[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g✝ f g : M[Λ^ι]→L[R]N
h : toAlternatingMap f = toAlternatingMap g
⊢ ↑f = ↑g
[PROOFSTEP]
convert FunLike.ext'_iff.1 h
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : M ≃L[R] M'
f : M[Λ^ι]→L[R]N
⊢ (fun f => compContinuousLinearMap f ↑e) ((fun f => compContinuousLinearMap f ↑(ContinuousLinearEquiv.symm e)) f) = f
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : M ≃L[R] M'
f : M[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑((fun f => compContinuousLinearMap f ↑e) ((fun f => compContinuousLinearMap f ↑(ContinuousLinearEquiv.symm e)) f))
      x✝ =
    ↑f x✝
[PROOFSTEP]
simp [(· ∘ ·)]
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : M ≃L[R] M'
f : M'[Λ^ι]→L[R]N
⊢ (fun f => compContinuousLinearMap f ↑(ContinuousLinearEquiv.symm e)) ((fun f => compContinuousLinearMap f ↑e) f) = f
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : M ≃L[R] M'
f : M'[Λ^ι]→L[R]N
x✝ : ι → M'
⊢ ↑((fun f => compContinuousLinearMap f ↑(ContinuousLinearEquiv.symm e)) ((fun f => compContinuousLinearMap f ↑e) f))
      x✝ =
    ↑f x✝
[PROOFSTEP]
simp [(· ∘ ·)]
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : N ≃L[R] N'
f : M[Λ^ι]→L[R]N
⊢ ContinuousLinearMap.compContinuousAlternatingMap (↑(ContinuousLinearEquiv.symm e))
      (ContinuousLinearMap.compContinuousAlternatingMap (↑e) f) =
    f
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : N ≃L[R] N'
f : M[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑(ContinuousLinearMap.compContinuousAlternatingMap (↑(ContinuousLinearEquiv.symm e))
          (ContinuousLinearMap.compContinuousAlternatingMap (↑e) f))
      x✝ =
    ↑f x✝
[PROOFSTEP]
simp [(· ∘ ·)]
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : N ≃L[R] N'
f : M[Λ^ι]→L[R]N'
⊢ ContinuousLinearMap.compContinuousAlternatingMap (↑e)
      (ContinuousLinearMap.compContinuousAlternatingMap (↑(ContinuousLinearEquiv.symm e)) f) =
    f
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹² : Semiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : Module R M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : AddCommMonoid M'
inst✝⁷ : Module R M'
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : AddCommMonoid N'
inst✝¹ : Module R N'
inst✝ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N
e : N ≃L[R] N'
f : M[Λ^ι]→L[R]N'
x✝ : ι → M
⊢ ↑(ContinuousLinearMap.compContinuousAlternatingMap (↑e)
          (ContinuousLinearMap.compContinuousAlternatingMap (↑(ContinuousLinearEquiv.symm e)) f))
      x✝ =
    ↑f x✝
[PROOFSTEP]
simp [(· ∘ ·)]
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N✝ : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁵ : Semiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommMonoid M'
inst✝¹⁰ : Module R M'
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : AddCommMonoid N✝
inst✝⁷ : Module R N✝
inst✝⁶ : TopologicalSpace N✝
inst✝⁵ : AddCommMonoid N'
inst✝⁴ : Module R N'
inst✝³ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N✝
ι' : Type u_7
N : ι' → Type u_8
inst✝² : (i : ι') → AddCommMonoid (N i)
inst✝¹ : (i : ι') → TopologicalSpace (N i)
inst✝ : (i : ι') → Module R (N i)
f : (i : ι') → M[Λ^ι]→L[R]N i
⊢ (fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) (pi f) = f
[PROOFSTEP]
ext
[GOAL]
case h.H
R : Type u_1
M : Type u_2
M' : Type u_3
N✝ : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁵ : Semiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommMonoid M'
inst✝¹⁰ : Module R M'
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : AddCommMonoid N✝
inst✝⁷ : Module R N✝
inst✝⁶ : TopologicalSpace N✝
inst✝⁵ : AddCommMonoid N'
inst✝⁴ : Module R N'
inst✝³ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N✝
ι' : Type u_7
N : ι' → Type u_8
inst✝² : (i : ι') → AddCommMonoid (N i)
inst✝¹ : (i : ι') → TopologicalSpace (N i)
inst✝ : (i : ι') → Module R (N i)
f : (i : ι') → M[Λ^ι]→L[R]N i
x✝¹ : ι'
x✝ : ι → M
⊢ ↑((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) (pi f) x✝¹) x✝ =
    ↑(f x✝¹) x✝
[PROOFSTEP]
rfl
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N✝ : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁵ : Semiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommMonoid M'
inst✝¹⁰ : Module R M'
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : AddCommMonoid N✝
inst✝⁷ : Module R N✝
inst✝⁶ : TopologicalSpace N✝
inst✝⁵ : AddCommMonoid N'
inst✝⁴ : Module R N'
inst✝³ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N✝
ι' : Type u_7
N : ι' → Type u_8
inst✝² : (i : ι') → AddCommMonoid (N i)
inst✝¹ : (i : ι') → TopologicalSpace (N i)
inst✝ : (i : ι') → Module R (N i)
f : M[Λ^ι]→L[R]((i : ι') → N i)
⊢ pi ((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) f) = f
[PROOFSTEP]
ext
[GOAL]
case H.h
R : Type u_1
M : Type u_2
M' : Type u_3
N✝ : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁵ : Semiring R
inst✝¹⁴ : AddCommMonoid M
inst✝¹³ : Module R M
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommMonoid M'
inst✝¹⁰ : Module R M'
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : AddCommMonoid N✝
inst✝⁷ : Module R N✝
inst✝⁶ : TopologicalSpace N✝
inst✝⁵ : AddCommMonoid N'
inst✝⁴ : Module R N'
inst✝³ : TopologicalSpace N'
n : ℕ
f✝ g : M[Λ^ι]→L[R]N✝
ι' : Type u_7
N : ι' → Type u_8
inst✝² : (i : ι') → AddCommMonoid (N i)
inst✝¹ : (i : ι') → TopologicalSpace (N i)
inst✝ : (i : ι') → Module R (N i)
f : M[Λ^ι]→L[R]((i : ι') → N i)
x✝¹ : ι → M
x✝ : ι'
⊢ ↑(pi ((fun f i => ContinuousLinearMap.compContinuousAlternatingMap (ContinuousLinearMap.proj i) f) f)) x✝¹ x✝ =
    ↑f x✝¹ x✝
[PROOFSTEP]
rfl
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
g g' : M'[Λ^ι]→L[R]N
⊢ (fun g => compContinuousLinearMap g f) (g + g') =
    (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g'
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
g g' : M'[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑((fun g => compContinuousLinearMap g f) (g + g')) x✝ =
    ↑((fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') x✝
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
c : R
g : M'[Λ^ι]→L[R]N
⊢ AddHom.toFun
      { toFun := fun g => compContinuousLinearMap g f,
        map_add' :=
          (_ :
            ∀ (g g' : M'[Λ^ι]→L[R]N),
              (fun g => compContinuousLinearMap g f) (g + g') =
                (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }
      (c • g) =
    ↑(RingHom.id R) c •
      AddHom.toFun
        { toFun := fun g => compContinuousLinearMap g f,
          map_add' :=
            (_ :
              ∀ (g g' : M'[Λ^ι]→L[R]N),
                (fun g => compContinuousLinearMap g f) (g + g') =
                  (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }
        g
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : M →L[R] M'
c : R
g : M'[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑(AddHom.toFun
          { toFun := fun g => compContinuousLinearMap g f,
            map_add' :=
              (_ :
                ∀ (g g' : M'[Λ^ι]→L[R]N),
                  (fun g => compContinuousLinearMap g f) (g + g') =
                    (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }
          (c • g))
      x✝ =
    ↑(↑(RingHom.id R) c •
          AddHom.toFun
            { toFun := fun g => compContinuousLinearMap g f,
              map_add' :=
                (_ :
                  ∀ (g g' : M'[Λ^ι]→L[R]N),
                    (fun g => compContinuousLinearMap g f) (g + g') =
                      (fun g => compContinuousLinearMap g f) g + (fun g => compContinuousLinearMap g f) g') }
            g)
      x✝
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : N →L[R] N'
g₁ g₂ : M[Λ^ι]→L[R]N
⊢ ContinuousLinearMap.compContinuousAlternatingMap f (g₁ + g₂) =
    ContinuousLinearMap.compContinuousAlternatingMap f g₁ + ContinuousLinearMap.compContinuousAlternatingMap f g₂
[PROOFSTEP]
ext1
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
f : N →L[R] N'
g₁ g₂ : M[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑(ContinuousLinearMap.compContinuousAlternatingMap f (g₁ + g₂)) x✝ =
    ↑(ContinuousLinearMap.compContinuousAlternatingMap f g₁ + ContinuousLinearMap.compContinuousAlternatingMap f g₂) x✝
[PROOFSTEP]
apply f.map_add
[GOAL]
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
c : R
f : N →L[R] N'
g : M[Λ^ι]→L[R]N
⊢ ContinuousLinearMap.compContinuousAlternatingMap f (c • g) = c • ContinuousLinearMap.compContinuousAlternatingMap f g
[PROOFSTEP]
ext1
[GOAL]
case H
R : Type u_1
M : Type u_2
M' : Type u_3
N : Type u_4
N' : Type u_5
ι : Type u_6
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : AddCommMonoid M
inst✝¹⁴ : Module R M
inst✝¹³ : TopologicalSpace M
inst✝¹² : AddCommMonoid M'
inst✝¹¹ : Module R M'
inst✝¹⁰ : TopologicalSpace M'
inst✝⁹ : AddCommMonoid N
inst✝⁸ : Module R N
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ContinuousAdd N
inst✝⁵ : ContinuousConstSMul R N
inst✝⁴ : AddCommMonoid N'
inst✝³ : Module R N'
inst✝² : TopologicalSpace N'
inst✝¹ : ContinuousAdd N'
inst✝ : ContinuousConstSMul R N'
c : R
f : N →L[R] N'
g : M[Λ^ι]→L[R]N
x✝ : ι → M
⊢ ↑(ContinuousLinearMap.compContinuousAlternatingMap f (c • g)) x✝ =
    ↑(c • ContinuousLinearMap.compContinuousAlternatingMap f g) x✝
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f✝ g f : ContinuousMultilinearMap R (fun x => M) N
v : ι → M
i j : ι
hv : v i = v j
hne : i ≠ j
⊢ MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v = 0
[PROOFSTEP]
simpa [MultilinearMap.alternatization_apply] using f.1.alternatization.map_eq_zero_of_eq' v i j hv hne
[GOAL]
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g : ContinuousMultilinearMap R (fun x => M) N
⊢ (fun f =>
        { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
          map_eq_zero_of_eq' :=
            (_ :
              ∀ (v : ι → M) (i j : ι),
                v i = v j →
                  i ≠ j →
                    MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                      0) })
      0 =
    0
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g : ContinuousMultilinearMap R (fun x => M) N
x✝ : ι → M
⊢ ↑((fun f =>
            { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
              map_eq_zero_of_eq' :=
                (_ :
                  ∀ (v : ι → M) (i j : ι),
                    v i = v j →
                      i ≠ j →
                        MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap
                            v =
                          0) })
          0)
      x✝ =
    ↑0 x✝
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g x✝¹ x✝ : ContinuousMultilinearMap R (fun x => M) N
⊢ ZeroHom.toFun
      {
        toFun := fun f =>
          { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
            map_eq_zero_of_eq' :=
              (_ :
                ∀ (v : ι → M) (i j : ι),
                  v i = v j →
                    i ≠ j →
                      MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap
                          v =
                        0) },
        map_zero' :=
          (_ :
            (fun f =>
                  { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                    map_eq_zero_of_eq' :=
                      (_ :
                        ∀ (v : ι → M) (i j : ι),
                          v i = v j →
                            i ≠ j →
                              MultilinearMap.toFun
                                  (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                0) })
                0 =
              0) }
      (x✝¹ + x✝) =
    ZeroHom.toFun
        {
          toFun := fun f =>
            { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
              map_eq_zero_of_eq' :=
                (_ :
                  ∀ (v : ι → M) (i j : ι),
                    v i = v j →
                      i ≠ j →
                        MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap
                            v =
                          0) },
          map_zero' :=
            (_ :
              (fun f =>
                    { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                      map_eq_zero_of_eq' :=
                        (_ :
                          ∀ (v : ι → M) (i j : ι),
                            v i = v j →
                              i ≠ j →
                                MultilinearMap.toFun
                                    (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                  0) })
                  0 =
                0) }
        x✝¹ +
      ZeroHom.toFun
        {
          toFun := fun f =>
            { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
              map_eq_zero_of_eq' :=
                (_ :
                  ∀ (v : ι → M) (i j : ι),
                    v i = v j →
                      i ≠ j →
                        MultilinearMap.toFun (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap
                            v =
                          0) },
          map_zero' :=
            (_ :
              (fun f =>
                    { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                      map_eq_zero_of_eq' :=
                        (_ :
                          ∀ (v : ι → M) (i j : ι),
                            v i = v j →
                              i ≠ j →
                                MultilinearMap.toFun
                                    (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                  0) })
                  0 =
                0) }
        x✝
[PROOFSTEP]
ext
[GOAL]
case H
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g x✝² x✝¹ : ContinuousMultilinearMap R (fun x => M) N
x✝ : ι → M
⊢ ↑(ZeroHom.toFun
          {
            toFun := fun f =>
              { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                map_eq_zero_of_eq' :=
                  (_ :
                    ∀ (v : ι → M) (i j : ι),
                      v i = v j →
                        i ≠ j →
                          MultilinearMap.toFun
                              (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                            0) },
            map_zero' :=
              (_ :
                (fun f =>
                      { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                        map_eq_zero_of_eq' :=
                          (_ :
                            ∀ (v : ι → M) (i j : ι),
                              v i = v j →
                                i ≠ j →
                                  MultilinearMap.toFun
                                      (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                    0) })
                    0 =
                  0) }
          (x✝² + x✝¹))
      x✝ =
    ↑(ZeroHom.toFun
            {
              toFun := fun f =>
                { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                  map_eq_zero_of_eq' :=
                    (_ :
                      ∀ (v : ι → M) (i j : ι),
                        v i = v j →
                          i ≠ j →
                            MultilinearMap.toFun
                                (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                              0) },
              map_zero' :=
                (_ :
                  (fun f =>
                        { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                          map_eq_zero_of_eq' :=
                            (_ :
                              ∀ (v : ι → M) (i j : ι),
                                v i = v j →
                                  i ≠ j →
                                    MultilinearMap.toFun
                                        (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                      0) })
                      0 =
                    0) }
            x✝² +
          ZeroHom.toFun
            {
              toFun := fun f =>
                { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                  map_eq_zero_of_eq' :=
                    (_ :
                      ∀ (v : ι → M) (i j : ι),
                        v i = v j →
                          i ≠ j →
                            MultilinearMap.toFun
                                (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                              0) },
              map_zero' :=
                (_ :
                  (fun f =>
                        { toContinuousMultilinearMap := ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f,
                          map_eq_zero_of_eq' :=
                            (_ :
                              ∀ (v : ι → M) (i j : ι),
                                v i = v j →
                                  i ≠ j →
                                    MultilinearMap.toFun
                                        (∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • domDomCongr σ f).toMultilinearMap v =
                                      0) })
                      0 =
                    0) }
            x✝¹)
      x✝
[PROOFSTEP]
simp [Finset.sum_add_distrib]
[GOAL]
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g : ContinuousMultilinearMap R (fun x => M) N
v : ι → M
⊢ ↑(↑alternatization f) v = ∑ σ : Equiv.Perm ι, ↑Equiv.Perm.sign σ • ↑f (v ∘ ↑σ)
[PROOFSTEP]
simp [alternatization, (· ∘ ·)]
[GOAL]
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g : ContinuousMultilinearMap R (fun x => M) N
⊢ ContinuousAlternatingMap.toAlternatingMap (↑alternatization f) = ↑MultilinearMap.alternatization f.toMultilinearMap
[PROOFSTEP]
ext v
[GOAL]
case H
R : Type u_1
M : Type u_2
N : Type u_3
ι : Type u_4
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : Module R M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
inst✝³ : TopologicalSpace N
inst✝² : TopologicalAddGroup N
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f g : ContinuousMultilinearMap R (fun x => M) N
v : ι → M
⊢ ↑(ContinuousAlternatingMap.toAlternatingMap (↑alternatization f)) v =
    ↑(↑MultilinearMap.alternatization f.toMultilinearMap) v
[PROOFSTEP]
simp [alternatization_apply_apply, MultilinearMap.alternatization_apply, (· ∘ ·)]
