fix: denote alternating map by ⋀, not Λ (#11064)

That is, `\bigwedge`, not `\Lambda`




Co-authored-by: Richard Copley <rcopley@gmail.com>
Co-authored-by: Patrick Massot <patrickmassot@free.fr>

Mathlib/Analysis/InnerProductSpace/Orientation.lean

@@ -93,7 +93,7 @@ theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.
   rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
   -- Porting note: added `neg_one_smul` with explicit type
   simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
-    neg_one_smul ℝ (M := E [Λ^ι]→ₗ[ℝ] ℝ)]
+    neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
 #align orthonormal_basis.det_eq_neg_det_of_opposite_orientation OrthonormalBasis.det_eq_neg_det_of_opposite_orientation
 
 section AdjustToOrientation
@@ -178,10 +178,10 @@ variable [_i : Fact (finrank ℝ E = n)] (o : Orientation ℝ E (Fin n))
 /-- The volume form on an oriented real inner product space, a nonvanishing top-dimensional
 alternating form uniquely defined by compatibility with the orientation and inner product structure.
 -/
-irreducible_def volumeForm : E [Λ^Fin n]→ₗ[ℝ] ℝ := by
+irreducible_def volumeForm : E [⋀^Fin n]→ₗ[ℝ] ℝ := by
   classical
     cases' n with n
-    · let opos : E [Λ^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
+    · let opos : E [⋀^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
       exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos
     · exact (o.finOrthonormalBasis n.succ_pos _i.out).toBasis.det
 #align orientation.volume_form Orientation.volumeForm

Mathlib/Analysis/InnerProductSpace/TwoDim.lean

@@ -93,10 +93,10 @@ namespace Orientation
 notation `ω`). When evaluated on two vectors, it gives the oriented area of the parallelogram they
 span. -/
 irreducible_def areaForm : E →ₗ[ℝ] E →ₗ[ℝ] ℝ := by
-  let z : E [Λ^Fin 0]→ₗ[ℝ] ℝ ≃ₗ[ℝ] ℝ :=
+  let z : E [⋀^Fin 0]→ₗ[ℝ] ℝ ≃ₗ[ℝ] ℝ :=
     AlternatingMap.constLinearEquivOfIsEmpty.symm
-  let y : E [Λ^Fin 1]→ₗ[ℝ] ℝ →ₗ[ℝ] E →ₗ[ℝ] ℝ :=
-    LinearMap.llcomp ℝ E (E [Λ^Fin 0]→ₗ[ℝ] ℝ) ℝ z ∘ₗ AlternatingMap.curryLeftLinearMap
+  let y : E [⋀^Fin 1]→ₗ[ℝ] ℝ →ₗ[ℝ] E →ₗ[ℝ] ℝ :=
+    LinearMap.llcomp ℝ E (E [⋀^Fin 0]→ₗ[ℝ] ℝ) ℝ z ∘ₗ AlternatingMap.curryLeftLinearMap
   exact y ∘ₗ AlternatingMap.curryLeftLinearMap (R' := ℝ) o.volumeForm
 #align orientation.area_form Orientation.areaForm
 

Mathlib/LinearAlgebra/Alternating/DomCoprod.lean

@@ -43,7 +43,7 @@ open Equiv
 variable [DecidableEq ιa] [DecidableEq ιb]
 
 /-- summand used in `AlternatingMap.domCoprod` -/
-def domCoprod.summand (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂)
+def domCoprod.summand (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂)
     (σ : Perm.ModSumCongr ιa ιb) : MultilinearMap R' (fun _ : Sum ιa ιb => Mᵢ) (N₁ ⊗[R'] N₂) :=
   Quotient.liftOn' σ
     (fun σ =>
@@ -66,7 +66,7 @@ def domCoprod.summand (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→
     erw [← a.map_congr_perm fun i => v (σ₁ _), ← b.map_congr_perm fun i => v (σ₁ _)]
 #align alternating_map.dom_coprod.summand AlternatingMap.domCoprod.summand
 
-theorem domCoprod.summand_mk'' (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂)
+theorem domCoprod.summand_mk'' (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂)
     (σ : Equiv.Perm (Sum ιa ιb)) :
     domCoprod.summand a b (Quotient.mk'' σ) =
       Equiv.Perm.sign σ •
@@ -76,8 +76,8 @@ theorem domCoprod.summand_mk'' (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^
 #align alternating_map.dom_coprod.summand_mk' AlternatingMap.domCoprod.summand_mk''
 
 /-- Swapping elements in `σ` with equal values in `v` results in an addition that cancels -/
-theorem domCoprod.summand_add_swap_smul_eq_zero (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁)
-    (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
+theorem domCoprod.summand_add_swap_smul_eq_zero (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁)
+    (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
     {i j : Sum ιa ιb} (hv : v i = v j) (hij : i ≠ j) :
     domCoprod.summand a b σ v + domCoprod.summand a b (swap i j • σ) v = 0 := by
   refine Quotient.inductionOn' σ fun σ => ?_
@@ -94,8 +94,8 @@ theorem domCoprod.summand_add_swap_smul_eq_zero (a : Mᵢ [Λ^ιa]→ₗ[R'] N
 
 /-- Swapping elements in `σ` with equal values in `v` result in zero if the swap has no effect
 on the quotient. -/
-theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁)
-    (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
+theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁)
+    (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
     {i j : Sum ιa ιb} (hv : v i = v j) (hij : i ≠ j) :
     swap i j • σ = σ → domCoprod.summand a b σ v = 0 := by
   refine Quotient.inductionOn' σ fun σ => ?_
@@ -154,8 +154,8 @@ The specialized version can be obtained by combining this definition with `finSu
 `LinearMap.mul'`.
 -/
 @[simps]
-def domCoprod (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) :
-    Mᵢ [Λ^ιa ⊕ ιb]→ₗ[R'] (N₁ ⊗[R'] N₂) :=
+def domCoprod (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) :
+    Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] (N₁ ⊗[R'] N₂) :=
   { ∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand a b σ with
     toFun := fun v => (⇑(∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand a b σ)) v
     map_eq_zero_of_eq' := fun v i j hv hij => by
@@ -170,7 +170,7 @@ def domCoprod (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N
 #align alternating_map.dom_coprod AlternatingMap.domCoprod
 #align alternating_map.dom_coprod_apply AlternatingMap.domCoprod_apply
 
-theorem domCoprod_coe (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) :
+theorem domCoprod_coe (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) :
     (↑(a.domCoprod b) : MultilinearMap R' (fun _ => Mᵢ) _) =
       ∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand a b σ :=
   MultilinearMap.ext fun _ => rfl
@@ -179,8 +179,8 @@ theorem domCoprod_coe (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→
 /-- A more bundled version of `AlternatingMap.domCoprod` that maps
 `((ι₁ → N) → N₁) ⊗ ((ι₂ → N) → N₂)` to `(ι₁ ⊕ ι₂ → N) → N₁ ⊗ N₂`. -/
 def domCoprod' :
-    (Mᵢ [Λ^ιa]→ₗ[R'] N₁) ⊗[R'] (Mᵢ [Λ^ιb]→ₗ[R'] N₂) →ₗ[R']
-      (Mᵢ [Λ^ιa ⊕ ιb]→ₗ[R'] (N₁ ⊗[R'] N₂)) :=
+    (Mᵢ [⋀^ιa]→ₗ[R'] N₁) ⊗[R'] (Mᵢ [⋀^ιb]→ₗ[R'] N₂) →ₗ[R']
+      (Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] (N₁ ⊗[R'] N₂)) :=
   TensorProduct.lift <| by
     refine'
       LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => _) (fun c m n => _) (fun m n₁ n₂ => _)
@@ -200,7 +200,7 @@ def domCoprod' :
 #align alternating_map.dom_coprod' AlternatingMap.domCoprod'
 
 @[simp]
-theorem domCoprod'_apply (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) :
+theorem domCoprod'_apply (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) :
     domCoprod' (a ⊗ₜ[R'] b) = domCoprod a b :=
   rfl
 #align alternating_map.dom_coprod'_apply AlternatingMap.domCoprod'_apply
@@ -271,7 +271,7 @@ theorem MultilinearMap.domCoprod_alternization [DecidableEq ιa] [DecidableEq ι
 `AlternatingMap`s gives a scaled version of the `AlternatingMap.coprod` of those maps.
 -/
 theorem MultilinearMap.domCoprod_alternization_eq [DecidableEq ιa] [DecidableEq ιb]
-    (a : Mᵢ [Λ^ιa]→ₗ[R'] N₁) (b : Mᵢ [Λ^ιb]→ₗ[R'] N₂) :
+    (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) :
     MultilinearMap.alternatization
       (MultilinearMap.domCoprod a b : MultilinearMap R' (fun _ : Sum ιa ιb => Mᵢ) (N₁ ⊗ N₂)) =
       ((Fintype.card ιa).factorial * (Fintype.card ιb).factorial) • a.domCoprod b := by

Mathlib/LinearAlgebra/Determinant.lean

@@ -504,7 +504,7 @@ theorem LinearMap.associated_det_comp_equiv {N : Type*} [AddCommGroup N] [Module
 
 /-- The determinant of a family of vectors with respect to some basis, as an alternating
 multilinear map. -/
-nonrec def Basis.det : M [Λ^ι]→ₗ[R] R where
+nonrec def Basis.det : M [⋀^ι]→ₗ[R] R where
   toFun v := det (e.toMatrix v)
   map_add' := by
     intro inst v i x y
@@ -570,7 +570,7 @@ theorem Basis.isUnit_det (e' : Basis ι R M) : IsUnit (e.det e') :=
 
 /-- Any alternating map to `R` where `ι` has the cardinality of a basis equals the determinant
 map with respect to that basis, multiplied by the value of that alternating map on that basis. -/
-theorem AlternatingMap.eq_smul_basis_det (f : M [Λ^ι]→ₗ[R] R) : f = f e • e.det := by
+theorem AlternatingMap.eq_smul_basis_det (f : M [⋀^ι]→ₗ[R] R) : f = f e • e.det := by
   refine' Basis.ext_alternating e fun i h => _
   let σ : Equiv.Perm ι := Equiv.ofBijective i (Finite.injective_iff_bijective.1 h)
   change f (e ∘ σ) = (f e • e.det) (e ∘ σ)
@@ -579,7 +579,7 @@ theorem AlternatingMap.eq_smul_basis_det (f : M [Λ^ι]→ₗ[R] R) : f = f e 
 
 @[simp]
 theorem AlternatingMap.map_basis_eq_zero_iff {ι : Type*} [Finite ι] (e : Basis ι R M)
-    (f : M [Λ^ι]→ₗ[R] R) : f e = 0 ↔ f = 0 :=
+    (f : M [⋀^ι]→ₗ[R] R) : f e = 0 ↔ f = 0 :=
   ⟨fun h => by
     cases nonempty_fintype ι
     letI := Classical.decEq ι
@@ -588,7 +588,7 @@ theorem AlternatingMap.map_basis_eq_zero_iff {ι : Type*} [Finite ι] (e : Basis
 #align alternating_map.map_basis_eq_zero_iff AlternatingMap.map_basis_eq_zero_iff
 
 theorem AlternatingMap.map_basis_ne_zero_iff {ι : Type*} [Finite ι] (e : Basis ι R M)
-    (f : M [Λ^ι]→ₗ[R] R) : f e ≠ 0 ↔ f ≠ 0 :=
+    (f : M [⋀^ι]→ₗ[R] R) : f e ≠ 0 ↔ f ≠ 0 :=
   not_congr <| f.map_basis_eq_zero_iff e
 #align alternating_map.map_basis_ne_zero_iff AlternatingMap.map_basis_ne_zero_iff
 

Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean

@@ -295,7 +295,7 @@ variable (R)
 
 This is a special case of `MultilinearMap.mkPiAlgebraFin`, and the exterior algebra version of
 `TensorAlgebra.tprod`. -/
-def ιMulti (n : ℕ) : M [Λ^Fin n]→ₗ[R] ExteriorAlgebra R M :=
+def ιMulti (n : ℕ) : M [⋀^Fin n]→ₗ[R] ExteriorAlgebra R M :=
   let F := (MultilinearMap.mkPiAlgebraFin R n (ExteriorAlgebra R M)).compLinearMap fun _ => ι R
   { F with
     map_eq_zero_of_eq' := fun f x y hfxy hxy => by

Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean

@@ -33,7 +33,7 @@ variable [Module R M] [Module R N] [Module R N']
 
 -- This instance can't be found where it's needed if we don't remind lean that it exists.
 instance AlternatingMap.instModuleAddCommGroup {ι : Type*} :
-    Module R (M [Λ^ι]→ₗ[R] N) := by
+    Module R (M [⋀^ι]→ₗ[R] N) := by
   infer_instance
 #align alternating_map.module_add_comm_group AlternatingMap.instModuleAddCommGroup
 
@@ -43,10 +43,10 @@ open CliffordAlgebra hiding ι
 
 /-- Build a map out of the exterior algebra given a collection of alternating maps acting on each
 exterior power -/
-def liftAlternating : (∀ i, M [Λ^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N := by
+def liftAlternating : (∀ i, M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N := by
   suffices
-    (∀ i, M [Λ^Fin i]→ₗ[R] N) →ₗ[R]
-      ExteriorAlgebra R M →ₗ[R] ∀ i, M [Λ^Fin i]→ₗ[R] N by
+    (∀ i, M [⋀^Fin i]→ₗ[R] N) →ₗ[R]
+      ExteriorAlgebra R M →ₗ[R] ∀ i, M [⋀^Fin i]→ₗ[R] N by
     refine' LinearMap.compr₂ this _
     refine' (LinearEquiv.toLinearMap _).comp (LinearMap.proj 0)
     exact AlternatingMap.constLinearEquivOfIsEmpty.symm
@@ -67,7 +67,7 @@ def liftAlternating : (∀ i, M [Λ^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra
 #align exterior_algebra.lift_alternating ExteriorAlgebra.liftAlternating
 
 @[simp]
-theorem liftAlternating_ι (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) (m : M) :
+theorem liftAlternating_ι (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M) :
     liftAlternating (R := R) (M := M) (N := N) f (ι R m) = f 1 ![m] := by
   dsimp [liftAlternating]
   rw [foldl_ι, LinearMap.mk₂_apply, AlternatingMap.curryLeft_apply_apply]
@@ -78,7 +78,7 @@ theorem liftAlternating_ι (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) (m : M) :
   rw [Matrix.zero_empty]
 #align exterior_algebra.lift_alternating_ι ExteriorAlgebra.liftAlternating_ι
 
-theorem liftAlternating_ι_mul (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) (m : M)
+theorem liftAlternating_ι_mul (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M)
     (x : ExteriorAlgebra R M) :
     liftAlternating (R := R) (M := M) (N := N) f (ι R m * x) =
     liftAlternating (R := R) (M := M) (N := N) (fun i => (f i.succ).curryLeft m) x := by
@@ -88,21 +88,21 @@ theorem liftAlternating_ι_mul (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) (m : M)
 #align exterior_algebra.lift_alternating_ι_mul ExteriorAlgebra.liftAlternating_ι_mul
 
 @[simp]
-theorem liftAlternating_one (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) :
+theorem liftAlternating_one (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) :
     liftAlternating (R := R) (M := M) (N := N) f (1 : ExteriorAlgebra R M) = f 0 0 := by
   dsimp [liftAlternating]
   rw [foldl_one]
 #align exterior_algebra.lift_alternating_one ExteriorAlgebra.liftAlternating_one
 
 @[simp]
-theorem liftAlternating_algebraMap (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) (r : R) :
+theorem liftAlternating_algebraMap (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (r : R) :
     liftAlternating (R := R) (M := M) (N := N) f (algebraMap _ (ExteriorAlgebra R M) r) =
     r • f 0 0 := by
   rw [Algebra.algebraMap_eq_smul_one, map_smul, liftAlternating_one]
 #align exterior_algebra.lift_alternating_algebra_map ExteriorAlgebra.liftAlternating_algebraMap
 
 @[simp]
-theorem liftAlternating_apply_ιMulti {n : ℕ} (f : ∀ i, M [Λ^Fin i]→ₗ[R] N)
+theorem liftAlternating_apply_ιMulti {n : ℕ} (f : ∀ i, M [⋀^Fin i]→ₗ[R] N)
     (v : Fin n → M) : liftAlternating (R := R) (M := M) (N := N) f (ιMulti R n v) = f n v := by
   rw [ιMulti_apply]
   -- porting note: `v` is generalized automatically so it was removed from the next line
@@ -118,13 +118,13 @@ theorem liftAlternating_apply_ιMulti {n : ℕ} (f : ∀ i, M [Λ^Fin i]→ₗ[R
 #align exterior_algebra.lift_alternating_apply_ι_multi ExteriorAlgebra.liftAlternating_apply_ιMulti
 
 @[simp]
-theorem liftAlternating_comp_ιMulti {n : ℕ} (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) :
+theorem liftAlternating_comp_ιMulti {n : ℕ} (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) :
     (liftAlternating (R := R) (M := M) (N := N) f).compAlternatingMap (ιMulti R n) = f n :=
   AlternatingMap.ext <| liftAlternating_apply_ιMulti f
 #align exterior_algebra.lift_alternating_comp_ι_multi ExteriorAlgebra.liftAlternating_comp_ιMulti
 
 @[simp]
-theorem liftAlternating_comp (g : N →ₗ[R] N') (f : ∀ i, M [Λ^Fin i]→ₗ[R] N) :
+theorem liftAlternating_comp (g : N →ₗ[R] N') (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) :
     (liftAlternating (R := R) (M := M) (N := N') fun i => g.compAlternatingMap (f i)) =
     g ∘ₗ liftAlternating (R := R) (M := M) (N := N) f := by
   ext v
@@ -152,7 +152,7 @@ theorem liftAlternating_ιMulti :
 
 /-- `ExteriorAlgebra.liftAlternating` is an equivalence. -/
 @[simps apply symm_apply]
-def liftAlternatingEquiv : (∀ i, M [Λ^Fin i]→ₗ[R] N) ≃ₗ[R] ExteriorAlgebra R M →ₗ[R] N
+def liftAlternatingEquiv : (∀ i, M [⋀^Fin i]→ₗ[R] N) ≃ₗ[R] ExteriorAlgebra R M →ₗ[R] N
     where
   toFun := liftAlternating (R := R)
   map_add' := map_add _

Mathlib/LinearAlgebra/Matrix/Determinant.lean

@@ -58,7 +58,7 @@ variable {R : Type v} [CommRing R]
 local notation "ε " σ:arg => ((sign σ : ℤ) : R)
 
 /-- `det` is an `AlternatingMap` in the rows of the matrix. -/
-def detRowAlternating : (n → R) [Λ^n]→ₗ[R] R :=
+def detRowAlternating : (n → R) [⋀^n]→ₗ[R] R :=
   MultilinearMap.alternatization ((MultilinearMap.mkPiAlgebra R n R).compLinearMap LinearMap.proj)
 #align matrix.det_row_alternating Matrix.detRowAlternating
 
@@ -91,7 +91,7 @@ theorem det_diagonal {d : n → R} : det (diagonal d) = ∏ i, d i := by
 
 -- @[simp] -- Porting note (#10618): simp can prove this
 theorem det_zero (_ : Nonempty n) : det (0 : Matrix n n R) = 0 :=
-  (detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_zero
+  (detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_zero
 #align matrix.det_zero Matrix.det_zero
 
 @[simp]
@@ -240,7 +240,7 @@ theorem det_transpose (M : Matrix n n R) : Mᵀ.det = M.det := by
 /-- Permuting the columns changes the sign of the determinant. -/
 theorem det_permute (σ : Perm n) (M : Matrix n n R) :
     (Matrix.det fun i => M (σ i)) = Perm.sign σ * M.det :=
-  ((detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_perm M σ).trans (by simp [Units.smul_def])
+  ((detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_perm M σ).trans (by simp [Units.smul_def])
 #align matrix.det_permute Matrix.det_permute
 
 /-- Permuting rows and columns with the same equivalence has no effect. -/
@@ -362,7 +362,7 @@ Prove that a matrix with a repeated column has determinant equal to zero.
 
 
 theorem det_eq_zero_of_row_eq_zero {A : Matrix n n R} (i : n) (h : ∀ j, A i j = 0) : det A = 0 :=
-  (detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_coord_zero i (funext h)
+  (detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_coord_zero i (funext h)
 #align matrix.det_eq_zero_of_row_eq_zero Matrix.det_eq_zero_of_row_eq_zero
 
 theorem det_eq_zero_of_column_eq_zero {A : Matrix n n R} (j : n) (h : ∀ i, A i j = 0) :
@@ -375,7 +375,7 @@ variable {M : Matrix n n R} {i j : n}
 
 /-- If a matrix has a repeated row, the determinant will be zero. -/
 theorem det_zero_of_row_eq (i_ne_j : i ≠ j) (hij : M i = M j) : M.det = 0 :=
-  (detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_eq_zero_of_eq M hij i_ne_j
+  (detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_eq_zero_of_eq M hij i_ne_j
 #align matrix.det_zero_of_row_eq Matrix.det_zero_of_row_eq
 
 /-- If a matrix has a repeated column, the determinant will be zero. -/
@@ -388,7 +388,7 @@ end DetZero
 
 theorem det_updateRow_add (M : Matrix n n R) (j : n) (u v : n → R) :
     det (updateRow M j <| u + v) = det (updateRow M j u) + det (updateRow M j v) :=
-  (detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_add M j u v
+  (detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_add M j u v
 #align matrix.det_update_row_add Matrix.det_updateRow_add
 
 theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
@@ -399,7 +399,7 @@ theorem det_updateColumn_add (M : Matrix n n R) (j : n) (u v : n → R) :
 
 theorem det_updateRow_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
     det (updateRow M j <| s • u) = s * det (updateRow M j u) :=
-  (detRowAlternating : (n → R) [Λ^n]→ₗ[R] R).map_smul M j s u
+  (detRowAlternating : (n → R) [⋀^n]→ₗ[R] R).map_smul M j s u
 #align matrix.det_update_row_smul Matrix.det_updateRow_smul
 
 theorem det_updateColumn_smul (M : Matrix n n R) (j : n) (s : R) (u : n → R) :