ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlternatingMap.coe_prod
@[simp]
theorem coe_prod (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P) :
    (f.prod g : MultilinearMap R (fun _ : ι => M) (N × P)) = MultilinearMap.prod f g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {P : Type u_4} [AddCommMonoid P] [Module R P] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (g : M [⋀^ι]→ₗ[R] P)
Goal: ↑(f.prod g) = (↑f).prod ↑g
-- AlternatingMap.pi_apply
@[simps!]
def pi {ι' : Type*} {N : ι' → Type*} [∀ i, AddCommMonoid (N i)] [∀ i, Module R (N i)]
    (f : ∀ i, M [⋀^ι]→ₗ[R] N i) : M [⋀^ι]→ₗ[R] (∀ i, N i) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_7} {ι' : Type u_10} {N : ι' → Type u_11} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→ₗ[R] N i) (m : (i : ι) → (fun x => M) i) (i : ι')
Goal: (AlternatingMap.pi f) m i = (f i) m
-- AlternatingMap.coe_pi
@[simp]
theorem coe_pi {ι' : Type*} {N : ι' → Type*} [∀ i, AddCommMonoid (N i)] [∀ i, Module R (N i)]
    (f : ∀ i, M [⋀^ι]→ₗ[R] N i) :
    (pi f : MultilinearMap R (fun _ : ι => M) (∀ i, N i)) = MultilinearMap.pi fun a => f a 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {ι : Type u_7} {ι' : Type u_10} {N : ι' → Type u_11} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→ₗ[R] N i)
Goal: ↑(AlternatingMap.pi f) = MultilinearMap.pi fun a => ↑(f a)
-- AlternatingMap.smulRight_apply
@[simps!]
def smulRight {R M₁ M₂ ι : Type*} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂]
    [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) : M₁ [⋀^ι]→ₗ[R] M₂ 
Arguments: {R : Type u_10} {M₁ : Type u_11} {M₂ : Type u_12} {ι : Type u_13} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂)
Goal: ∀ (a : (i : ι) → (fun x => M₁) i), (f.smulRight z) a = f a • z
-- AlternatingMap.coe_smulRight
@[simp]
theorem coe_smulRight {R M₁ M₂ ι : Type*} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂]
    [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) :
    (f.smulRight z : MultilinearMap R (fun _ : ι => M₁) M₂) = MultilinearMap.smulRight f z 
Arguments: {R : Type u_10} {M₁ : Type u_11} {M₂ : Type u_12} {ι : Type u_13} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂)
Goal: ↑(f.smulRight z) = (↑f).smulRight z
-- AlternatingMap.add_apply
@[simp]
theorem add_apply : (f + f') v = f v + f' v 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f f' : M [⋀^ι]→ₗ[R] N) (v : ι → M)
Goal: (f + f') v = f v + f' v
-- AlternatingMap.coe_add
@[norm_cast]
theorem coe_add : (↑(f + f') : MultilinearMap R (fun _ : ι => M) N) = f + f' 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f f' : M [⋀^ι]→ₗ[R] N)
Goal: ↑(f + f') = ↑f + ↑f'
-- AlternatingMap.zero_apply
@[simp]
theorem zero_apply : (0 : M [⋀^ι]→ₗ[R] N) v = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (v : ι → M)
Goal: 0 v = 0
-- AlternatingMap.coe_zero
@[norm_cast]
theorem coe_zero : ((0 : M [⋀^ι]→ₗ[R] N) : MultilinearMap R (fun _ : ι => M) N) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7}
Goal: ↑0 = 0
-- AlternatingMap.mk_zero
@[simp]
theorem mk_zero :
    mk (0 : MultilinearMap R (fun _ : ι ↦ M) N) (0 : M [⋀^ι]→ₗ[R] N).2 = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7}
Goal: { toMultilinearMap := 0, map_eq_zero_of_eq' := ⋯ } = 0
-- AlternatingMap.neg_apply
@[simp]
theorem neg_apply (m : ι → M) : (-g) m = -g m 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (m : ι → M)
Goal: (-g) m = -g m
-- AlternatingMap.coe_neg
@[norm_cast]
theorem coe_neg : ((-g : M [⋀^ι]→ₗ[R] N') : MultilinearMap R (fun _ : ι => M) N') = -g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N')
Goal: ↑(-g) = -↑g
-- AlternatingMap.sub_apply
@[simp]
theorem sub_apply (m : ι → M) : (g - g₂) m = g m - g₂ m 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g g₂ : M [⋀^ι]→ₗ[R] N') (m : ι → M)
Goal: (g - g₂) m = g m - g₂ m
-- AlternatingMap.coe_sub
@[norm_cast]
theorem coe_sub : (↑(g - g₂) : MultilinearMap R (fun _ : ι => M) N') = g - g₂ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g g₂ : M [⋀^ι]→ₗ[R] N')
Goal: ↑(g - g₂) = ↑g - ↑g₂
-- AlternatingMap.ofSubsingleton_apply_apply
@[simps!]
def ofSubsingleton [Subsingleton ι] (i : ι) : (M →ₗ[R] N) ≃ (M [⋀^ι]→ₗ[R] N) 
Arguments: (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] {ι : Type u_7} [Subsingleton ι] (i : ι) (f : M →ₗ[R] N) (x : ι → M)
Goal: ((AlternatingMap.ofSubsingleton R M N i) f) x = f (x i)
-- AlternatingMap.ofSubsingleton_symm_apply_apply
@[simps!]
def ofSubsingleton [Subsingleton ι] (i : ι) : (M →ₗ[R] N) ≃ (M [⋀^ι]→ₗ[R] N) 
Arguments: (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] {ι : Type u_7} [Subsingleton ι] (i : ι) (f : M [⋀^ι]→ₗ[R] N) (x : M)
Goal: ((AlternatingMap.ofSubsingleton R M N i).symm f) x = f fun x_1 => x
-- AlternatingMap.constOfIsEmpty_apply
@[simps (config := .asFn)]
def constOfIsEmpty [IsEmpty ι] (m : N) : M [⋀^ι]→ₗ[R] N 
Arguments: (R : Type u_1) [Semiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_7) [IsEmpty ι] (m : N)
Goal: ⇑(AlternatingMap.constOfIsEmpty R M ι m) = Function.const (ι → M) m
-- AlternatingMap.codRestrict_apply_coe
@[simps]
def codRestrict (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ v, f v ∈ p) :
    M [⋀^ι]→ₗ[R] p 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p) (v : ι → M)
Goal: ↑((f.codRestrict p h) v) = f v
-- LinearMap.coe_compAlternatingMap
@[simp]
theorem coe_compAlternatingMap (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) :
    ⇑(g.compAlternatingMap f) = g ∘ f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N)
Goal: ⇑(g.compAlternatingMap f) = ⇑g ∘ ⇑f
-- LinearMap.compAlternatingMap_apply
@[simp]
theorem compAlternatingMap_apply (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) (m : ι → M) :
    g.compAlternatingMap f m = g (f m) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) (m : ι → M)
Goal: (g.compAlternatingMap f) m = g (f m)
-- LinearMap.smulRight_eq_comp
theorem smulRight_eq_comp {R M₁ M₂ ι : Type*} [CommSemiring R] [AddCommMonoid M₁]
    [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂) :
    f.smulRight z = (LinearMap.id.smulRight z).compAlternatingMap f 
Arguments: {R : Type u_11} {M₁ : Type u_12} {M₂ : Type u_13} {ι : Type u_14} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] (f : M₁ [⋀^ι]→ₗ[R] R) (z : M₂)
Goal: f.smulRight z = (LinearMap.id.smulRight z).compAlternatingMap f
-- LinearMap.subtype_compAlternatingMap_codRestrict
@[simp]
theorem subtype_compAlternatingMap_codRestrict (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N)
    (h) : p.subtype.compAlternatingMap (f.codRestrict p h) = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N) (h : ∀ (v : ι → M), f v ∈ p)
Goal: p.subtype.compAlternatingMap (f.codRestrict p h) = f
-- LinearMap.compAlternatingMap_codRestrict
@[simp]
theorem compAlternatingMap_codRestrict (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N)
    (p : Submodule R N₂) (h) :
    (g.codRestrict p h).compAlternatingMap f =
      (g.compAlternatingMap f).codRestrict p fun v => h (f v) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {N₂ : Type u_10} [AddCommMonoid N₂] [Module R N₂] (g : N →ₗ[R] N₂) (f : M [⋀^ι]→ₗ[R] N) (p : Submodule R N₂) (h : ∀ (c : N), g c ∈ p)
Goal: (LinearMap.codRestrict p g h).compAlternatingMap f = (g.compAlternatingMap f).codRestrict p ⋯
-- AlternatingMap.coe_compLinearMap
theorem coe_compLinearMap (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) :
    ⇑(f.compLinearMap g) = f ∘ (g ∘ ·) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M)
Goal: ⇑(f.compLinearMap g) = ⇑f ∘ fun x => ⇑g ∘ x
-- AlternatingMap.compLinearMap_apply
@[simp]
theorem compLinearMap_apply (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) (v : ι → M₂) :
    f.compLinearMap g v = f fun i => g (v i) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) (v : ι → M₂)
Goal: (f.compLinearMap g) v = f fun i => g (v i)
-- AlternatingMap.compLinearMap_assoc
theorem compLinearMap_assoc (f : M [⋀^ι]→ₗ[R] N) (g₁ : M₂ →ₗ[R] M) (g₂ : M₃ →ₗ[R] M₂) :
    (f.compLinearMap g₁).compLinearMap g₂ = f.compLinearMap (g₁ ∘ₗ g₂) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_11} [AddCommMonoid M₃] [Module R M₃] (f : M [⋀^ι]→ₗ[R] N) (g₁ : M₂ →ₗ[R] M) (g₂ : M₃ →ₗ[R] M₂)
Goal: (f.compLinearMap g₁).compLinearMap g₂ = f.compLinearMap (g₁ ∘ₗ g₂)
-- AlternatingMap.zero_compLinearMap
@[simp]
theorem zero_compLinearMap (g : M₂ →ₗ[R] M) : (0 : M [⋀^ι]→ₗ[R] N).compLinearMap g = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (g : M₂ →ₗ[R] M)
Goal: AlternatingMap.compLinearMap 0 g = 0
-- AlternatingMap.add_compLinearMap
@[simp]
theorem add_compLinearMap (f₁ f₂ : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M) :
    (f₁ + f₂).compLinearMap g = f₁.compLinearMap g + f₂.compLinearMap g 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f₁ f₂ : M [⋀^ι]→ₗ[R] N) (g : M₂ →ₗ[R] M)
Goal: (f₁ + f₂).compLinearMap g = f₁.compLinearMap g + f₂.compLinearMap g
-- AlternatingMap.compLinearMap_zero
@[simp]
theorem compLinearMap_zero [Nonempty ι] (f : M [⋀^ι]→ₗ[R] N) :
    f.compLinearMap (0 : M₂ →ₗ[R] M) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] [Nonempty ι] (f : M [⋀^ι]→ₗ[R] N)
Goal: f.compLinearMap 0 = 0
-- AlternatingMap.compLinearMap_id
@[simp]
theorem compLinearMap_id (f : M [⋀^ι]→ₗ[R] N) : f.compLinearMap LinearMap.id = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N)
Goal: f.compLinearMap LinearMap.id = f
-- AlternatingMap.compLinearMap_injective
theorem compLinearMap_injective (f : M₂ →ₗ[R] M) (hf : Function.Surjective f) :
    Function.Injective fun g : M [⋀^ι]→ₗ[R] N => g.compLinearMap f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M₂ →ₗ[R] M) (hf : Function.Surjective ⇑f)
Goal: Function.Injective fun g => g.compLinearMap f
-- AlternatingMap.compLinearMap_inj
theorem compLinearMap_inj (f : M₂ →ₗ[R] M) (hf : Function.Surjective f)
    (g₁ g₂ : M [⋀^ι]→ₗ[R] N) : g₁.compLinearMap f = g₂.compLinearMap f ↔ g₁ = g₂ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M₂ →ₗ[R] M) (hf : Function.Surjective ⇑f) (g₁ g₂ : M [⋀^ι]→ₗ[R] N)
Goal: g₁.compLinearMap f = g₂.compLinearMap f ↔ g₁ = g₂
-- AlternatingMap.domLCongr_apply
@[simps apply]
def domLCongr (e : M ≃ₗ[R] M₂) : M [⋀^ι]→ₗ[R] N ≃ₗ[S] (M₂ [⋀^ι]→ₗ[R] N) 
Arguments: (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) (f : M [⋀^ι]→ₗ[R] N)
Goal: (AlternatingMap.domLCongr R N ι S e) f = f.compLinearMap ↑e.symm
-- AlternatingMap.domLCongr_refl
@[simp]
theorem domLCongr_refl : domLCongr R N ι S (LinearEquiv.refl R M) = LinearEquiv.refl S _ 
Arguments: (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N]
Goal: AlternatingMap.domLCongr R N ι S (LinearEquiv.refl R M) = LinearEquiv.refl S (M [⋀^ι]→ₗ[R] N)
-- AlternatingMap.domLCongr_symm
@[simp]
theorem domLCongr_symm (e : M ≃ₗ[R] M₂) : (domLCongr R N ι S e).symm = domLCongr R N ι S e.symm 
Arguments: (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂)
Goal: (AlternatingMap.domLCongr R N ι S e).symm = AlternatingMap.domLCongr R N ι S e.symm
-- AlternatingMap.domLCongr_trans
theorem domLCongr_trans (e : M ≃ₗ[R] M₂) (f : M₂ ≃ₗ[R] M₃) :
    (domLCongr R N ι S e).trans (domLCongr R N ι S f) = domLCongr R N ι S (e.trans f) 
Arguments: (R : Type u_1) [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (N : Type u_3) [AddCommMonoid N] [Module R N] (ι : Type u_7) {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] {M₃ : Type u_11} [AddCommMonoid M₃] [Module R M₃] (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (e : M ≃ₗ[R] M₂) (f : M₂ ≃ₗ[R] M₃)
Goal: AlternatingMap.domLCongr R N ι S e ≪≫ₗ AlternatingMap.domLCongr R N ι S f = AlternatingMap.domLCongr R N ι S (e ≪≫ₗ f)
-- AlternatingMap.compLinearEquiv_eq_zero_iff
@[simp]
theorem compLinearEquiv_eq_zero_iff (f : M [⋀^ι]→ₗ[R] N) (g : M₂ ≃ₗ[R] M) :
    f.compLinearMap (g : M₂ →ₗ[R] M) = 0 ↔ f = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {M₂ : Type u_10} [AddCommMonoid M₂] [Module R M₂] (f : M [⋀^ι]→ₗ[R] N) (g : M₂ ≃ₗ[R] M)
Goal: f.compLinearMap ↑g = 0 ↔ f = 0
-- AlternatingMap.map_update_sum
theorem map_update_sum {α : Type*} [DecidableEq ι] (t : Finset α) (i : ι) (g : α → M) (m : ι → M) :
    f (update m i (∑ a ∈ t, g a)) = ∑ a ∈ t, f (update m i (g a)) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) {α : Type u_12} [DecidableEq ι] (t : Finset α) (i : ι) (g : α → M) (m : ι → M)
Goal: f (Function.update m i (∑ a ∈ t, g a)) = ∑ a ∈ t, f (Function.update m i (g a))
-- AlternatingMap.map_update_self
theorem map_update_self [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
    f (Function.update v i (v j)) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i j : ι} (hij : i ≠ j)
Goal: f (Function.update v i (v j)) = 0
-- AlternatingMap.map_update_update
theorem map_update_update [DecidableEq ι] {i j : ι} (hij : i ≠ j) (m : M) :
    f (Function.update (Function.update v i m) j m) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i j : ι} (hij : i ≠ j) (m : M)
Goal: f (Function.update (Function.update v i m) j m) = 0
-- AlternatingMap.map_swap_add
theorem map_swap_add [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
    f (v ∘ Equiv.swap i j) + f v = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i j : ι} (hij : i ≠ j)
Goal: f (v ∘ ⇑(Equiv.swap i j)) + f v = 0
-- AlternatingMap.map_add_swap
theorem map_add_swap [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
    f v + f (v ∘ Equiv.swap i j) = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N) (v : ι → M) [DecidableEq ι] {i j : ι} (hij : i ≠ j)
Goal: f v + f (v ∘ ⇑(Equiv.swap i j)) = 0
-- AlternatingMap.map_swap
theorem map_swap [DecidableEq ι] {i j : ι} (hij : i ≠ j) : g (v ∘ Equiv.swap i j) = -g v 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (v : ι → M) [DecidableEq ι] {i j : ι} (hij : i ≠ j)
Goal: g (v ∘ ⇑(Equiv.swap i j)) = -g v
-- AlternatingMap.map_perm
theorem map_perm [DecidableEq ι] [Fintype ι] (v : ι → M) (σ : Equiv.Perm ι) :
    g (v ∘ σ) = Equiv.Perm.sign σ • g v 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') [DecidableEq ι] [Fintype ι] (v : ι → M) (σ : Equiv.Perm ι)
Goal: g (v ∘ ⇑σ) = Equiv.Perm.sign σ • g v
-- AlternatingMap.map_congr_perm
theorem map_congr_perm [DecidableEq ι] [Fintype ι] (σ : Equiv.Perm ι) :
    g v = Equiv.Perm.sign σ • g (v ∘ σ) 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N' : Type u_6} [AddCommGroup N'] [Module R N'] {ι : Type u_7} (g : M [⋀^ι]→ₗ[R] N') (v : ι → M) [DecidableEq ι] [Fintype ι] (σ : Equiv.Perm ι)
Goal: g v = Equiv.Perm.sign σ • g (v ∘ ⇑σ)
-- AlternatingMap.domDomCongr_apply
@[simps]
def domDomCongr (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) : M [⋀^ι']→ₗ[R] N 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N) (v : ι' → M)
Goal: (AlternatingMap.domDomCongr σ f) v = f (v ∘ ⇑σ)
-- AlternatingMap.domDomCongr_refl
@[simp]
theorem domDomCongr_refl (f : M [⋀^ι]→ₗ[R] N) : f.domDomCongr (Equiv.refl ι) = f 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (f : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr (Equiv.refl ι) f = f
-- AlternatingMap.domDomCongr_trans
theorem domDomCongr_trans (σ₁ : ι ≃ ι') (σ₂ : ι' ≃ ι'') (f : M [⋀^ι]→ₗ[R] N) :
    f.domDomCongr (σ₁.trans σ₂) = (f.domDomCongr σ₁).domDomCongr σ₂ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} {ι'' : Type u_9} (σ₁ : ι ≃ ι') (σ₂ : ι' ≃ ι'') (f : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr (σ₁.trans σ₂) f = AlternatingMap.domDomCongr σ₂ (AlternatingMap.domDomCongr σ₁ f)
-- AlternatingMap.domDomCongr_zero
@[simp]
theorem domDomCongr_zero (σ : ι ≃ ι') : (0 : M [⋀^ι]→ₗ[R] N).domDomCongr σ = 0 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι')
Goal: AlternatingMap.domDomCongr σ 0 = 0
-- AlternatingMap.domDomCongr_add
@[simp]
theorem domDomCongr_add (σ : ι ≃ ι') (f g : M [⋀^ι]→ₗ[R] N) :
    (f + g).domDomCongr σ = f.domDomCongr σ + g.domDomCongr σ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f g : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr σ (f + g) = AlternatingMap.domDomCongr σ f + AlternatingMap.domDomCongr σ g
-- AlternatingMap.domDomCongr_smul
@[simp]
theorem domDomCongr_smul {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N]
    (σ : ι ≃ ι') (c : S) (f : M [⋀^ι]→ₗ[R] N) :
    (c • f).domDomCongr σ = c • f.domDomCongr σ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} {S : Type u_12} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N] (σ : ι ≃ ι') (c : S) (f : M [⋀^ι]→ₗ[R] N)
Goal: AlternatingMap.domDomCongr σ (c • f) = c • AlternatingMap.domDomCongr σ f
-- AlternatingMap.domDomCongrEquiv_apply
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']→ₗ[R] N 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N)
Goal: (AlternatingMap.domDomCongrEquiv σ) f = AlternatingMap.domDomCongr σ f
-- AlternatingMap.domDomCongrEquiv_symm_apply
@[simps apply symm_apply]
def domDomCongrEquiv (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']→ₗ[R] N 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (σ : ι ≃ ι') (f : M [⋀^ι']→ₗ[R] N)
Goal: (AlternatingMap.domDomCongrEquiv σ).symm f = AlternatingMap.domDomCongr σ.symm f
-- AlternatingMap.domDomCongrₗ_apply
@[simps apply symm_apply]
def domDomCongrₗ (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M [⋀^ι']→ₗ[R] N 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') (f : M [⋀^ι]→ₗ[R] N)
Goal: (AlternatingMap.domDomCongrₗ S σ) f = AlternatingMap.domDomCongr σ f
-- AlternatingMap.domDomCongrₗ_symm_apply
@[simps apply symm_apply]
def domDomCongrₗ (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M [⋀^ι']→ₗ[R] N 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} {ι' : Type u_8} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N] (σ : ι ≃ ι') (f : M [⋀^ι']→ₗ[R] N)
Goal: (AlternatingMap.domDomCongrₗ S σ).symm f = AlternatingMap.domDomCongr σ.symm f
-- AlternatingMap.domDomCongrₗ_refl
@[simp]
theorem domDomCongrₗ_refl :
    (domDomCongrₗ S (Equiv.refl ι) : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M [⋀^ι]→ₗ[R] N) =
      LinearEquiv.refl _ _ 
Arguments: {R : Type u_1} [Semiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] {ι : Type u_7} (S : Type u_12) [Semiring S] [Module S N] [SMulCommClass R S N]
Goal: AlternatingMap.domDomCongrₗ S (Equiv.refl ι) = LinearEquiv.refl S (M [⋀^ι]→ₗ[R] N)