u3} (Filter.{u3} M₁) (Filter.map.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ 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(ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e x))) (nhds.{u3} M₁ _inst_12 x) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u2} (Filter.{u2} M₁) (Filter.map.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (nhds.{u1} M₂ _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x))) (nhds.{u2} M₁ _inst_12 x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_map_nhds_eq ContinuousLinearEquiv.symm_map_nhds_eqₓ'. -/ theorem symm_map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e.symm (𝓝 (e x)) = 𝓝 x := e.toHomeomorph.symm_map_nhds_eq x #align continuous_linear_equiv.symm_map_nhds_eq ContinuousLinearEquiv.symm_map_nhds_eq @@ -2068,6 +3322,7 @@ omit σ₂₁ include σ₂₁ σ₃₂ σ₃₁ +#print ContinuousLinearEquiv.trans /- /-- The composition of two continuous linear equivalences as a continuous linear equivalence. -/ @[trans] protected def trans (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : M₁ ≃SL[σ₁₃] M₃ := @@ -2077,9 +3332,16 @@ protected def trans (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂ continuous_toFun := e₂.continuous_toFun.comp e₁.continuous_toFun continuous_invFun := e₁.continuous_invFun.comp e₂.continuous_invFun } #align continuous_linear_equiv.trans ContinuousLinearEquiv.trans +-/ include σ₁₃ +/- warning: continuous_linear_equiv.trans_to_linear_equiv -> ContinuousLinearEquiv.trans_toLinearEquiv is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₂ : RingHom.{u3, u2} R₃ R₂ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u2, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u3, u2} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₁ : RingHom.{u3, u1} R₃ R₁ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u3, u1} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u3, u2, u1} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_16 : TopologicalSpace.{u5} M₂] [_inst_17 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_18 : TopologicalSpace.{u6} M₃] [_inst_19 : AddCommMonoid.{u6} M₃] [_inst_22 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ 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σ₃₁ _inst_10 _inst_11 _inst_4 _inst_6 _inst_8 _inst_5 _inst_7 _inst_9 (ContinuousLinearEquiv.toLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e₁) (ContinuousLinearEquiv.toLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 e₂)) +but is expected to have type + forall {R₁ : Type.{u6}} {R₂ : Type.{u5}} {R₃ : Type.{u2}} [_inst_1 : Semiring.{u6} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u2} R₃] {σ₁₂ : RingHom.{u6, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₁ : RingHom.{u5, u6} R₂ R₁ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u6, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u5, u6} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u5, u2} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₂ : RingHom.{u2, u5} R₃ R₂ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u5, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u2, u5} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u6, u2} R₁ R₃ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₁ : RingHom.{u2, u6} R₃ R₁ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u6, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u2, u6} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u6, u5, u2} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u2, u5, u6} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u1}} [_inst_18 : TopologicalSpace.{u1} M₃] [_inst_19 : AddCommMonoid.{u1} M₃] [_inst_22 : Module.{u6, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u5, u3} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u2, u1} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (e₂ : ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25), Eq.{max (succ u4) (succ u1)} (LinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ M₃ _inst_13 _inst_19 _inst_22 _inst_25) (ContinuousLinearEquiv.toLinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousLinearEquiv.trans.{u6, u5, u2, u4, u3, u1} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₁ _inst_4 _inst_5 σ₂₃ σ₃₂ _inst_6 _inst_7 σ₁₃ σ₃₁ _inst_8 _inst_9 _inst_10 _inst_11 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 e₁ e₂)) (LinearEquiv.trans.{u6, u5, u2, u4, u3, u1} R₁ R₂ R₃ M₁ M₂ M₃ _inst_1 _inst_2 _inst_3 _inst_13 _inst_17 _inst_19 _inst_22 _inst_24 _inst_25 σ₁₂ σ₂₃ σ₁₃ σ₂₁ σ₃₂ σ₃₁ _inst_10 _inst_11 _inst_4 _inst_6 _inst_8 _inst_5 _inst_7 _inst_9 (ContinuousLinearEquiv.toLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e₁) (ContinuousLinearEquiv.toLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 e₂)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.trans_to_linear_equiv ContinuousLinearEquiv.trans_toLinearEquivₓ'. -/ @[simp] theorem trans_toLinearEquiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : (e₁.trans e₂).toLinearEquiv = e₁.toLinearEquiv.trans e₂.toLinearEquiv := @@ -2090,6 +3352,12 @@ theorem trans_toLinearEquiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃S omit σ₁₃ σ₂₁ σ₃₂ σ₃₁ +/- warning: continuous_linear_equiv.prod -> ContinuousLinearEquiv.prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} M₃] [_inst_19 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_20 : TopologicalSpace.{u5} M₄] [_inst_21 : AddCommMonoid.{u5} M₄] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_17] [_inst_27 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_19] [_inst_28 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_21], (ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) -> (ContinuousLinearEquiv.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28) -> (ContinuousLinearEquiv.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_12 _inst_18) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_13 _inst_19) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_16 _inst_20) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_17 _inst_21) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_13 _inst_19 _inst_22 _inst_27) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_17 _inst_21 _inst_26 _inst_28)) +but is expected to have type + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} M₃] [_inst_19 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_20 : TopologicalSpace.{u5} M₄] [_inst_21 : AddCommMonoid.{u5} M₄] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_17] [_inst_27 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_19] [_inst_28 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_21], (ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) -> (ContinuousLinearEquiv.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28) -> (ContinuousLinearEquiv.{u1, u1, max u4 u2, max u5 u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) (Prod.{u2, u4} M₁ M₃) (instTopologicalSpaceProd.{u2, u4} M₁ M₃ _inst_12 _inst_18) (Prod.instAddCommMonoidSum.{u2, u4} M₁ M₃ _inst_13 _inst_19) (Prod.{u3, u5} M₂ M₄) (instTopologicalSpaceProd.{u3, u5} M₂ M₄ _inst_16 _inst_20) (Prod.instAddCommMonoidSum.{u3, u5} M₂ M₄ _inst_17 _inst_21) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_13 _inst_19 _inst_22 _inst_27) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_17 _inst_21 _inst_26 _inst_28)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.prod ContinuousLinearEquiv.prodₓ'. -/ /-- Product of two continuous linear equivalences. The map comes from `equiv.prod_congr`. -/ def prod [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : (M₁ × M₃) ≃L[R₁] M₂ × M₄ := @@ -2100,12 +3368,24 @@ def prod [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L continuous_invFun := e.continuous_invFun.Prod_map e'.continuous_invFun } #align continuous_linear_equiv.prod ContinuousLinearEquiv.prod +/- warning: continuous_linear_equiv.prod_apply -> ContinuousLinearEquiv.prod_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} M₃] [_inst_19 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_20 : TopologicalSpace.{u5} M₄] [_inst_21 : AddCommMonoid.{u5} M₄] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 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R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u3 u2, u5, u5, u3, u2} (ContinuousLinearEquiv.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28) R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28)))) e' (Prod.snd.{u1, u3} M₁ M₃ x))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.prod_apply ContinuousLinearEquiv.prod_applyₓ'. -/ @[simp, norm_cast] theorem prod_apply [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) (x) : e.Prod e' x = (e x.1, e' x.2) := rfl #align continuous_linear_equiv.prod_apply ContinuousLinearEquiv.prod_apply +/- warning: continuous_linear_equiv.coe_prod -> ContinuousLinearEquiv.coe_prod is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} M₃] [_inst_19 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_20 : TopologicalSpace.{u5} M₄] [_inst_21 : AddCommMonoid.{u5} M₄] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_17] [_inst_27 : 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[_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_18 : TopologicalSpace.{u3} M₃] [_inst_19 : AddCommMonoid.{u3} M₃] {M₄ : Type.{u2}} [_inst_20 : TopologicalSpace.{u2} M₄] [_inst_21 : AddCommMonoid.{u2} M₄] [_inst_22 : Module.{u5, u1} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u5, u4} R₁ M₂ _inst_1 _inst_17] [_inst_27 : Module.{u5, u3} R₁ M₃ _inst_1 _inst_19] [_inst_28 : Module.{u5, u2} R₁ M₄ _inst_1 _inst_21] (e : ContinuousLinearEquiv.{u5, u5, u1, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) (e' : ContinuousLinearEquiv.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) 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Consider using '#align continuous_linear_equiv.coe_prod ContinuousLinearEquiv.coe_prodₓ'. -/ @[simp, norm_cast] theorem coe_prod [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : @@ -2113,6 +3393,12 @@ theorem coe_prod [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M rfl #align continuous_linear_equiv.coe_prod ContinuousLinearEquiv.coe_prod +/- warning: continuous_linear_equiv.prod_symm -> ContinuousLinearEquiv.prod_symm is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} [_inst_1 : Semiring.{u1} R₁] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u4}} [_inst_18 : TopologicalSpace.{u4} M₃] [_inst_19 : AddCommMonoid.{u4} M₃] {M₄ : Type.{u5}} [_inst_20 : TopologicalSpace.{u5} M₄] [_inst_21 : AddCommMonoid.{u5} M₄] [_inst_22 : Module.{u1, u2} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u1, u3} R₁ M₂ _inst_1 _inst_17] [_inst_27 : Module.{u1, u4} R₁ M₃ _inst_1 _inst_19] [_inst_28 : Module.{u1, u5} R₁ M₄ _inst_1 _inst_21] (e : ContinuousLinearEquiv.{u1, u1, u2, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26) (e' : ContinuousLinearEquiv.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28), Eq.{max (succ (max u3 u5)) (succ (max u2 u4))} (ContinuousLinearEquiv.{u1, u1, max u3 u5, max u2 u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) (Prod.{u3, u5} M₂ M₄) (Prod.topologicalSpace.{u3, u5} M₂ M₄ _inst_16 _inst_20) (Prod.addCommMonoid.{u3, u5} M₂ M₄ _inst_17 _inst_21) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_12 _inst_18) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_13 _inst_19) (Prod.module.{u1, u3, u5} R₁ M₂ M₄ _inst_1 _inst_17 _inst_21 _inst_26 _inst_28) (Prod.module.{u1, u2, u4} R₁ M₁ M₃ _inst_1 _inst_13 _inst_19 _inst_22 _inst_27)) (ContinuousLinearEquiv.symm.{u1, u1, max u2 u4, max u3 u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) (Prod.{u2, u4} M₁ M₃) (Prod.topologicalSpace.{u2, u4} M₁ M₃ _inst_12 _inst_18) (Prod.addCommMonoid.{u2, u4} M₁ M₃ _inst_13 _inst_19) 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(ContinuousLinearEquiv.symm.{u1, u1, u4, u5} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) (RingHomInvPair.ids.{u1} R₁ _inst_1) (RingHomInvPair.ids.{u1} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28 e')) +but is expected to have type + forall {R₁ : Type.{u5}} [_inst_1 : Semiring.{u5} R₁] {M₁ : Type.{u1}} [_inst_12 : TopologicalSpace.{u1} M₁] [_inst_13 : AddCommMonoid.{u1} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u3}} [_inst_18 : TopologicalSpace.{u3} M₃] [_inst_19 : AddCommMonoid.{u3} M₃] {M₄ : Type.{u2}} [_inst_20 : TopologicalSpace.{u2} M₄] [_inst_21 : AddCommMonoid.{u2} M₄] [_inst_22 : Module.{u5, u1} R₁ M₁ _inst_1 _inst_13] [_inst_26 : Module.{u5, u4} R₁ M₂ _inst_1 _inst_17] [_inst_27 : Module.{u5, u3} R₁ M₃ _inst_1 _inst_19] [_inst_28 : Module.{u5, u2} R₁ M₄ _inst_1 _inst_21] (e : 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(Prod.instAddCommMonoidSum.{u4, u2} M₂ M₄ _inst_17 _inst_21) (Prod.module.{u5, u1, u3} R₁ M₁ M₃ _inst_1 _inst_13 _inst_19 _inst_22 _inst_27) (Prod.module.{u5, u4, u2} R₁ M₂ M₄ _inst_1 _inst_17 _inst_21 _inst_26 _inst_28) (ContinuousLinearEquiv.prod.{u5, u1, u4, u3, u2} R₁ _inst_1 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_22 _inst_26 _inst_27 _inst_28 e e')) (ContinuousLinearEquiv.prod.{u5, u4, u1, u2, u3} R₁ _inst_1 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 M₄ _inst_20 _inst_21 M₃ _inst_18 _inst_19 _inst_26 _inst_22 _inst_28 _inst_27 (ContinuousLinearEquiv.symm.{u5, u5, u1, u4} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_26 e) (ContinuousLinearEquiv.symm.{u5, u5, u3, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHom.id.{u5} R₁ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)) (RingHomInvPair.ids.{u5} R₁ _inst_1) (RingHomInvPair.ids.{u5} R₁ _inst_1) M₃ _inst_18 _inst_19 M₄ _inst_20 _inst_21 _inst_27 _inst_28 e')) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.prod_symm ContinuousLinearEquiv.prod_symmₓ'. -/ theorem prod_symm [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) : (e.Prod e').symm = e.symm.Prod e'.symm := rfl @@ -2120,20 +3406,44 @@ theorem prod_symm [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : include σ₂₁ +/- warning: continuous_linear_equiv.bijective -> ContinuousLinearEquiv.bijective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Bijective.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Bijective.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.bijective ContinuousLinearEquiv.bijectiveₓ'. -/ protected theorem bijective (e : M₁ ≃SL[σ₁₂] M₂) : Function.Bijective e := e.toLinearEquiv.toEquiv.Bijective #align continuous_linear_equiv.bijective ContinuousLinearEquiv.bijective +/- warning: continuous_linear_equiv.injective -> ContinuousLinearEquiv.injective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Injective.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Injective.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.injective ContinuousLinearEquiv.injectiveₓ'. -/ protected theorem injective (e : M₁ ≃SL[σ₁₂] M₂) : Function.Injective e := e.toLinearEquiv.toEquiv.Injective #align continuous_linear_equiv.injective ContinuousLinearEquiv.injective +/- warning: continuous_linear_equiv.surjective -> ContinuousLinearEquiv.surjective is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Surjective.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Function.Surjective.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.surjective ContinuousLinearEquiv.surjectiveₓ'. -/ protected theorem surjective (e : M₁ ≃SL[σ₁₂] M₂) : Function.Surjective e := e.toLinearEquiv.toEquiv.Surjective #align continuous_linear_equiv.surjective ContinuousLinearEquiv.surjective include σ₃₂ σ₃₁ σ₁₃ +/- warning: continuous_linear_equiv.trans_apply -> ContinuousLinearEquiv.trans_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₂ : RingHom.{u3, u2} R₃ R₂ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u2, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u3, u2} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₁ : RingHom.{u3, u1} R₃ R₁ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u3, u1} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u3, u2, u1} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_16 : TopologicalSpace.{u5} M₂] [_inst_17 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_18 : TopologicalSpace.{u6} M₃] [_inst_19 : AddCommMonoid.{u6} M₃] [_inst_22 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (e₂ : ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) (c : M₁), Eq.{succ u6} M₃ (coeFn.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (fun (_x : ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) => M₁ -> M₃) (ContinuousLinearEquiv.hasCoeToFun.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearEquiv.trans.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₁ _inst_4 _inst_5 σ₂₃ σ₃₂ _inst_6 _inst_7 σ₁₃ σ₃₁ _inst_8 _inst_9 _inst_10 _inst_11 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 e₁ e₂) c) (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) (fun (_x : ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) => M₂ -> M₃) (ContinuousLinearEquiv.hasCoeToFun.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) e₂ (coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e₁ c)) +but is expected to have type + forall {R₁ : Type.{u6}} {R₂ : Type.{u5}} {R₃ : Type.{u2}} [_inst_1 : Semiring.{u6} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u2} R₃] {σ₁₂ : RingHom.{u6, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₁ : RingHom.{u5, u6} R₂ R₁ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u6, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u5, u6} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u5, u2} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₂ : RingHom.{u2, u5} R₃ R₂ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u5, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u2, u5} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u6, u2} R₁ R₃ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₁ : RingHom.{u2, u6} R₃ R₁ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u6, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u2, u6} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u6, u5, u2} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u2, u5, u6} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u1}} [_inst_18 : TopologicalSpace.{u1} M₃] [_inst_19 : AddCommMonoid.{u1} M₃] [_inst_22 : Module.{u6, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u5, u3} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u2, u1} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (e₂ : ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) (c : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) c) (FunLike.coe.{max (succ u4) (succ u1), succ u4, succ u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) _x) (ContinuousMapClass.toFunLike.{max u4 u1, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) M₁ M₃ _inst_12 _inst_18 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u1, u6, u2, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u4 u1, u6, u2, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25)))) (ContinuousLinearEquiv.trans.{u6, u5, u2, u4, u3, u1} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₁ _inst_4 _inst_5 σ₂₃ σ₃₂ _inst_6 _inst_7 σ₁₃ σ₃₁ _inst_8 _inst_9 _inst_10 _inst_11 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 e₁ e₂) c) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₃) _x) (ContinuousMapClass.toFunLike.{max u3 u1, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) M₂ M₃ _inst_16 _inst_18 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u1, u5, u2, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u3 u1, u5, u2, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25)))) e₂ (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u6, u5, u4, u3} (ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u4 u3, u6, u5, u4, u3} (ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e₁ c)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.trans_apply ContinuousLinearEquiv.trans_applyₓ'. -/ @[simp] theorem trans_apply (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) (c : M₁) : (e₁.trans e₂) c = e₂ (e₁ c) := @@ -2142,11 +3452,23 @@ theorem trans_apply (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂ omit σ₃₂ σ₃₁ σ₁₃ +/- warning: continuous_linear_equiv.apply_symm_apply -> ContinuousLinearEquiv.apply_symm_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (c : M₂), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) c)) c +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (c : M₂), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (a : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) a) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) c)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) c)) c +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.apply_symm_apply ContinuousLinearEquiv.apply_symm_applyₓ'. -/ @[simp] theorem apply_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (c : M₂) : e (e.symm c) = c := e.1.right_inv c #align continuous_linear_equiv.apply_symm_apply ContinuousLinearEquiv.apply_symm_apply +/- warning: continuous_linear_equiv.symm_apply_apply -> ContinuousLinearEquiv.symm_apply_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (b : M₁), Eq.{succ u3} M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e b)) b +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (b : M₁), Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (a : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) a) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e b)) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e b)) b +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_apply_apply ContinuousLinearEquiv.symm_apply_applyₓ'. -/ @[simp] theorem symm_apply_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : e.symm (e b) = b := e.1.left_inv b @@ -2154,6 +3476,12 @@ theorem symm_apply_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : e.symm (e include σ₁₂ σ₂₃ σ₁₃ σ₃₁ +/- warning: continuous_linear_equiv.symm_trans_apply -> ContinuousLinearEquiv.symm_trans_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₂ : RingHom.{u3, u2} R₃ R₂ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u2, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u3, u2} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₁ : RingHom.{u3, u1} R₃ R₁ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u3, u1} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u3, u2, u1} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_16 : TopologicalSpace.{u5} M₂] [_inst_17 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_18 : TopologicalSpace.{u6} M₃] [_inst_19 : AddCommMonoid.{u6} M₃] [_inst_22 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u2, u1, u5, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (e₂ : ContinuousLinearEquiv.{u3, u2, u6, u5} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃ _inst_7 _inst_6 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 _inst_25 _inst_24) (c : M₁), Eq.{succ u6} M₃ (coeFn.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (fun (_x : ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) => M₁ -> M₃) (ContinuousLinearEquiv.hasCoeToFun.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearEquiv.symm.{u3, u1, u6, u4} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃ _inst_9 _inst_8 M₃ _inst_18 _inst_19 M₁ _inst_12 _inst_13 _inst_25 _inst_22 (ContinuousLinearEquiv.trans.{u3, u2, u1, u6, u5, u4} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₃ _inst_7 _inst_6 σ₂₁ σ₁₂ _inst_5 _inst_4 σ₃₁ σ₁₃ _inst_9 _inst_8 _inst_11 _inst_10 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_25 _inst_24 _inst_22 e₂ e₁)) c) (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) (fun (_x : ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) => M₂ -> M₃) (ContinuousLinearEquiv.hasCoeToFun.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) (ContinuousLinearEquiv.symm.{u3, u2, u6, u5} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃ _inst_7 _inst_6 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 _inst_25 _inst_24 e₂) (coeFn.{max (succ u4) (succ u5), max (succ u4) (succ u5)} (ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.symm.{u2, u1, u5, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 e₁) c)) +but is expected to have type + forall {R₁ : Type.{u5}} {R₂ : Type.{u6}} {R₃ : Type.{u2}} [_inst_1 : Semiring.{u5} R₁] [_inst_2 : Semiring.{u6} R₂] [_inst_3 : Semiring.{u2} R₃] {σ₁₂ : RingHom.{u5, u6} R₁ R₂ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)} {σ₂₁ : RingHom.{u6, u5} R₂ R₁ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u5, u6} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u6, u5} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u6, u2} R₂ R₃ (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₂ : RingHom.{u2, u6} R₃ R₂ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u6} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u6, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u2, u6} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u5, u2} R₁ R₃ (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₁ : RingHom.{u2, u5} R₃ R₁ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u5} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u5, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u2, u5} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u5, u6, u2} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u2, u6, u5} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] {M₃ : Type.{u1}} [_inst_18 : TopologicalSpace.{u1} M₃] [_inst_19 : AddCommMonoid.{u1} M₃] [_inst_22 : Module.{u5, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u6, u4} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u2, u1} R₃ M₃ _inst_3 _inst_19] (e₁ : ContinuousLinearEquiv.{u6, u5, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (e₂ : ContinuousLinearEquiv.{u2, u6, u1, u4} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃ _inst_7 _inst_6 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 _inst_25 _inst_24) (c : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) c) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₃) _x) (ContinuousMapClass.toFunLike.{max u3 u1, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) M₁ M₃ _inst_12 _inst_18 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u1, u5, u2, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u3 u1, u5, u2, u3, u1} (ContinuousLinearEquiv.{u5, u2, u3, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u5, u2, u3, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25)))) (ContinuousLinearEquiv.symm.{u2, u5, u1, u3} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃ _inst_9 _inst_8 M₃ _inst_18 _inst_19 M₁ _inst_12 _inst_13 _inst_25 _inst_22 (ContinuousLinearEquiv.trans.{u2, u6, u5, u1, u4, u3} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₃ _inst_7 _inst_6 σ₂₁ σ₁₂ _inst_5 _inst_4 σ₃₁ σ₁₃ _inst_9 _inst_8 _inst_11 _inst_10 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_25 _inst_24 _inst_22 e₂ e₁)) c) (FunLike.coe.{max (succ u4) (succ u1), succ u4, succ u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₃) _x) (ContinuousMapClass.toFunLike.{max u4 u1, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) M₂ M₃ _inst_16 _inst_18 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u1, u6, u2, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) R₂ R₃ _inst_2 _inst_3 σ₂₃ M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u4 u1, u6, u2, u4, u1} (ContinuousLinearEquiv.{u6, u2, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25) R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u6, u2, u4, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25)))) (ContinuousLinearEquiv.symm.{u2, u6, u1, u4} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃ _inst_7 _inst_6 M₃ _inst_18 _inst_19 M₂ _inst_16 _inst_17 _inst_25 _inst_24 e₂) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearEquiv.{u5, u6, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearEquiv.{u5, u6, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u5, u6, u3, u4} (ContinuousLinearEquiv.{u5, u6, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u3 u4, u5, u6, u3, u4} (ContinuousLinearEquiv.{u5, u6, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u5, u6, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) (ContinuousLinearEquiv.symm.{u6, u5, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 e₁) c)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_trans_apply ContinuousLinearEquiv.symm_trans_applyₓ'. -/ @[simp] theorem symm_trans_apply (e₁ : M₂ ≃SL[σ₂₁] M₁) (e₂ : M₃ ≃SL[σ₃₂] M₂) (c : M₁) : (e₂.trans e₁).symm c = e₂.symm (e₁.symm c) := @@ -2162,11 +3490,23 @@ theorem symm_trans_apply (e₁ : M₂ ≃SL[σ₂₁] M₁) (e₂ : M₃ ≃SL[ omit σ₁₂ σ₂₃ σ₁₃ σ₃₁ +/- warning: continuous_linear_equiv.symm_image_image -> ContinuousLinearEquiv.symm_image_image is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u3} M₁), Eq.{succ u3} (Set.{u3} M₁) (Set.image.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (Set.image.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s)) s +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u2} M₁), Eq.{succ u2} (Set.{u2} M₁) (Set.image.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s)) s +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_image_image ContinuousLinearEquiv.symm_image_imageₓ'. -/ @[simp] theorem symm_image_image (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e.symm '' (e '' s) = s := e.toLinearEquiv.toEquiv.symm_image_image s #align continuous_linear_equiv.symm_image_image ContinuousLinearEquiv.symm_image_image +/- warning: continuous_linear_equiv.image_symm_image -> ContinuousLinearEquiv.image_symm_image is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u4} M₂), Eq.{succ u4} (Set.{u4} M₂) (Set.image.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (Set.image.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s)) s +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u1} M₂), Eq.{succ u1} (Set.{u1} M₂) (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) (Set.image.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s)) s +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.image_symm_image ContinuousLinearEquiv.image_symm_imageₓ'. -/ @[simp] theorem image_symm_image (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e '' (e.symm '' s) = s := e.symm.symm_image_image s @@ -2174,6 +3514,12 @@ theorem image_symm_image (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e '' ( include σ₃₂ σ₃₁ +/- warning: continuous_linear_equiv.comp_coe -> ContinuousLinearEquiv.comp_coe is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} {R₃ : Type.{u3}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] [_inst_3 : Semiring.{u3} R₃] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u2, u3} R₂ R₃ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₂ : RingHom.{u3, u2} R₃ R₂ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u2, u3} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u3, u2} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u1, u3} R₁ R₃ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3)} {σ₃₁ : RingHom.{u3, u1} R₃ R₁ (Semiring.toNonAssocSemiring.{u3} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u1, u3} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u3, u1} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u1, u2, u3} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u3, u2, u1} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u5}} [_inst_16 : TopologicalSpace.{u5} M₂] [_inst_17 : AddCommMonoid.{u5} M₂] {M₃ : Type.{u6}} [_inst_18 : TopologicalSpace.{u6} M₃] [_inst_19 : AddCommMonoid.{u6} M₃] [_inst_22 : Module.{u1, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u5} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u3, u6} R₃ M₃ _inst_3 _inst_19] (f : ContinuousLinearEquiv.{u1, u2, u4, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f' : ContinuousLinearEquiv.{u2, u3, u5, u6} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25), Eq.{max (succ u4) (succ u6)} (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearMap.comp.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 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_inst_18 _inst_19 _inst_22 _inst_25) (CoeTCₓ.coe.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (coeBase.{max (succ u4) (succ u6), max (succ u4) (succ u6)} (ContinuousLinearEquiv.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearMap.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u1, u3, u4, u6} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25)))) (ContinuousLinearEquiv.trans.{u1, u2, u3, u4, u5, u6} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₁ _inst_4 _inst_5 σ₂₃ σ₃₂ _inst_6 _inst_7 σ₁₃ σ₃₁ _inst_8 _inst_9 _inst_10 _inst_11 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 f f')) +but is expected to have type + forall {R₁ : Type.{u6}} {R₂ : Type.{u5}} {R₃ : Type.{u2}} [_inst_1 : Semiring.{u6} R₁] [_inst_2 : Semiring.{u5} R₂] [_inst_3 : Semiring.{u2} R₃] {σ₁₂ : RingHom.{u6, u5} R₁ R₂ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} {σ₂₁ : RingHom.{u5, u6} R₂ R₁ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u6, u5} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u5, u6} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {σ₂₃ : RingHom.{u5, u2} R₂ R₃ (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₂ : RingHom.{u2, u5} R₃ R₂ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u5} R₂ _inst_2)} [_inst_6 : RingHomInvPair.{u5, u2} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂] [_inst_7 : RingHomInvPair.{u2, u5} R₃ R₂ _inst_3 _inst_2 σ₃₂ σ₂₃] {σ₁₃ : RingHom.{u6, u2} R₁ R₃ (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3)} {σ₃₁ : RingHom.{u2, u6} R₃ R₁ (Semiring.toNonAssocSemiring.{u2} R₃ _inst_3) (Semiring.toNonAssocSemiring.{u6} R₁ _inst_1)} [_inst_8 : RingHomInvPair.{u6, u2} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁] [_inst_9 : RingHomInvPair.{u2, u6} R₃ R₁ _inst_3 _inst_1 σ₃₁ σ₁₃] [_inst_10 : RingHomCompTriple.{u6, u5, u2} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃] [_inst_11 : RingHomCompTriple.{u2, u5, u6} R₃ R₂ R₁ _inst_3 _inst_2 _inst_1 σ₃₂ σ₂₁ σ₃₁] {M₁ : Type.{u4}} [_inst_12 : TopologicalSpace.{u4} M₁] [_inst_13 : AddCommMonoid.{u4} M₁] {M₂ : Type.{u3}} [_inst_16 : TopologicalSpace.{u3} M₂] [_inst_17 : AddCommMonoid.{u3} M₂] {M₃ : Type.{u1}} [_inst_18 : TopologicalSpace.{u1} M₃] [_inst_19 : AddCommMonoid.{u1} M₃] [_inst_22 : Module.{u6, u4} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u5, u3} R₂ M₂ _inst_2 _inst_17] [_inst_25 : Module.{u2, u1} R₃ M₃ _inst_3 _inst_19] (f : ContinuousLinearEquiv.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f' : ContinuousLinearEquiv.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25), Eq.{max (succ u4) (succ u1)} (ContinuousLinearMap.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25) (ContinuousLinearMap.comp.{u6, u5, u2, u4, u3, u1} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₃ σ₁₃ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 _inst_10 (ContinuousLinearEquiv.toContinuousLinearMap.{u5, u2, u3, u1} R₂ R₃ _inst_2 _inst_3 σ₂₃ σ₃₂ _inst_6 _inst_7 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_24 _inst_25 f') (ContinuousLinearEquiv.toContinuousLinearMap.{u6, u5, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f)) (ContinuousLinearEquiv.toContinuousLinearMap.{u6, u2, u4, u1} R₁ R₃ _inst_1 _inst_3 σ₁₃ σ₃₁ _inst_8 _inst_9 M₁ _inst_12 _inst_13 M₃ _inst_18 _inst_19 _inst_22 _inst_25 (ContinuousLinearEquiv.trans.{u6, u5, u2, u4, u3, u1} R₁ R₂ R₃ _inst_1 _inst_2 _inst_3 σ₁₂ σ₂₁ _inst_4 _inst_5 σ₂₃ σ₃₂ _inst_6 _inst_7 σ₁₃ σ₃₁ _inst_8 _inst_9 _inst_10 _inst_11 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₃ _inst_18 _inst_19 _inst_22 _inst_24 _inst_25 f f')) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.comp_coe ContinuousLinearEquiv.comp_coeₓ'. -/ @[simp, norm_cast] theorem comp_coe (f : M₁ ≃SL[σ₁₂] M₂) (f' : M₂ ≃SL[σ₂₃] M₃) : (f' : M₂ →SL[σ₂₃] M₃).comp (f : M₁ →SL[σ₁₂] M₂) = (f.trans f' : M₁ →SL[σ₁₃] M₃) := @@ -2182,12 +3528,24 @@ theorem comp_coe (f : M₁ ≃SL[σ₁₂] M₂) (f' : M₂ ≃SL[σ₂₃] M₃ omit σ₃₂ σ₃₁ σ₂₁ +/- warning: continuous_linear_equiv.coe_comp_coe_symm -> ContinuousLinearEquiv.coe_comp_coe_symm is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u4} (ContinuousLinearMap.{u2, u2, u4, u4} R₂ R₂ _inst_2 _inst_2 (RingHom.id.{u2} R₂ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)) M₂ _inst_16 _inst_17 M₂ _inst_16 _inst_17 _inst_24 _inst_24) (ContinuousLinearMap.comp.{u2, u1, u2, u4, u3, u4} R₂ R₁ R₂ _inst_2 _inst_1 _inst_2 σ₂₁ σ₁₂ (RingHom.id.{u2} R₂ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)) M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_24 _inst_22 _inst_24 (RingHomInvPair.triples₂.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4) ((fun (a : Sort.{max (succ u3) (succ u4)}) (b : Sort.{max 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_inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) ((fun (a : Sort.{max (succ u4) (succ u3)}) (b : Sort.{max (succ u4) (succ u3)}) [self : HasLiftT.{max (succ u4) (succ u3), max (succ u4) (succ u3)} a b] => self.0) (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (HasLiftT.mk.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (CoeTCₓ.coe.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (coeBase.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e))) (ContinuousLinearMap.id.{u2, u4} R₂ _inst_2 M₂ _inst_16 _inst_17 _inst_24) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u1} (ContinuousLinearMap.{u3, u3, u1, u1} R₂ R₂ _inst_2 _inst_2 (RingHom.id.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) M₂ _inst_16 _inst_17 M₂ _inst_16 _inst_17 _inst_24 _inst_24) (ContinuousLinearMap.comp.{u3, u4, u3, u1, u2, u1} R₂ R₁ R₂ _inst_2 _inst_1 _inst_2 σ₂₁ σ₁₂ (RingHom.id.{u3} R₂ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)) M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_24 _inst_22 _inst_24 (RingHomInvPair.triples₂.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) (ContinuousLinearEquiv.toContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e))) (ContinuousLinearMap.id.{u3, u1} R₂ _inst_2 M₂ _inst_16 _inst_17 _inst_24) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_comp_coe_symm ContinuousLinearEquiv.coe_comp_coe_symmₓ'. -/ @[simp] theorem coe_comp_coe_symm (e : M₁ ≃SL[σ₁₂] M₂) : (e : M₁ →SL[σ₁₂] M₂).comp (e.symm : M₂ →SL[σ₂₁] M₁) = ContinuousLinearMap.id R₂ M₂ := ContinuousLinearMap.ext e.apply_symm_apply #align continuous_linear_equiv.coe_comp_coe_symm ContinuousLinearEquiv.coe_comp_coe_symm +/- warning: continuous_linear_equiv.coe_symm_comp_coe -> ContinuousLinearEquiv.coe_symm_comp_coe is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u3} (ContinuousLinearMap.{u1, u1, u3, u3} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₁ _inst_12 _inst_13 _inst_22 _inst_22) (ContinuousLinearMap.comp.{u1, u2, u1, u3, u4, u3} R₁ R₂ R₁ _inst_1 _inst_2 _inst_1 σ₁₂ σ₂₁ (RingHom.id.{u1} R₁ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_22 _inst_24 _inst_22 (RingHomInvPair.triples₂.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5) ((fun (a : Sort.{max 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_inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (HasLiftT.mk.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (CoeTCₓ.coe.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (coeBase.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e)) (ContinuousLinearMap.id.{u1, u3} R₁ _inst_1 M₁ _inst_12 _inst_13 _inst_22) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u2} (ContinuousLinearMap.{u4, u4, u2, u2} R₁ R₁ _inst_1 _inst_1 (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₁ _inst_12 _inst_13 _inst_22 _inst_22) (ContinuousLinearMap.comp.{u4, u3, u4, u2, u1, u2} R₁ R₂ R₁ _inst_1 _inst_2 _inst_1 σ₁₂ σ₂₁ (RingHom.id.{u4} R₁ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)) M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_22 _inst_24 _inst_22 (RingHomInvPair.triples₂.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5) (ContinuousLinearEquiv.toContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (ContinuousLinearEquiv.toContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (ContinuousLinearMap.id.{u4, u2} R₁ _inst_1 M₁ _inst_12 _inst_13 _inst_22) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.coe_symm_comp_coe ContinuousLinearEquiv.coe_symm_comp_coeₓ'. -/ @[simp] theorem coe_symm_comp_coe (e : M₁ ≃SL[σ₁₂] M₂) : (e.symm : M₂ →SL[σ₂₁] M₁).comp (e : M₁ →SL[σ₁₂] M₂) = ContinuousLinearMap.id R₁ M₁ := @@ -2196,6 +3554,12 @@ theorem coe_symm_comp_coe (e : M₁ ≃SL[σ₁₂] M₂) : include σ₂₁ +/- warning: continuous_linear_equiv.symm_comp_self -> ContinuousLinearEquiv.symm_comp_self is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u3} (M₁ -> M₁) (Function.comp.{succ u3, succ u4, succ u3} M₁ M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e)) (id.{succ u3} M₁) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u2} (M₁ -> M₁) (Function.comp.{succ u2, succ u1, succ u2} M₁ M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e)) (id.{succ u2} M₁) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_comp_self ContinuousLinearEquiv.symm_comp_selfₓ'. -/ @[simp] theorem symm_comp_self (e : M₁ ≃SL[σ₁₂] M₂) : (e.symm : M₂ → M₁) ∘ (e : M₁ → M₂) = id := by @@ -2203,6 +3567,12 @@ theorem symm_comp_self (e : M₁ ≃SL[σ₁₂] M₂) : (e.symm : M₂ → M₁ exact symm_apply_apply e x #align continuous_linear_equiv.symm_comp_self ContinuousLinearEquiv.symm_comp_self +/- warning: continuous_linear_equiv.self_comp_symm -> ContinuousLinearEquiv.self_comp_symm is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u4} (M₂ -> M₂) (Function.comp.{succ u4, succ u3, succ u4} M₂ M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e))) (id.{succ u4} M₂) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{succ u1} (M₂ -> M₂) (Function.comp.{succ u1, succ u2, succ u1} M₂ M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e))) (id.{succ u1} M₂) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.self_comp_symm ContinuousLinearEquiv.self_comp_symmₓ'. -/ @[simp] theorem self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) : (e : M₁ → M₂) ∘ (e.symm : M₂ → M₁) = id := by @@ -2210,6 +3580,12 @@ theorem self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) : (e : M₁ → M₂) ∘ exact apply_symm_apply e x #align continuous_linear_equiv.self_comp_symm ContinuousLinearEquiv.self_comp_symm +/- warning: continuous_linear_equiv.symm_symm -> ContinuousLinearEquiv.symm_symm is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.symm.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) e +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24), Eq.{max (succ u2) (succ u1)} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.symm.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) e +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_symm ContinuousLinearEquiv.symm_symmₓ'. -/ @[simp] theorem symm_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e := by @@ -2219,45 +3595,95 @@ theorem symm_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e := omit σ₂₁ +#print ContinuousLinearEquiv.refl_symm /- @[simp] theorem refl_symm : (ContinuousLinearEquiv.refl R₁ M₁).symm = ContinuousLinearEquiv.refl R₁ M₁ := rfl #align continuous_linear_equiv.refl_symm ContinuousLinearEquiv.refl_symm +-/ include σ₂₁ +/- warning: continuous_linear_equiv.symm_symm_apply -> ContinuousLinearEquiv.symm_symm_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, 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forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (x : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) (ContinuousLinearEquiv.symm.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_symm_apply ContinuousLinearEquiv.symm_symm_applyₓ'. -/ theorem symm_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : e.symm.symm x = e x := rfl #align continuous_linear_equiv.symm_symm_apply ContinuousLinearEquiv.symm_symm_apply +/- warning: continuous_linear_equiv.symm_apply_eq -> ContinuousLinearEquiv.symm_apply_eq is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₂} {y : M₁}, Iff (Eq.{succ u3} M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) x) y) (Eq.{succ u4} M₂ x (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e y)) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₂} {y : (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) x}, Iff (Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) x) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) x) y) (Eq.{succ u1} M₂ x (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e y)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_apply_eq ContinuousLinearEquiv.symm_apply_eqₓ'. -/ theorem symm_apply_eq (e : M₁ ≃SL[σ₁₂] M₂) {x y} : e.symm x = y ↔ x = e y := e.toLinearEquiv.symm_apply_eq #align continuous_linear_equiv.symm_apply_eq ContinuousLinearEquiv.symm_apply_eq +/- warning: continuous_linear_equiv.eq_symm_apply -> ContinuousLinearEquiv.eq_symm_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₂} {y : M₁}, Iff (Eq.{succ u3} M₁ y (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) x)) (Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e y) x) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) {x : M₂} {y : (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) x}, Iff (Eq.{succ u2} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) x) y (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e) x)) (Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) y) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e y) x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.eq_symm_apply ContinuousLinearEquiv.eq_symm_applyₓ'. -/ theorem eq_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) {x y} : y = e.symm x ↔ e y = x := e.toLinearEquiv.eq_symm_apply #align continuous_linear_equiv.eq_symm_apply ContinuousLinearEquiv.eq_symm_apply +/- warning: continuous_linear_equiv.image_eq_preimage -> ContinuousLinearEquiv.image_eq_preimage is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u3} M₁), Eq.{succ u4} (Set.{u4} M₂) (Set.image.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s) (Set.preimage.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u2} M₁), Eq.{succ u1} (Set.{u1} M₂) (Set.image.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s) (Set.preimage.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.image_eq_preimage ContinuousLinearEquiv.image_eq_preimageₓ'. -/ protected theorem image_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e '' s = e.symm ⁻¹' s := e.toLinearEquiv.toEquiv.image_eq_preimage s #align continuous_linear_equiv.image_eq_preimage ContinuousLinearEquiv.image_eq_preimage +/- warning: continuous_linear_equiv.image_symm_eq_preimage -> ContinuousLinearEquiv.image_symm_eq_preimage is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u4} M₂), Eq.{succ u3} (Set.{u3} M₁) (Set.image.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s) (Set.preimage.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u1} M₂), Eq.{succ u2} (Set.{u2} M₁) (Set.image.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s) (Set.preimage.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.image_symm_eq_preimage ContinuousLinearEquiv.image_symm_eq_preimageₓ'. -/ protected theorem image_symm_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e.symm '' s = e ⁻¹' s := by rw [e.symm.image_eq_preimage, e.symm_symm] #align continuous_linear_equiv.image_symm_eq_preimage ContinuousLinearEquiv.image_symm_eq_preimage +/- warning: continuous_linear_equiv.symm_preimage_preimage -> ContinuousLinearEquiv.symm_preimage_preimage is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u4} M₂), Eq.{succ u4} (Set.{u4} M₂) (Set.preimage.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (Set.preimage.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) s)) s +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u1} M₂), Eq.{succ u1} (Set.{u1} M₂) (Set.preimage.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) (Set.preimage.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) s)) s +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.symm_preimage_preimage ContinuousLinearEquiv.symm_preimage_preimageₓ'. -/ @[simp] protected theorem symm_preimage_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e.symm ⁻¹' (e ⁻¹' s) = s := e.toLinearEquiv.toEquiv.symm_preimage_preimage s #align continuous_linear_equiv.symm_preimage_preimage ContinuousLinearEquiv.symm_preimage_preimage +/- warning: continuous_linear_equiv.preimage_symm_preimage -> ContinuousLinearEquiv.preimage_symm_preimage is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u3} M₁), Eq.{succ u3} (Set.{u3} M₁) (Set.preimage.{u3, u4} M₁ M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) e) (Set.preimage.{u4, u3} M₂ M₁ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearEquiv.hasCoeToFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s)) s +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (e : ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (s : Set.{u2} M₁), Eq.{succ u2} (Set.{u2} M₁) (Set.preimage.{u2, u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) e) (Set.preimage.{u1, u2} M₂ M₁ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearEquiv.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22)))) (ContinuousLinearEquiv.symm.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 e)) s)) s +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.preimage_symm_preimage ContinuousLinearEquiv.preimage_symm_preimageₓ'. -/ @[simp] protected theorem preimage_symm_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e ⁻¹' (e.symm ⁻¹' s) = s := e.symm.symm_preimage_preimage s #align continuous_linear_equiv.preimage_symm_preimage ContinuousLinearEquiv.preimage_symm_preimage +/- warning: continuous_linear_equiv.uniform_embedding -> ContinuousLinearEquiv.uniformEmbedding is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {E₁ : Type.{u3}} {E₂ : Type.{u4}} [_inst_26 : UniformSpace.{u3} E₁] [_inst_27 : UniformSpace.{u4} E₂] [_inst_28 : AddCommGroup.{u3} E₁] [_inst_29 : AddCommGroup.{u4} E₂] [_inst_30 : Module.{u1, u3} R₁ E₁ _inst_1 (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28)] [_inst_31 : Module.{u2, u4} R₂ E₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29)] [_inst_32 : UniformAddGroup.{u3} E₁ _inst_26 (AddCommGroup.toAddGroup.{u3} E₁ _inst_28)] [_inst_33 : UniformAddGroup.{u4} E₂ _inst_27 (AddCommGroup.toAddGroup.{u4} E₂ _inst_29)] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u3} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u4} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29) _inst_30 _inst_31), UniformEmbedding.{u3, u4} E₁ E₂ _inst_26 _inst_27 (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u3} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u4} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29) _inst_30 _inst_31) (fun (_x : ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u3} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u4} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29) _inst_30 _inst_31) => E₁ -> E₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u3} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u4} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29) _inst_30 _inst_31) e) +but is expected to have type + forall {R₁ : Type.{u2}} {R₂ : Type.{u1}} [_inst_1 : Semiring.{u2} R₁] [_inst_2 : Semiring.{u1} R₂] {σ₁₂ : RingHom.{u2, u1} R₁ R₂ (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2)} {σ₂₁ : RingHom.{u1, u2} R₂ R₁ (Semiring.toNonAssocSemiring.{u1} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u2} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {E₁ : Type.{u4}} {E₂ : Type.{u3}} [_inst_26 : UniformSpace.{u4} E₁] [_inst_27 : UniformSpace.{u3} E₂] [_inst_28 : AddCommGroup.{u4} E₁] [_inst_29 : AddCommGroup.{u3} E₂] [_inst_30 : Module.{u2, u4} R₁ E₁ _inst_1 (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28)] [_inst_31 : Module.{u1, u3} R₂ E₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29)] [_inst_32 : UniformAddGroup.{u4} E₁ _inst_26 (AddCommGroup.toAddGroup.{u4} E₁ _inst_28)] [_inst_33 : UniformAddGroup.{u3} E₂ _inst_27 (AddCommGroup.toAddGroup.{u3} E₂ _inst_29)] (e : ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31), UniformEmbedding.{u4, u3} E₁ E₂ _inst_26 _inst_27 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31) E₁ (fun (_x : E₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E₁) => E₂) _x) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31) E₁ E₂ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u2, u1, u4, u3} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31) R₁ R₂ _inst_1 _inst_2 σ₁₂ E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u4 u3, u2, u1, u4, u3} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) E₂ (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31)))) e) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.uniform_embedding ContinuousLinearEquiv.uniformEmbeddingₓ'. -/ protected theorem uniformEmbedding {E₁ E₂ : Type _} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂) : UniformEmbedding e := @@ -2265,6 +3691,12 @@ protected theorem uniformEmbedding {E₁ E₂ : Type _} [UniformSpace E₁] [Uni e.symm.toContinuousLinearMap.UniformContinuous #align continuous_linear_equiv.uniform_embedding ContinuousLinearEquiv.uniformEmbedding +/- warning: linear_equiv.uniform_embedding -> LinearEquiv.uniformEmbedding is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {E₁ : Type.{u3}} {E₂ : Type.{u4}} [_inst_26 : UniformSpace.{u3} E₁] [_inst_27 : UniformSpace.{u4} E₂] [_inst_28 : AddCommGroup.{u3} E₁] [_inst_29 : AddCommGroup.{u4} E₂] [_inst_30 : Module.{u1, u3} R₁ E₁ _inst_1 (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28)] [_inst_31 : Module.{u2, u4} R₂ E₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29)] [_inst_32 : UniformAddGroup.{u3} E₁ _inst_26 (AddCommGroup.toAddGroup.{u3} E₁ _inst_28)] [_inst_33 : UniformAddGroup.{u4} E₂ _inst_27 (AddCommGroup.toAddGroup.{u4} E₂ _inst_29)] (e : LinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ E₂ (AddCommGroup.toAddCommMonoid.{u3} E₁ _inst_28) (AddCommGroup.toAddCommMonoid.{u4} E₂ _inst_29) _inst_30 _inst_31), (Continuous.{u3, u4} E₁ E₂ (UniformSpace.toTopologicalSpace.{u3} E₁ _inst_26) 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_inst_2 _inst_1 σ₂₁ σ₁₂] {E₁ : Type.{u4}} {E₂ : Type.{u3}} [_inst_26 : UniformSpace.{u4} E₁] [_inst_27 : UniformSpace.{u3} E₂] [_inst_28 : AddCommGroup.{u4} E₁] [_inst_29 : AddCommGroup.{u3} E₂] [_inst_30 : Module.{u2, u4} R₁ E₁ _inst_1 (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28)] [_inst_31 : Module.{u1, u3} R₂ E₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29)] [_inst_32 : UniformAddGroup.{u4} E₁ _inst_26 (AddCommGroup.toAddGroup.{u4} E₁ _inst_28)] [_inst_33 : UniformAddGroup.{u3} E₂ _inst_27 (AddCommGroup.toAddGroup.{u3} E₂ _inst_29)] (e : LinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 E₁ E₂ (AddCommGroup.toAddCommMonoid.{u4} E₁ _inst_28) (AddCommGroup.toAddCommMonoid.{u3} E₂ _inst_29) _inst_30 _inst_31), (Continuous.{u4, u3} E₁ E₂ (UniformSpace.toTopologicalSpace.{u4} E₁ _inst_26) (UniformSpace.toTopologicalSpace.{u3} E₂ _inst_27) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (LinearEquiv.{u2, u1, u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ 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Consider using '#align linear_equiv.uniform_embedding LinearEquiv.uniformEmbeddingₓ'. -/ protected theorem LinearEquiv.uniformEmbedding {E₁ E₂ : Type _} [UniformSpace E₁] [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃ₛₗ[σ₁₂] E₂) (h₁ : Continuous e) (h₂ : Continuous e.symm) : @@ -2278,6 +3710,12 @@ protected theorem LinearEquiv.uniformEmbedding {E₁ E₂ : Type _} [UniformSpac omit σ₂₁ +/- warning: continuous_linear_equiv.equiv_of_inverse -> ContinuousLinearEquiv.equivOfInverse is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22), (Function.LeftInverse.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 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-> M₁) (ContinuousLinearMap.toFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) f₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) f₁)) -> (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) +but is expected to have type + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22), (Function.LeftInverse.{succ u3, succ u4} M₁ M₂ (FunLike.coe.{max (succ u3) (succ u4), succ u4, succ u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u2, u1, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22))) f₂) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁)) -> (Function.RightInverse.{succ u3, succ u4} M₁ M₂ (FunLike.coe.{max (succ u3) (succ u4), succ u4, succ u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u2, u1, u4, u3} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22))) f₂) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u3 u4, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u4, u1, u2, u3, u4} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁)) -> (ContinuousLinearEquiv.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.equiv_of_inverse ContinuousLinearEquiv.equivOfInverseₓ'. -/ /-- Create a `continuous_linear_equiv` from two `continuous_linear_map`s that are inverse of each other. -/ def equivOfInverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse f₂ f₁) @@ -2293,12 +3731,24 @@ def equivOfInverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂ include σ₂₁ +/- warning: continuous_linear_equiv.equiv_of_inverse_apply -> ContinuousLinearEquiv.equivOfInverse_apply is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (h₁ : Function.LeftInverse.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) 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M₂) (ContinuousLinearEquiv.hasCoeToFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (ContinuousLinearEquiv.equivOfInverse.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f₁ f₂ h₁ h₂) x) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) f₁ x) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (h₁ : Function.LeftInverse.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} 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_inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁)) (h₂ : Function.RightInverse.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22))) f₂) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁)) (x : M₁), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24)))) (ContinuousLinearEquiv.equivOfInverse.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f₁ f₂ h₁ h₂) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁ x) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.equiv_of_inverse_apply ContinuousLinearEquiv.equivOfInverse_applyₓ'. -/ @[simp] theorem equivOfInverse_apply (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂ x) : equivOfInverse f₁ f₂ h₁ h₂ x = f₁ x := rfl #align continuous_linear_equiv.equiv_of_inverse_apply ContinuousLinearEquiv.equivOfInverse_apply +/- warning: continuous_linear_equiv.symm_equiv_of_inverse -> ContinuousLinearEquiv.symm_equivOfInverse is a dubious translation: +lean 3 declaration is + forall {R₁ : Type.{u1}} {R₂ : Type.{u2}} [_inst_1 : Semiring.{u1} R₁] [_inst_2 : Semiring.{u2} R₂] {σ₁₂ : RingHom.{u1, u2} R₁ R₂ (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2)} {σ₂₁ : RingHom.{u2, u1} R₂ R₁ (Semiring.toNonAssocSemiring.{u2} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u1} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u1, u2} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u2, u1} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u3}} [_inst_12 : TopologicalSpace.{u3} M₁] [_inst_13 : AddCommMonoid.{u3} M₁] {M₂ : Type.{u4}} [_inst_16 : TopologicalSpace.{u4} M₂] [_inst_17 : AddCommMonoid.{u4} M₂] [_inst_22 : Module.{u1, u3} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u2, u4} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (h₁ : Function.LeftInverse.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearMap.toFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) f₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) f₁)) (h₂ : Function.RightInverse.{succ u3, succ u4} M₁ M₂ (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (fun (_x : ContinuousLinearMap.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) => M₂ -> M₁) (ContinuousLinearMap.toFun.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) f₂) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (fun (_x : ContinuousLinearMap.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) => M₁ -> M₂) (ContinuousLinearMap.toFun.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) f₁)), Eq.{max (succ u4) (succ u3)} (ContinuousLinearEquiv.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (ContinuousLinearEquiv.symm.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearEquiv.equivOfInverse.{u1, u2, u3, u4} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁ _inst_4 _inst_5 M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 f₁ f₂ h₁ h₂)) (ContinuousLinearEquiv.equivOfInverse.{u2, u1, u4, u3} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂ _inst_5 _inst_4 M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 f₂ f₁ h₂ h₁) +but is expected to have type + forall {R₁ : Type.{u4}} {R₂ : Type.{u3}} [_inst_1 : Semiring.{u4} R₁] [_inst_2 : Semiring.{u3} R₂] {σ₁₂ : RingHom.{u4, u3} R₁ R₂ (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1) (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2)} {σ₂₁ : RingHom.{u3, u4} R₂ R₁ (Semiring.toNonAssocSemiring.{u3} R₂ _inst_2) (Semiring.toNonAssocSemiring.{u4} R₁ _inst_1)} [_inst_4 : RingHomInvPair.{u4, u3} R₁ R₂ _inst_1 _inst_2 σ₁₂ σ₂₁] [_inst_5 : RingHomInvPair.{u3, u4} R₂ R₁ _inst_2 _inst_1 σ₂₁ σ₁₂] {M₁ : Type.{u2}} [_inst_12 : TopologicalSpace.{u2} M₁] [_inst_13 : AddCommMonoid.{u2} M₁] {M₂ : Type.{u1}} [_inst_16 : TopologicalSpace.{u1} M₂] [_inst_17 : AddCommMonoid.{u1} M₂] [_inst_22 : Module.{u4, u2} R₁ M₁ _inst_1 _inst_13] [_inst_24 : Module.{u3, u1} R₂ M₂ _inst_2 _inst_17] (f₁ : ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) (f₂ : ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) (h₁ : Function.LeftInverse.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22))) f₂) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ (fun (_x : M₁) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₁) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) M₁ M₂ _inst_12 _inst_16 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearMap.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24) R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24 (ContinuousLinearMap.continuousSemilinearMapClass.{u4, u3, u2, u1} R₁ R₂ _inst_1 _inst_2 σ₁₂ M₁ _inst_12 _inst_13 M₂ _inst_16 _inst_17 _inst_22 _inst_24))) f₁)) (h₂ : Function.RightInverse.{succ u2, succ u1} M₁ M₂ (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M₁) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) M₂ M₁ _inst_16 _inst_12 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u4, u1, u2} (ContinuousLinearMap.{u3, u4, u1, u2} R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22) R₂ R₁ _inst_2 _inst_1 σ₂₁ M₂ _inst_16 _inst_17 M₁ _inst_12 _inst_13 _inst_24 _inst_22 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Consider using '#align continuous_linear_equiv.symm_equiv_of_inverse ContinuousLinearEquiv.symm_equivOfInverseₓ'. -/ @[simp] theorem symm_equivOfInverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂) : (equivOfInverse f₁ f₂ h₁ h₂).symm = equivOfInverse f₂ f₁ h₂ h₁ := @@ -2309,6 +3759,7 @@ omit σ₂₁ variable (M₁) +#print ContinuousLinearEquiv.automorphismGroup /- /-- The continuous linear equivalences from `M` to itself form a group under composition. -/ instance automorphismGroup : Group (M₁ ≃L[R₁] M₁) where @@ -2328,11 +3779,13 @@ instance automorphismGroup : Group (M₁ ≃L[R₁] M₁) ext exact f.left_inv x #align continuous_linear_equiv.automorphism_group ContinuousLinearEquiv.automorphismGroup +-/ variable {M₁} {R₄ : Type _} [Semiring R₄] [Module R₄ M₄] {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [RingHomInvPair σ₃₄ σ₄₃] [RingHomInvPair σ₄₃ σ₃₄] {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄} [RingHomCompTriple σ₂₁ σ₁₄ σ₂₄] [RingHomCompTriple σ₂₄ σ₄₃ σ₂₃] [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] +#print ContinuousLinearEquiv.ulift /- /-- The continuous linear equivalence between `ulift M₁` and `M₁`. -/ def ulift : ULift M₁ ≃L[R₁] M₁ := { Equiv.ulift with @@ -2341,9 +3794,11 @@ def ulift : ULift M₁ ≃L[R₁] M₁ := continuous_toFun := continuous_uLift_down continuous_invFun := continuous_uLift_up } #align continuous_linear_equiv.ulift ContinuousLinearEquiv.ulift +-/ include σ₂₁ σ₃₄ σ₂₃ σ₂₄ σ₁₃ +#print ContinuousLinearEquiv.arrowCongrEquiv /- /-- A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of continuous linear maps. See also `continuous_linear_equiv.arrow_congr`. -/ @[simps] @@ -2359,6 +3814,7 @@ def arrowCongrEquiv (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[ ContinuousLinearMap.ext fun x => by simp only [ContinuousLinearMap.comp_apply, apply_symm_apply, coe_coe] #align continuous_linear_equiv.arrow_congr_equiv ContinuousLinearEquiv.arrowCongrEquiv +-/ end AddCommMonoid @@ -2371,6 +3827,12 @@ variable {R : Type _} [Semiring R] {M : Type _} [TopologicalSpace M] [AddCommGro variable [TopologicalAddGroup M₄] +/- warning: continuous_linear_equiv.skew_prod -> ContinuousLinearEquiv.skewProd is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₂] [_inst_5 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₃] [_inst_7 : AddCommGroup.{u4} M₃] {M₄ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₄] [_inst_9 : AddCommGroup.{u5} M₄] [_inst_10 : Module.{u1, u2} R M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)] [_inst_11 : Module.{u1, u3} R M₂ _inst_1 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5)] [_inst_12 : Module.{u1, u4} R M₃ _inst_1 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_7)] [_inst_13 : Module.{u1, u5} R M₄ _inst_1 (AddCommGroup.toAddCommMonoid.{u5} M₄ _inst_9)] [_inst_14 : TopologicalAddGroup.{u5} M₄ _inst_8 (AddCommGroup.toAddGroup.{u5} M₄ _inst_9)], (ContinuousLinearEquiv.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R _inst_1) (RingHomInvPair.ids.{u1} R _inst_1) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5) _inst_10 _inst_11) -> (ContinuousLinearEquiv.{u1, u1, u4, u5} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) 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Consider using '#align continuous_linear_equiv.skew_prod ContinuousLinearEquiv.skewProdₓ'. -/ /-- Equivalence given by a block lower diagonal matrix. `e` and `e'` are diagonal square blocks, and `f` is a rectangular block below the diagonal. -/ def skewProd (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) : (M × M₃) ≃L[R] M₂ × M₄ := @@ -2386,12 +3848,24 @@ def skewProd (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) continuous_snd.sub <| f.Continuous.comp <| e.continuous_invFun.comp continuous_fst) } #align continuous_linear_equiv.skew_prod ContinuousLinearEquiv.skewProd +/- warning: continuous_linear_equiv.skew_prod_apply -> ContinuousLinearEquiv.skewProd_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₂] [_inst_5 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₃] [_inst_7 : AddCommGroup.{u4} M₃] {M₄ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₄] [_inst_9 : AddCommGroup.{u5} M₄] [_inst_10 : Module.{u1, u2} R M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)] [_inst_11 : Module.{u1, u3} R M₂ _inst_1 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5)] [_inst_12 : Module.{u1, u4} R M₃ _inst_1 (AddCommGroup.toAddCommMonoid.{u4} M₃ _inst_7)] [_inst_13 : Module.{u1, u5} R M₄ _inst_1 (AddCommGroup.toAddCommMonoid.{u5} M₄ _inst_9)] [_inst_14 : TopologicalAddGroup.{u5} M₄ _inst_8 (AddCommGroup.toAddGroup.{u5} M₄ _inst_9)] (e : ContinuousLinearEquiv.{u1, u1, u2, u3} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R _inst_1) (RingHomInvPair.ids.{u1} R _inst_1) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5) _inst_10 _inst_11) (e' : 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be inaccurate. Consider using '#align continuous_linear_equiv.skew_prod_apply ContinuousLinearEquiv.skewProd_applyₓ'. -/ @[simp] theorem skewProd_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) : e.skewProd e' f x = (e x.1, e' x.2 + f x.1) := rfl #align continuous_linear_equiv.skew_prod_apply ContinuousLinearEquiv.skewProd_apply +/- warning: continuous_linear_equiv.skew_prod_symm_apply -> ContinuousLinearEquiv.skewProd_symm_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₂] [_inst_5 : AddCommGroup.{u3} M₂] {M₃ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₃] [_inst_7 : AddCommGroup.{u4} M₃] {M₄ : Type.{u5}} [_inst_8 : TopologicalSpace.{u5} M₄] [_inst_9 : AddCommGroup.{u5} M₄] [_inst_10 : Module.{u1, u2} R M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)] [_inst_11 : Module.{u1, u3} R M₂ 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Consider using '#align continuous_linear_equiv.skew_prod_symm_apply ContinuousLinearEquiv.skewProd_symm_applyₓ'. -/ @[simp] theorem skewProd_symm_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) : (e.skewProd e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) := @@ -2409,11 +3883,23 @@ variable {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [RingHomInvPair σ include σ₂₁ +/- warning: continuous_linear_equiv.map_sub -> ContinuousLinearEquiv.map_sub is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} M] [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_8 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_9 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_10 : RingHomInvPair.{u2, u1} R₂ R (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u1} R _inst_1) σ₂₁ σ₁₂] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_5 _inst_8) (x : M) (y : M), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, 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Ring.{u4} R] {R₂ : Type.{u3}} [_inst_2 : Ring.{u3} R₂] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u4, u2} R M (Ring.toSemiring.{u4} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] [_inst_8 : Module.{u3, u1} R₂ M₂ (Ring.toSemiring.{u3} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] {σ₁₂ : RingHom.{u4, u3} R R₂ (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u3} R₂ (Ring.toNonAssocRing.{u3} R₂ _inst_2))} {σ₂₁ : RingHom.{u3, u4} R₂ R (NonAssocRing.toNonAssocSemiring.{u3} R₂ (Ring.toNonAssocRing.{u3} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u4} R (Ring.toNonAssocRing.{u4} R _inst_1))} [_inst_9 : RingHomInvPair.{u4, u3} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_10 : RingHomInvPair.{u3, u4} R₂ R 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σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_8)))) e y)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_sub ContinuousLinearEquiv.map_subₓ'. -/ @[simp] theorem map_sub (e : M ≃SL[σ₁₂] M₂) (x y : M) : e (x - y) = e x - e y := (e : M →SL[σ₁₂] M₂).map_sub x y #align continuous_linear_equiv.map_sub ContinuousLinearEquiv.map_sub +/- warning: continuous_linear_equiv.map_neg -> ContinuousLinearEquiv.map_neg is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {R₂ : Type.{u2}} [_inst_2 : Ring.{u2} R₂] {M : Type.{u3}} [_inst_3 : TopologicalSpace.{u3} M] [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] {M₂ : Type.{u4}} [_inst_6 : TopologicalSpace.{u4} M₂] [_inst_7 : AddCommGroup.{u4} M₂] [_inst_8 : Module.{u2, u4} R₂ M₂ (Ring.toSemiring.{u2} R₂ _inst_2) (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7)] {σ₁₂ : RingHom.{u1, u2} R R₂ (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2))} {σ₂₁ : RingHom.{u2, u1} R₂ R (NonAssocRing.toNonAssocSemiring.{u2} R₂ (Ring.toNonAssocRing.{u2} R₂ _inst_2)) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))} [_inst_9 : RingHomInvPair.{u1, u2} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁] [_inst_10 : RingHomInvPair.{u2, u1} R₂ R (Ring.toSemiring.{u2} R₂ _inst_2) (Ring.toSemiring.{u1} R _inst_1) σ₂₁ σ₁₂] (e : ContinuousLinearEquiv.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u4} M₂ _inst_7) _inst_5 _inst_8) (x : M), Eq.{succ u4} M₂ (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearEquiv.{u1, u2, u3, u4} R R₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M 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R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_8 (ContinuousSemilinearEquivClass.continuousSemilinearMapClass.{max u2 u1, u4, u3, u2, u1} (ContinuousLinearEquiv.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_8) R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_8 (ContinuousLinearEquiv.continuousSemilinearEquivClass.{u4, u3, u2, u1} R R₂ (Ring.toSemiring.{u4} R _inst_1) (Ring.toSemiring.{u3} R₂ _inst_2) σ₁₂ σ₂₁ _inst_9 _inst_10 M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_8)))) e x)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.map_neg ContinuousLinearEquiv.map_negₓ'. -/ @[simp] theorem map_neg (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x := (e : M →SL[σ₁₂] M₂).map_neg x @@ -2429,6 +3915,12 @@ section variable [TopologicalAddGroup M] +/- warning: continuous_linear_equiv.of_unit -> ContinuousLinearEquiv.ofUnit is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_11 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)], (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) -> (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.ofUnit._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.ofUnit._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) -> (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.of_unit ContinuousLinearEquiv.ofUnitₓ'. -/ /-- An invertible continuous linear map `f` determines a continuous equivalence from `M` to itself. -/ def ofUnit (f : (M →L[R] M)ˣ) : M ≃L[R] M @@ -2450,6 +3942,12 @@ def ofUnit (f : (M →L[R] M)ˣ) : M ≃L[R] M continuous_invFun := f.inv.Continuous #align continuous_linear_equiv.of_unit ContinuousLinearEquiv.ofUnit +/- warning: continuous_linear_equiv.to_unit -> ContinuousLinearEquiv.toUnit is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_11 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)], (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.toUnit._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.toUnit._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) -> (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) -> (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.to_unit ContinuousLinearEquiv.toUnitₓ'. -/ /-- A continuous equivalence from `M` to itself determines an invertible continuous linear map. -/ def toUnit (f : M ≃L[R] M) : (M →L[R] M)ˣ where val := f @@ -2464,6 +3962,12 @@ def toUnit (f : M ≃L[R] M) : (M →L[R] M)ˣ where variable (R M) +/- warning: continuous_linear_equiv.units_equiv -> ContinuousLinearEquiv.unitsEquiv is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) [_inst_1 : Ring.{u1} R] (M : Type.{u2}) [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_11 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)], MulEquiv.{u2, u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toHasMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (Units.mulOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R 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(Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Monoid.toMulOneClass.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (DivInvMonoid.toMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) +but is expected to have type + forall (R : Type.{u1}) [_inst_1 : Ring.{u1} R] (M : Type.{u2}) [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], MulEquiv.{u2, u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Units.instMulOneClassUnits.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) (MulOneClass.toMul.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Monoid.toMulOneClass.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (DivInvMonoid.toMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.units_equiv ContinuousLinearEquiv.unitsEquivₓ'. -/ /-- The units of the algebra of continuous `R`-linear endomorphisms of `M` is multiplicatively equivalent to the type of continuous linear equivalences between `M` and itself. -/ def unitsEquiv : (M →L[R] M)ˣ ≃* M ≃L[R] M @@ -2481,6 +3985,12 @@ def unitsEquiv : (M →L[R] M)ˣ ≃* M ≃L[R] M rfl #align continuous_linear_equiv.units_equiv ContinuousLinearEquiv.unitsEquiv +/- warning: continuous_linear_equiv.units_equiv_apply -> ContinuousLinearEquiv.unitsEquiv_apply is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) [_inst_1 : Ring.{u1} R] (M : Type.{u2}) [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_11 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)] (f : Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (x : M), Eq.{succ u2} M (coeFn.{succ u2, succ u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R 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(Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toHasMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (Units.mulOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11)))) (MulOneClass.toHasMul.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Monoid.toMulOneClass.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (DivInvMonoid.toMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))))) => (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) -> (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5)) (MulEquiv.hasCoeToFun.{u2, u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toHasMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11))) (Units.mulOneClass.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.ring.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11)))) (MulOneClass.toHasMul.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) 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M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquiv._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquiv._proof_2.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))))) (ContinuousLinearEquiv.unitsEquiv.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5 _inst_11) f) x) (coeFn.{succ u2, succ u2} (Units.{u2} 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Units.instMulOneClassUnits.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) (MulOneClass.toMul.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Monoid.toMulOneClass.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (DivInvMonoid.toMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) (MulEquiv.instMulEquivClassMulEquiv.{u2, u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MulOneClass.toMul.{u2} (Units.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Units.instMulOneClassUnits.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) (MulOneClass.toMul.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Monoid.toMulOneClass.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (DivInvMonoid.toMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (Group.toDivInvMonoid.{u2} (ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearEquiv.automorphismGroup.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))))))) (ContinuousLinearEquiv.unitsEquiv.{u1, u2} R _inst_1 M _inst_3 _inst_4 _inst_5) _inst_11) f) (FunLike.coe.{succ u2, succ u2, succ u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) M (fun (a : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M) a) (ContinuousMapClass.toFunLike.{u2, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) M M _inst_3 _inst_3 (ContinuousSemilinearMapClass.toContinuousMapClass.{u2, u1, u1, u2, u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5))) (Units.val.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (MonoidWithZero.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_1) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) _inst_11) f) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.units_equiv_apply ContinuousLinearEquiv.unitsEquiv_applyₓ'. -/ @[simp] theorem unitsEquiv_apply (f : (M →L[R] M)ˣ) (x : M) : unitsEquiv R M f x = f x := rfl @@ -2492,6 +4002,12 @@ section variable (R) [TopologicalSpace R] [ContinuousMul R] +/- warning: continuous_linear_equiv.units_equiv_aut -> ContinuousLinearEquiv.unitsEquivAut is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))], Equiv.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquivAut._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquivAut._proof_2.{u1} R _inst_1) R _inst_11 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) R _inst_11 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) +but is expected to have type + forall (R : Type.{u1}) [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))], Equiv.{succ u1, succ u1} (Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) R _inst_11 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) R _inst_11 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.units_equiv_aut ContinuousLinearEquiv.unitsEquivAutₓ'. -/ /-- Continuous linear equivalences `R ≃L[R] R` are enumerated by `Rˣ`. -/ def unitsEquivAut : Rˣ ≃ R ≃L[R] R where @@ -2507,16 +4023,34 @@ def unitsEquivAut : Rˣ ≃ R ≃L[R] R variable {R} +/- warning: continuous_linear_equiv.units_equiv_aut_apply -> ContinuousLinearEquiv.unitsEquivAut_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))] (u : Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (x : R), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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(NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) R _inst_11 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (ContinuousLinearEquiv.unitsEquivAut.{u1} R _inst_1 _inst_11 _inst_12) u) x) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) R (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) R (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) R (coeBase.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) R (Units.hasCoe.{u1} R (Ring.toMonoid.{u1} R _inst_1))))) u)) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))] (u : Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (x : R), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : R) => R) x) (FunLike.coe.{succ u1, succ u1, succ u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) => ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (RingHom.id.{u1} R 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(Ring.toSemiring.{u1} R _inst_1))) u)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.units_equiv_aut_apply ContinuousLinearEquiv.unitsEquivAut_applyₓ'. -/ @[simp] theorem unitsEquivAut_apply (u : Rˣ) (x : R) : unitsEquivAut R u x = x * u := rfl #align continuous_linear_equiv.units_equiv_aut_apply ContinuousLinearEquiv.unitsEquivAut_apply +/- warning: continuous_linear_equiv.units_equiv_aut_apply_symm -> ContinuousLinearEquiv.unitsEquivAut_apply_symm is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))] (u : Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (x : R), Eq.{succ u1} R (coeFn.{succ u1, succ u1} (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R 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(NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) R _inst_11 (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Equiv.hasCoeToFun.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (ContinuousLinearEquiv.unitsEquivAut._proof_1.{u1} R _inst_1) (ContinuousLinearEquiv.unitsEquivAut._proof_2.{u1} R _inst_1) R _inst_11 (AddCommGroup.toAddCommMonoid.{u1} R 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(Units.hasCoe.{u1} R (Ring.toMonoid.{u1} R _inst_1))))) (Inv.inv.{u1} (Units.{u1} R (Ring.toMonoid.{u1} R _inst_1)) (Units.hasInv.{u1} R (Ring.toMonoid.{u1} R _inst_1)) u))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))] (u : Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (x : R), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : R) => R) x) (FunLike.coe.{succ u1, succ u1, succ u1} (ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R 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(Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (ContinuousLinearEquiv.unitsEquivAut.{u1} R _inst_1 _inst_11 _inst_12) u)) x) (HMul.hMul.{u1, u1, u1} R ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : R) => R) x) R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) x (Units.val.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Inv.inv.{u1} (Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Units.instInvUnits.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) u))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.units_equiv_aut_apply_symm ContinuousLinearEquiv.unitsEquivAut_apply_symmₓ'. -/ @[simp] theorem unitsEquivAut_apply_symm (u : Rˣ) (x : R) : (unitsEquivAut R u).symm x = x * ↑u⁻¹ := rfl #align continuous_linear_equiv.units_equiv_aut_apply_symm ContinuousLinearEquiv.unitsEquivAut_apply_symm +/- warning: continuous_linear_equiv.units_equiv_aut_symm_apply -> ContinuousLinearEquiv.unitsEquivAut_symm_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] [_inst_11 : TopologicalSpace.{u1} R] [_inst_12 : ContinuousMul.{u1} R _inst_11 (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))] (e : ContinuousLinearEquiv.{u1, u1, u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) 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Consider using '#align continuous_linear_equiv.units_equiv_aut_symm_apply ContinuousLinearEquiv.unitsEquivAut_symm_applyₓ'. -/ @[simp] theorem unitsEquivAut_symm_apply (e : R ≃L[R] R) : ↑((unitsEquivAut R).symm e) = e 1 := rfl @@ -2530,6 +4064,12 @@ open _Root_.ContinuousLinearMap (id fst snd) open _Root_.LinearMap (mem_ker) +/- warning: continuous_linear_equiv.equiv_of_right_inverse -> ContinuousLinearEquiv.equivOfRightInverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_11 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_12 : TopologicalAddGroup.{u2} M _inst_3 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(AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11)) f₁)))) +but is expected to have type + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_11 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_12 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)] (f₁ : ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11) (f₂ : ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_11 _inst_5), (Function.RightInverse.{succ u2, succ u3} M M₂ (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_11 _inst_5) M₂ (fun (_x : M₂) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M₂) => M) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u3, u2} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R 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_inst_5 (ContinuousLinearMap.continuousSemilinearMapClass.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_11 _inst_5))) f₂) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u3, u2, u3} (ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) 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_inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11)) f₁)))) (instTopologicalSpaceProd.{u3, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u1, u1, u2, u3, max u2 u3} R R M M₂ (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R 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inaccurate. Consider using '#align continuous_linear_equiv.equiv_of_right_inverse ContinuousLinearEquiv.equivOfRightInverseₓ'. -/ /-- A pair of continuous linear maps such that `f₁ ∘ f₂ = id` generates a continuous linear equivalence `e` between `M` and `M₂ × f₁.ker` such that `(e x).2 = x` for `x ∈ f₁.ker`, `(e x).1 = f₁ x`, and `(e (f₂ y)).2 = 0`. The map is given by `e x = (f₁ x, x - f₂ (f₁ x))`. -/ @@ -2539,12 +4079,24 @@ def equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Fun (fun x => by simp) fun ⟨x, y⟩ => by simp [h x] #align continuous_linear_equiv.equiv_of_right_inverse ContinuousLinearEquiv.equivOfRightInverse +/- warning: continuous_linear_equiv.fst_equiv_of_right_inverse -> ContinuousLinearEquiv.fst_equivOfRightInverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_11 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7)] [_inst_12 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)] (f₁ : 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_inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_5 _inst_11) f₁ x) +but is expected to have type + forall {R : Type.{u3}} [_inst_1 : Ring.{u3} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u1}} [_inst_6 : TopologicalSpace.{u1} M₂] [_inst_7 : AddCommGroup.{u1} M₂] [_inst_11 : Module.{u3, u1} R M₂ (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7)] [_inst_12 : TopologicalAddGroup.{u2} M _inst_3 (AddCommGroup.toAddGroup.{u2} M _inst_4)] (f₁ : ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) 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_inst_11)) f₁)) _inst_3)) (Prod.instAddCommMonoidSum.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} 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inaccurate. Consider using '#align continuous_linear_equiv.fst_equiv_of_right_inverse ContinuousLinearEquiv.fst_equivOfRightInverseₓ'. -/ @[simp] theorem fst_equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse f₂ f₁) (x : M) : (equivOfRightInverse f₁ f₂ h x).1 = f₁ x := rfl #align continuous_linear_equiv.fst_equiv_of_right_inverse ContinuousLinearEquiv.fst_equivOfRightInverse +/- warning: continuous_linear_equiv.snd_equiv_of_right_inverse -> ContinuousLinearEquiv.snd_equivOfRightInverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_11 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R 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_inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)) _inst_3)) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u3, u2, max u2 u1} (ContinuousLinearEquiv.{u3, u3, u2, max u2 u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_1)) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (Prod.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} 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(Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)))) (instTopologicalSpaceProd.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R 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(LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)) _inst_3)) (Prod.instAddCommMonoidSum.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) 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_inst_5 _inst_11)) f₁))) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (Submodule.addCommMonoid.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R 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_inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)))) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (Prod.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 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_inst_11)) f₁))) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (Submodule.addCommMonoid.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R 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(Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11))) f₁ x))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.snd_equiv_of_right_inverse ContinuousLinearEquiv.snd_equivOfRightInverseₓ'. -/ @[simp] theorem snd_equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse f₂ f₁) (x : M) : @@ -2552,6 +4104,12 @@ theorem snd_equivOfRightInverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) rfl #align continuous_linear_equiv.snd_equiv_of_right_inverse ContinuousLinearEquiv.snd_equivOfRightInverse +/- warning: continuous_linear_equiv.equiv_of_right_inverse_symm_apply -> ContinuousLinearEquiv.equivOfRightInverse_symm_apply is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_3 : TopologicalSpace.{u2} M] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {M₂ : Type.{u3}} [_inst_6 : TopologicalSpace.{u3} M₂] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_11 : Module.{u1, u3} R M₂ 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_inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁))) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (Submodule.addCommMonoid.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) 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_inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)) _inst_11 (Submodule.module.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R 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(NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁)))) M (instTopologicalSpaceProd.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R 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_inst_11)) f₁)) _inst_3)) (Prod.instAddCommMonoidSum.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} 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(Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11)) f₁))) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) (Submodule.addCommMonoid.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R 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(NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_3 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_6 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R 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_inst_5 _inst_11)) f₁)))) (instTopologicalSpaceProd.{u1, u2} M₂ (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_7) _inst_5 _inst_11 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} 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Consider using '#align continuous_linear_equiv.equiv_of_right_inverse_symm_apply ContinuousLinearEquiv.equivOfRightInverse_symm_applyₓ'. -/ @[simp] theorem equivOfRightInverse_symm_apply (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse f₂ f₁) (y : M₂ × ker f₁) : @@ -2566,18 +4124,32 @@ section variable (ι R M : Type _) [Unique ι] [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] +#print ContinuousLinearEquiv.funUnique /- /-- If `ι` has a unique element, then `ι → M` is continuously linear equivalent to `M`. -/ def funUnique : (ι → M) ≃L[R] M := { Homeomorph.funUnique ι M with toLinearEquiv := LinearEquiv.funUnique ι R M } #align continuous_linear_equiv.fun_unique ContinuousLinearEquiv.funUnique +-/ variable {ι R M} +/- warning: continuous_linear_equiv.coe_fun_unique -> ContinuousLinearEquiv.coe_funUnique is a dubious translation: +lean 3 declaration is + forall {ι : Type.{u1}} {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Unique.{succ u1} ι] [_inst_2 : 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Consider using '#align continuous_linear_equiv.coe_fun_unique ContinuousLinearEquiv.coe_funUniqueₓ'. -/ @[simp] theorem coe_funUnique : ⇑(funUnique ι R M) = Function.eval default := rfl #align continuous_linear_equiv.coe_fun_unique ContinuousLinearEquiv.coe_funUnique +/- warning: continuous_linear_equiv.coe_fun_unique_symm -> ContinuousLinearEquiv.coe_funUnique_symm is a dubious translation: +lean 3 declaration is + forall {ι : Type.{u1}} {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Unique.{succ u1} ι] [_inst_2 : Semiring.{u2} R] [_inst_3 : AddCommMonoid.{u3} M] [_inst_4 : Module.{u2, u3} R M _inst_2 _inst_3] [_inst_5 : TopologicalSpace.{u3} M], Eq.{max (succ u3) (succ (max u1 u3))} (M -> ι -> M) (coeFn.{max (succ u3) (succ (max u1 u3)), max (succ u3) (succ (max u1 u3))} (ContinuousLinearEquiv.{u2, u2, u3, max u1 u3} R R _inst_2 _inst_2 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_2)) 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Consider using '#align continuous_linear_equiv.coe_fun_unique_symm ContinuousLinearEquiv.coe_funUnique_symmₓ'. -/ @[simp] theorem coe_funUnique_symm : ⇑(funUnique ι R M).symm = Function.const ι := rfl @@ -2585,6 +4157,12 @@ theorem coe_funUnique_symm : ⇑(funUnique ι R M).symm = Function.const ι := variable (R M) +/- warning: continuous_linear_equiv.pi_fin_two -> ContinuousLinearEquiv.piFinTwo is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) [_inst_2 : Semiring.{u1} R] (M : (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) -> Type.{u2}) [_inst_6 : forall (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))), AddCommMonoid.{u2} (M i)] [_inst_7 : forall (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))), Module.{u1, u2} R (M i) _inst_2 (_inst_6 i)] [_inst_8 : forall (i : Fin (OfNat.ofNat.{0} Nat 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Consider using '#align continuous_linear_equiv.pi_fin_two ContinuousLinearEquiv.piFinTwoₓ'. -/ /-- Continuous linear equivalence between dependent functions `Π i : fin 2, M i` and `M 0 × M 1`. -/ @[simps (config := { fullyApplied := false })] def piFinTwo (M : Fin 2 → Type _) [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)] @@ -2592,6 +4170,12 @@ def piFinTwo (M : Fin 2 → Type _) [∀ i, AddCommMonoid (M i)] [∀ i, Module { Homeomorph.piFinTwo M with toLinearEquiv := LinearEquiv.piFinTwo R M } #align continuous_linear_equiv.pi_fin_two ContinuousLinearEquiv.piFinTwo +/- warning: continuous_linear_equiv.fin_two_arrow -> ContinuousLinearEquiv.finTwoArrow is a dubious translation: +lean 3 declaration is + forall (R : Type.{u1}) (M : Type.{u2}) [_inst_2 : Semiring.{u1} R] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_2 _inst_3] [_inst_5 : TopologicalSpace.{u2} M], ContinuousLinearEquiv.{u1, u1, u2, u2} R R _inst_2 _inst_2 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_2)) (RingHomInvPair.ids.{u1} R _inst_2) (RingHomInvPair.ids.{u1} R _inst_2) ((Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) -> M) (Pi.topologicalSpace.{0, u2} (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (fun (ᾰ : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => M) (fun (a : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => _inst_5)) (Pi.addCommMonoid.{0, u2} (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) (fun (ᾰ : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => M) (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) => _inst_3)) (Prod.{u2, u2} M M) (Prod.topologicalSpace.{u2, u2} M M _inst_5 _inst_5) (Prod.addCommMonoid.{u2, u2} M M _inst_3 _inst_3) (Pi.Function.module.{0, u1, u2} (Fin (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne))))) R M _inst_2 _inst_3 _inst_4) (Prod.module.{u1, u2, u2} R M M _inst_2 _inst_3 _inst_3 _inst_4 _inst_4) +but is expected to have type + forall (R : Type.{u1}) (M : Type.{u2}) [_inst_2 : Semiring.{u1} R] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Module.{u1, u2} R M _inst_2 _inst_3] [_inst_5 : TopologicalSpace.{u2} M], ContinuousLinearEquiv.{u1, u1, u2, u2} R R _inst_2 _inst_2 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_2)) (RingHomInvPair.ids.{u1} R _inst_2) (RingHomInvPair.ids.{u1} R _inst_2) ((Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) -> M) (Pi.topologicalSpace.{0, u2} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (fun (ᾰ : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => M) (fun (a : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => _inst_5)) (Pi.addCommMonoid.{0, u2} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (fun (ᾰ : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => M) (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => _inst_3)) (Prod.{u2, u2} M M) (instTopologicalSpaceProd.{u2, u2} M M _inst_5 _inst_5) (Prod.instAddCommMonoidSum.{u2, u2} M M _inst_3 _inst_3) (Pi.module.{0, u2, u1} (Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (fun (a._@.Mathlib.Topology.Algebra.Module.Basic._hyg.61742 : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => M) R _inst_2 (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => _inst_3) (fun (i : Fin (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) => _inst_4)) (Prod.module.{u1, u2, u2} R M M _inst_2 _inst_3 _inst_3 _inst_4 _inst_4) +Case conversion may be inaccurate. Consider using '#align continuous_linear_equiv.fin_two_arrow ContinuousLinearEquiv.finTwoArrowₓ'. -/ /-- Continuous linear equivalence between vectors in `M² = fin 2 → M` and `M × M`. -/ @[simps (config := { fullyApplied := false })] def finTwoArrow : (Fin 2 → M) ≃L[R] M × M := @@ -2616,6 +4200,7 @@ variable [AddCommMonoid M₂] [Module R M₂] variable [AddCommMonoid M] [Module R M] +#print ContinuousLinearMap.inverse /- /-- Introduce a function `inverse` from `M →L[R] M₂` to `M₂ →L[R] M`, which sends `f` to `f.symm` if `f` is a continuous linear equivalence and to `0` otherwise. This definition is somewhat ad hoc, but one needs a fully (rather than partially) defined inverse function for some purposes, including @@ -2623,7 +4208,14 @@ for calculus. -/ noncomputable def inverse : (M →L[R] M₂) → M₂ →L[R] M := fun f => if h : ∃ e : M ≃L[R] M₂, (e : M →L[R] M₂) = f then ((Classical.choose h).symm : M₂ →L[R] M) else 0 #align continuous_linear_map.inverse ContinuousLinearMap.inverse +-/ +/- warning: continuous_linear_map.inverse_equiv -> ContinuousLinearMap.inverse_equiv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_2 : TopologicalSpace.{u3} M₂] [_inst_3 : Semiring.{u1} R] [_inst_4 : AddCommMonoid.{u3} M₂] [_inst_5 : Module.{u1, u3} R M₂ _inst_3 _inst_4] [_inst_6 : AddCommMonoid.{u2} M] [_inst_7 : Module.{u1, u2} R M _inst_3 _inst_6] (e : ContinuousLinearEquiv.{u1, u1, u2, u3} R R _inst_3 _inst_3 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_3)) 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Consider using '#align continuous_linear_map.inverse_equiv ContinuousLinearMap.inverse_equivₓ'. -/ /-- By definition, if `f` is invertible then `inverse f = f.symm`. -/ @[simp] theorem inverse_equiv (e : M ≃L[R] M₂) : inverse (e : M →L[R] M₂) = e.symm := @@ -2634,6 +4226,12 @@ theorem inverse_equiv (e : M ≃L[R] M₂) : inverse (e : M →L[R] M₂) = e.sy exact_mod_cast Classical.choose_spec h #align continuous_linear_map.inverse_equiv ContinuousLinearMap.inverse_equiv +/- warning: continuous_linear_map.inverse_non_equiv -> ContinuousLinearMap.inverse_non_equiv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_2 : TopologicalSpace.{u3} M₂] [_inst_3 : Semiring.{u1} R] [_inst_4 : AddCommMonoid.{u3} M₂] [_inst_5 : Module.{u1, u3} R M₂ _inst_3 _inst_4] [_inst_6 : AddCommMonoid.{u2} M] [_inst_7 : Module.{u1, u2} R M _inst_3 _inst_6] (f : ContinuousLinearMap.{u1, u1, u2, u3} R R _inst_3 _inst_3 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Consider using '#align continuous_linear_map.inverse_non_equiv ContinuousLinearMap.inverse_non_equivₓ'. -/ /-- By definition, if `f` is not invertible then `inverse f = 0`. -/ @[simp] theorem inverse_non_equiv (f : M →L[R] M₂) (h : ¬∃ e' : M ≃L[R] M₂, ↑e' = f) : inverse f = 0 := @@ -2650,6 +4248,12 @@ variable [AddCommGroup M] [TopologicalAddGroup M] [Module R M] variable [AddCommGroup M₂] [Module R M₂] +/- warning: continuous_linear_map.ring_inverse_equiv -> ContinuousLinearMap.ring_inverse_equiv is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_3 : Ring.{u1} R] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : TopologicalAddGroup.{u2} M _inst_1 (AddCommGroup.toAddGroup.{u2} M _inst_4)] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] (e : ContinuousLinearEquiv.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6) (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_3)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6)))) e)) +but is expected to have type + forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : TopologicalSpace.{u1} M] [_inst_3 : Ring.{u2} R] [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_3) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] (_inst_6 : ContinuousLinearEquiv.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_3) (Ring.toSemiring.{u2} R _inst_3) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_3))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_3))) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 _inst_5), Eq.{succ u1} (ContinuousLinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_3) (Ring.toSemiring.{u2} R _inst_3) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 _inst_5) (Ring.inverse.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_3) (Ring.toSemiring.{u2} R _inst_3) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u2, u1} R (Ring.toSemiring.{u2} R _inst_3) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (ContinuousLinearEquiv.toContinuousLinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_3) (Ring.toSemiring.{u2} R _inst_3) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_3))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_3))) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 _inst_5 _inst_6)) (ContinuousLinearMap.inverse.{u2, u1, u1} R M M _inst_1 _inst_1 (Ring.toSemiring.{u2} R _inst_3) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (ContinuousLinearEquiv.toContinuousLinearMap.{u2, u2, u1, u1} R R (Ring.toSemiring.{u2} R _inst_3) (Ring.toSemiring.{u2} R _inst_3) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_3))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_3))) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) (RingHomInvPair.ids.{u2} R (Ring.toSemiring.{u2} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 _inst_5 _inst_6)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ring_inverse_equiv ContinuousLinearMap.ring_inverse_equivₓ'. -/ @[simp] theorem ring_inverse_equiv (e : M ≃L[R] M) : Ring.inverse ↑e = inverse (e : M →L[R] M) := by @@ -2659,6 +4263,12 @@ theorem ring_inverse_equiv (e : M ≃L[R] M) : Ring.inverse ↑e = inverse (e : rfl #align continuous_linear_map.ring_inverse_equiv ContinuousLinearMap.ring_inverse_equiv +/- warning: continuous_linear_map.to_ring_inverse -> ContinuousLinearMap.to_ring_inverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} {M₂ : Type.{u3}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_2 : TopologicalSpace.{u3} M₂] [_inst_3 : Ring.{u1} R] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : TopologicalAddGroup.{u2} M _inst_1 (AddCommGroup.toAddGroup.{u2} M _inst_4)] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_7 : AddCommGroup.{u3} M₂] [_inst_8 : Module.{u1, u3} R M₂ 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_inst_2 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_8 _inst_6)))) (ContinuousLinearEquiv.symm.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_3)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_7) _inst_6 _inst_8 e))) +but is expected to have type + forall {R : Type.{u3}} {M : Type.{u2}} {M₂ : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_2 : TopologicalSpace.{u1} M₂] [_inst_3 : Ring.{u3} R] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u3, u2} R M (Ring.toSemiring.{u3} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : AddCommGroup.{u1} M₂] [_inst_7 : Module.{u3, u1} R M₂ (Ring.toSemiring.{u3} R _inst_3) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6)] (_inst_8 : ContinuousLinearEquiv.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) _inst_5 _inst_7) (e : ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) _inst_5 _inst_7), Eq.{max (succ u2) (succ u1)} (ContinuousLinearMap.{u3, u3, u1, u2} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_7 _inst_5) (ContinuousLinearMap.inverse.{u3, u2, u1} R M M₂ _inst_1 _inst_2 (Ring.toSemiring.{u3} R _inst_3) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) _inst_7 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 e) (ContinuousLinearMap.comp.{u3, u3, u3, u1, u2, u2} R R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_7 _inst_5 _inst_5 (RingHomCompTriple.ids.{u3, u3} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3)))) (Ring.inverse.{u2} (ContinuousLinearMap.{u3, u3, u2, u2} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u3, u2} R (Ring.toSemiring.{u3} R _inst_3) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (ContinuousLinearMap.comp.{u3, u3, u3, u2, u1, u2} R R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_7 _inst_5 (RingHomCompTriple.ids.{u3, u3} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3)))) (ContinuousLinearEquiv.toContinuousLinearMap.{u3, u3, u1, u2} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_7 _inst_5 (ContinuousLinearEquiv.symm.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) _inst_5 _inst_7 _inst_8)) e)) (ContinuousLinearEquiv.toContinuousLinearMap.{u3, u3, u1, u2} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_7 _inst_5 (ContinuousLinearEquiv.symm.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_3) (Ring.toSemiring.{u3} R _inst_3) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_3))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (Ring.toSemiring.{u3} R _inst_3))) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) (RingHomInvPair.ids.{u3} R (Ring.toSemiring.{u3} R _inst_3)) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M₂ _inst_2 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_6) _inst_5 _inst_7 _inst_8))) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.to_ring_inverse ContinuousLinearMap.to_ring_inverseₓ'. -/ /-- The function `continuous_linear_equiv.inverse` can be written in terms of `ring.inverse` for the ring of self-maps of the domain. -/ theorem to_ring_inverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) : @@ -2680,6 +4290,12 @@ theorem to_ring_inverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) : simp #align continuous_linear_map.to_ring_inverse ContinuousLinearMap.to_ring_inverse +/- warning: continuous_linear_map.ring_inverse_eq_map_inverse -> ContinuousLinearMap.ring_inverse_eq_map_inverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_3 : Ring.{u1} R] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : TopologicalAddGroup.{u2} M _inst_1 (AddCommGroup.toAddGroup.{u2} M _inst_4)] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], Eq.{succ u2} ((ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6) -> (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6)) (Ring.inverse.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 _inst_6) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_3) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6)) (ContinuousLinearMap.inverse.{u1, u2, u2} R M M _inst_1 _inst_1 (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_6) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : TopologicalSpace.{u2} M] [_inst_3 : Ring.{u1} R] [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], Eq.{succ u2} ((ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) -> (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5)) (Ring.inverse.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_3) (Ring.toSemiring.{u1} R _inst_3) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_3))) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 _inst_5) (ContinuousLinearMap.monoidWithZero.{u1, u2} R (Ring.toSemiring.{u1} R _inst_3) M _inst_1 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) (ContinuousLinearMap.inverse.{u1, u2, u2} R M M _inst_1 _inst_1 (Ring.toSemiring.{u1} R _inst_3) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.ring_inverse_eq_map_inverse ContinuousLinearMap.ring_inverse_eq_map_inverseₓ'. -/ theorem ring_inverse_eq_map_inverse : Ring.inverse = @inverse R M M _ _ _ _ _ _ _ := by ext @@ -2697,16 +4313,25 @@ variable {R : Type _} [Ring R] {M : Type _} [TopologicalSpace M] [AddCommGroup M open ContinuousLinearMap +#print Submodule.ClosedComplemented /- /-- A submodule `p` is called *complemented* if there exists a continuous projection `M →ₗ[R] p`. -/ def ClosedComplemented (p : Submodule R M) : Prop := ∃ f : M →L[R] p, ∀ x : p, f x = x #align submodule.closed_complemented Submodule.ClosedComplemented +-/ +/- warning: submodule.closed_complemented.has_closed_complement -> Submodule.ClosedComplemented.has_closed_complement is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] [_inst_4 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)] {p : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4} [_inst_8 : T1Space.{u2} (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) p) (Subtype.topologicalSpace.{u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) x p) _inst_2)], (Submodule.ClosedComplemented.{u1, u2} R _inst_1 M _inst_2 _inst_3 _inst_4 p) -> (Exists.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (fun (q : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) => Exists.{0} (IsClosed.{u2} M _inst_2 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Set.{u2} 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(Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)))) q)) => IsCompl.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) (CompleteLattice.toBoundedOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) p q))) +but is expected to have type + forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {M : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} M] [_inst_3 : AddCommGroup.{u1} M] [_inst_4 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3)] {p : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4} [_inst_8 : T1Space.{u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4)) x p)) (instTopologicalSpaceSubtype.{u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4)) x p) _inst_2)], (Submodule.ClosedComplemented.{u2, u1} R _inst_1 M _inst_2 _inst_3 _inst_4 p) -> (Exists.{succ u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (fun (q : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) => Exists.{0} (IsClosed.{u1} M _inst_2 (SetLike.coe.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) q)) (fun (hq : IsClosed.{u1} M _inst_2 (SetLike.coe.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) q)) => IsCompl.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4))) (CompleteLattice.toBoundedOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4)) p q))) +Case conversion may be inaccurate. Consider using '#align submodule.closed_complemented.has_closed_complement Submodule.ClosedComplemented.has_closed_complementₓ'. -/ theorem ClosedComplemented.has_closed_complement {p : Submodule R M} [T1Space p] (h : ClosedComplemented p) : ∃ (q : Submodule R M)(hq : IsClosed (q : Set M)), IsCompl p q := Exists.elim h fun f hf => ⟨ker f, f.isClosed_ker, LinearMap.isCompl_of_proj hf⟩ #align submodule.closed_complemented.has_closed_complement Submodule.ClosedComplemented.has_closed_complement +#print Submodule.ClosedComplemented.isClosed /- protected theorem ClosedComplemented.isClosed [TopologicalAddGroup M] [T1Space M] {p : Submodule R M} (h : ClosedComplemented p) : IsClosed (p : Set M) := by @@ -2714,12 +4339,25 @@ protected theorem ClosedComplemented.isClosed [TopologicalAddGroup M] [T1Space M have : ker (id R M - p.subtypeL.comp f) = p := LinearMap.ker_id_sub_eq_of_proj hf exact this ▸ is_closed_ker _ #align submodule.closed_complemented.is_closed Submodule.ClosedComplemented.isClosed +-/ +/- warning: submodule.closed_complemented_bot -> Submodule.closedComplemented_bot is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] [_inst_4 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)], Submodule.ClosedComplemented.{u1, u2} R _inst_1 M _inst_2 _inst_3 _inst_4 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) +but is expected to have type + forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {M : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} M] [_inst_3 : AddCommGroup.{u1} M] [_inst_4 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3)], Submodule.ClosedComplemented.{u2, u1} R _inst_1 M _inst_2 _inst_3 _inst_4 (Bot.bot.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (Submodule.instBotSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4)) +Case conversion may be inaccurate. Consider using '#align submodule.closed_complemented_bot Submodule.closedComplemented_botₓ'. -/ @[simp] theorem closedComplemented_bot : ClosedComplemented (⊥ : Submodule R M) := ⟨0, fun x => by simp only [zero_apply, eq_zero_of_bot_submodule x]⟩ #align submodule.closed_complemented_bot Submodule.closedComplemented_bot +/- warning: submodule.closed_complemented_top -> Submodule.closedComplemented_top is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] [_inst_4 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)], Submodule.ClosedComplemented.{u1, u2} R _inst_1 M _inst_2 _inst_3 _inst_4 (Top.top.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4) (Submodule.hasTop.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_4)) +but is expected to have type + forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {M : Type.{u1}} [_inst_2 : TopologicalSpace.{u1} M] [_inst_3 : AddCommGroup.{u1} M] [_inst_4 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3)], Submodule.ClosedComplemented.{u2, u1} R _inst_1 M _inst_2 _inst_3 _inst_4 (Top.top.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4) (Submodule.instTopSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_3) _inst_4)) +Case conversion may be inaccurate. Consider using '#align submodule.closed_complemented_top Submodule.closedComplemented_topₓ'. -/ @[simp] theorem closedComplemented_top : ClosedComplemented (⊤ : Submodule R M) := ⟨(id R M).codRestrict ⊤ fun x => trivial, fun x => Subtype.ext_iff_val.2 <| by simp⟩ @@ -2727,6 +4365,12 @@ theorem closedComplemented_top : ClosedComplemented (⊤ : Submodule R M) := end Submodule +/- warning: continuous_linear_map.closed_complemented_ker_of_right_inverse -> ContinuousLinearMap.closedComplemented_ker_of_rightInverse is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {M : Type.{u2}} [_inst_2 : TopologicalSpace.{u2} M] [_inst_3 : AddCommGroup.{u2} M] {M₂ : Type.{u3}} [_inst_4 : TopologicalSpace.{u3} M₂] [_inst_5 : AddCommGroup.{u3} M₂] [_inst_6 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3)] [_inst_7 : Module.{u1, u3} R M₂ (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5)] [_inst_8 : TopologicalAddGroup.{u2} M _inst_2 (AddCommGroup.toAddGroup.{u2} M _inst_3)] (f₁ : ContinuousLinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5) _inst_6 _inst_7) (f₂ : ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u3} M₂ _inst_5) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_7 _inst_6), (Function.RightInverse.{succ u2, succ u3} M M₂ (coeFn.{max (succ u3) (succ u2), max (succ u3) (succ u2)} (ContinuousLinearMap.{u1, u1, u3, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R 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_inst_5) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) _inst_7 _inst_6))) f₂) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7) M (fun (_x : M) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : M) => M₂) _x) (ContinuousMapClass.toFunLike.{max u2 u1, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7) M M₂ _inst_2 _inst_4 (ContinuousSemilinearMapClass.toContinuousMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7))) f₁)) -> (Submodule.ClosedComplemented.{u3, u2} R _inst_1 M _inst_2 _inst_3 _inst_6 (LinearMap.ker.{u3, u3, u2, u1, max u2 u1} R R M M₂ (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7 (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7) (ContinuousSemilinearMapClass.toSemilinearMapClass.{max u2 u1, u3, u3, u2, u1} (ContinuousLinearMap.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7) R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7 (ContinuousLinearMap.continuousSemilinearMapClass.{u3, u3, u2, u1} R R (Ring.toSemiring.{u3} R _inst_1) (Ring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (NonAssocRing.toNonAssocSemiring.{u3} R (Ring.toNonAssocRing.{u3} R _inst_1))) M _inst_2 (AddCommGroup.toAddCommMonoid.{u2} M _inst_3) M₂ _inst_4 (AddCommGroup.toAddCommMonoid.{u1} M₂ _inst_5) _inst_6 _inst_7)) f₁)) +Case conversion may be inaccurate. Consider using '#align continuous_linear_map.closed_complemented_ker_of_right_inverse ContinuousLinearMap.closedComplemented_ker_of_rightInverseₓ'. -/ theorem ContinuousLinearMap.closedComplemented_ker_of_rightInverse {R : Type _} [Ring R] {M : Type _} [TopologicalSpace M] [AddCommGroup M] {M₂ : Type _} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M] [Module R M₂] [TopologicalAddGroup M] (f₁ : M →L[R] M₂) @@ -2741,14 +4385,23 @@ namespace Submodule variable {R M : Type _} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] (S : Submodule R M) +/- warning: submodule.is_open_map_mkq -> Submodule.isOpenMap_mkQ is a dubious translation: +lean 3 declaration is + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_4 : TopologicalSpace.{u2} M] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) [_inst_5 : TopologicalAddGroup.{u2} M _inst_4 (AddCommGroup.toAddGroup.{u2} M _inst_2)], IsOpenMap.{u2, u2} M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) _inst_4 (Quotient.topologicalSpace.{u2} M (Submodule.quotientRel.{u1, u2} R M _inst_1 _inst_2 _inst_3 S) _inst_4) (coeFn.{succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) _inst_3 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) (fun (_x : LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) _inst_3 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) => M -> (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S)) (LinearMap.hasCoeToFun.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) _inst_3 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) +but is expected to have type + forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_4 : TopologicalSpace.{u2} M] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) [_inst_5 : TopologicalAddGroup.{u2} M _inst_4 (AddCommGroup.toAddGroup.{u2} M _inst_2)], IsOpenMap.{u2, u2} M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) _inst_4 (QuotientModule.Quotient.topologicalSpace.{u1, u2} R M _inst_1 _inst_2 _inst_3 _inst_4 S) (FunLike.coe.{succ u2, succ u2, succ u2} (LinearMap.{u1, u1, u2, u2} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) _inst_3 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u2} R R M (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M _inst_1 _inst_2 _inst_3) S) (Submodule.Quotient.addCommGroup.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) _inst_3 (Submodule.Quotient.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 S) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) (Submodule.mkQ.{u1, u2} R M _inst_1 _inst_2 _inst_3 S)) +Case conversion may be inaccurate. Consider using '#align submodule.is_open_map_mkq Submodule.isOpenMap_mkQₓ'. -/ theorem isOpenMap_mkQ [TopologicalAddGroup M] : IsOpenMap S.mkQ := QuotientAddGroup.isOpenMap_coe S.toAddSubgroup #align submodule.is_open_map_mkq Submodule.isOpenMap_mkQ +#print Submodule.topologicalAddGroup_quotient /- instance topologicalAddGroup_quotient [TopologicalAddGroup M] : TopologicalAddGroup (M ⧸ S) := topologicalAddGroup_quotient S.toAddSubgroup #align submodule.topological_add_group_quotient Submodule.topologicalAddGroup_quotient +-/ +#print Submodule.continuousSMul_quotient /- instance continuousSMul_quotient [TopologicalSpace R] [TopologicalAddGroup M] [ContinuousSMul R M] : ContinuousSMul R (M ⧸ S) := by constructor @@ -2759,11 +4412,14 @@ instance continuousSMul_quotient [TopologicalSpace R] [TopologicalAddGroup M] [C rw [quot.continuous_iff] exact continuous_quot_mk.comp continuous_smul #align submodule.has_continuous_smul_quotient Submodule.continuousSMul_quotient +-/ +#print Submodule.t3_quotient_of_isClosed /- instance t3_quotient_of_isClosed [TopologicalAddGroup M] [IsClosed (S : Set M)] : T3Space (M ⧸ S) := letI : IsClosed (S.to_add_subgroup : Set M) := ‹_› S.to_add_subgroup.t3_quotient_of_is_closed #align submodule.t3_quotient_of_is_closed Submodule.t3_quotient_of_isClosed +-/ end Submodule diff --git a/Mathbin/Topology/Algebra/Module/FiniteDimension.lean b/Mathbin/Topology/Algebra/Module/FiniteDimension.lean index e1202aed64..bfb3de687a 100644 --- a/Mathbin/Topology/Algebra/Module/FiniteDimension.lean +++ b/Mathbin/Topology/Algebra/Module/FiniteDimension.lean @@ -62,7 +62,7 @@ variable {𝕜 E F : Type _} [Field 𝕜] [TopologicalSpace 𝕜] [AddCommGroup /-- The space of continuous linear maps between finite-dimensional spaces is finite-dimensional. -/ instance [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] : FiniteDimensional 𝕜 (E →L[𝕜] F) := - FiniteDimensional.of_injective (ContinuousLinearMap.coeLm 𝕜 : (E →L[𝕜] F) →ₗ[𝕜] E →ₗ[𝕜] F) + FiniteDimensional.of_injective (ContinuousLinearMap.coeLM 𝕜 : (E →L[𝕜] F) →ₗ[𝕜] E →ₗ[𝕜] F) ContinuousLinearMap.coe_injective end Field diff --git a/Mathbin/Topology/Algebra/Module/WeakDual.lean b/Mathbin/Topology/Algebra/Module/WeakDual.lean index f1047bea0c..34e9618355 100644 --- a/Mathbin/Topology/Algebra/Module/WeakDual.lean +++ b/Mathbin/Topology/Algebra/Module/WeakDual.lean @@ -194,7 +194,7 @@ section WeakStarTopology /-- The canonical pairing of a vector space and its topological dual. -/ def topDualPairing (𝕜 E) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] : (E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜 := - ContinuousLinearMap.coeLm 𝕜 + ContinuousLinearMap.coeLM 𝕜 #align top_dual_pairing topDualPairing variable [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] diff --git a/Mathbin/Topology/VectorBundle/Constructions.lean b/Mathbin/Topology/VectorBundle/Constructions.lean index 9e26f48c2e..116c9e078a 100644 --- a/Mathbin/Topology/VectorBundle/Constructions.lean +++ b/Mathbin/Topology/VectorBundle/Constructions.lean @@ -104,7 +104,7 @@ theorem coordChangeL_prod [e₁.isLinear 𝕜] [e₁'.isLinear 𝕜] [e₂.isLin ((e₁.Prod e₂).coordChangeL 𝕜 (e₁'.Prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) = (e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).Prod_map (e₂.coordChangeL 𝕜 e₂' b) := by - rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prod_map'] + rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prod_map] rintro ⟨v₁, v₂⟩ show (e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b (v₁, v₂) = diff --git a/lake-manifest.json b/lake-manifest.json index 7cbf5493e6..538ca9dbe3 100644 --- a/lake-manifest.json +++ b/lake-manifest.json @@ -4,15 +4,15 @@ [{"git": {"url": "https://github.com/leanprover-community/lean3port.git", "subDir?": null, - "rev": "783824c20db13c2bc44f93873399cc4086e5f1e7", + "rev": "86cf6fb4ede10bdee889bb0a02a7ffb817802979", "name": "lean3port", - "inputRev?": "783824c20db13c2bc44f93873399cc4086e5f1e7"}}, + "inputRev?": "86cf6fb4ede10bdee889bb0a02a7ffb817802979"}}, {"git": {"url": "https://github.com/leanprover-community/mathlib4.git", "subDir?": null, - "rev": "d75ac7248d6337206b297a61c0162be27506d37a", + "rev": "2dd4be199951744baaac6c20439a1ed1555ae926", "name": "mathlib", - "inputRev?": "d75ac7248d6337206b297a61c0162be27506d37a"}}, + "inputRev?": "2dd4be199951744baaac6c20439a1ed1555ae926"}}, {"git": {"url": "https://github.com/gebner/quote4", "subDir?": null, diff --git a/lakefile.lean b/lakefile.lean index 65e52bed7e..b60959f5f5 100644 --- a/lakefile.lean +++ b/lakefile.lean @@ -4,7 +4,7 @@ open Lake DSL System -- Usually the `tag` will be of the form `nightly-2021-11-22`. -- If you would like to use an artifact from a PR build, -- it will be of the form `pr-branchname-sha`. -def tag : String := "nightly-2023-04-04-14" +def tag : String := "nightly-2023-04-04-16" def releaseRepo : String := "leanprover-community/mathport" def oleanTarName : String := "mathlib3-binport.tar.gz" @@ -38,7 +38,7 @@ target fetchOleans (_pkg : Package) : Unit := do untarReleaseArtifact releaseRepo tag oleanTarName libDir return .nil -require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"783824c20db13c2bc44f93873399cc4086e5f1e7" +require lean3port from git "https://github.com/leanprover-community/lean3port.git"@"86cf6fb4ede10bdee889bb0a02a7ffb817802979" @[default_target] lean_lib Mathbin where