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[Merged by Bors] - feat: A lin. indep. family of vectors in a fin. dim. space is finite #9103

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[Merged by Bors] - feat: A lin. indep. family of vectors in a fin. dim. space is finite #9103

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feat: A lin. indep. family of vectors in a fin. dim. space is finite
YaelDillies Dec 16, 2023
3c8dbcc
fix `Algebra.Module.Zlattice`
YaelDillies Dec 16, 2023
2e3bc30
Update Mathlib/LinearAlgebra/FiniteDimensional.lean
YaelDillies Dec 16, 2023
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4 changes: 2 additions & 2 deletions Mathlib/Algebra/Module/Zlattice.lean
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Expand Up @@ -388,7 +388,7 @@ theorem Zlattice.FG : AddSubgroup.FG L := by
rw [← hs, ← h_span]
exact span_mono (by simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq, subset_rfl]))
rw [show span ℤ s = span ℤ (Set.range b) by simp [Basis.coe_mk, Subtype.range_coe_subtype]]
have : Fintype s := Set.Finite.fintype h_lind.finite
have : Fintype s := h_lind.setFinite.fintype
refine Set.Finite.of_finite_image (f := ((↑) : _ → E) ∘ Zspan.quotientEquiv b) ?_
(Function.Injective.injOn (Subtype.coe_injective.comp (Zspan.quotientEquiv b).injective) _)
have : Set.Finite ((Zspan.fundamentalDomain b) ∩ L) :=
Expand All @@ -406,7 +406,7 @@ theorem Zlattice.FG : AddSubgroup.FG L := by
exact span_le.mpr h_incl
· -- `span ℤ s` is finitely generated because `s` is finite
rw [ker_mkQ, inf_of_le_right (span_le.mpr h_incl)]
exact fg_span (LinearIndependent.finite h_lind)
exact fg_span (LinearIndependent.setFinite h_lind)

theorem Zlattice.module_finite : Module.Finite ℤ L :=
Module.Finite.iff_addGroup_fg.mpr ((AddGroup.fg_iff_addSubgroup_fg L).mpr (FG K hs))
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2 changes: 1 addition & 1 deletion Mathlib/Analysis/InnerProductSpace/PiL2.lean
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Expand Up @@ -805,7 +805,7 @@ theorem Orthonormal.exists_orthonormalBasis_extension (hv : Orthonormal 𝕜 ((
∃ (u : Finset E) (b : OrthonormalBasis u 𝕜 E), v ⊆ u ∧ ⇑b = ((↑) : u → E) := by
obtain ⟨u₀, hu₀s, hu₀, hu₀_max⟩ := exists_maximal_orthonormal hv
rw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hu₀] at hu₀_max
have hu₀_finite : u₀.Finite := hu₀.linearIndependent.finite
have hu₀_finite : u₀.Finite := hu₀.linearIndependent.setFinite
let u : Finset E := hu₀_finite.toFinset
let fu : ↥u ≃ ↥u₀ := hu₀_finite.subtypeEquivToFinset.symm
have hu : Orthonormal 𝕜 ((↑) : u → E) := by simpa using hu₀.comp _ fu.injective
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20 changes: 10 additions & 10 deletions Mathlib/LinearAlgebra/FiniteDimensional.lean
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Expand Up @@ -299,18 +299,18 @@ theorem lt_aleph0_of_linearIndependent {ι : Type w} [FiniteDimensional K V] {v
apply Cardinal.nat_lt_aleph0
#align finite_dimensional.lt_aleph_0_of_linear_independent FiniteDimensional.lt_aleph0_of_linearIndependent

theorem _root_.LinearIndependent.finite [FiniteDimensional K V] {b : Set V}
lemma _root_.LinearIndependent.finite {ι : Type*} [FiniteDimensional K V] {f : ι → V}
(h : LinearIndependent K f) : Finite ι :=
Cardinal.lt_aleph0_iff_finite.1 $ FiniteDimensional.lt_aleph0_of_linearIndependent h

theorem not_linearIndependent_of_infinite {ι : Type*} [Infinite ι] [FiniteDimensional K V]
(v : ι → V) : ¬LinearIndependent K v := mt LinearIndependent.finite $ @not_finite _ _
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#align finite_dimensional.not_linear_independent_of_infinite FiniteDimensional.not_linearIndependent_of_infinite

theorem _root_.LinearIndependent.setFinite [FiniteDimensional K V] {b : Set V}
(h : LinearIndependent K fun x : b => (x : V)) : b.Finite :=
Cardinal.lt_aleph0_iff_set_finite.mp (FiniteDimensional.lt_aleph0_of_linearIndependent h)
#align linear_independent.finite LinearIndependent.finite

theorem not_linearIndependent_of_infinite {ι : Type w} [inf : Infinite ι] [FiniteDimensional K V]
(v : ι → V) : ¬LinearIndependent K v := by
intro h_lin_indep
have : ¬ℵ₀ ≤ #ι := not_le.mpr (lt_aleph0_of_linearIndependent h_lin_indep)
have : ℵ₀ ≤ #ι := infinite_iff.mp inf
contradiction
#align finite_dimensional.not_linear_independent_of_infinite FiniteDimensional.not_linearIndependent_of_infinite
#align linear_independent.finite LinearIndependent.setFinite

/-- A finite dimensional space has positive `finrank` iff it has a nonzero element. -/
theorem finrank_pos_iff_exists_ne_zero [FiniteDimensional K V] : 0 < finrank K V ↔ ∃ x : V, x ≠ 0 :=
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