-- Module: Mathlib.Topology.VectorBundle.ContinuousAlternatingMap
Goal:
𝕜 : Type u_1
ι : Type u_2
inst : NontriviallyNormedField 𝕜
B : Type u_3
inst_1 : TopologicalSpace B
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜 (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_1, u_7} 𝕜 (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : (x : B) → TopologicalSpace (E₁ x)
inst_13 : FiberBundle F₁ E₁
inst_14 : (x : B) → TopologicalSpace (E₂ x)
inst_15 : FiberBundle F₂ E₂
e₁ : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ : Trivialization (B := B) F₂ (Z := TotalSpace F₂ E₂) (TotalSpace.proj (E := E₂))
inst_16 : Trivialization.IsLinear 𝕜 (E := E₁) e₁
inst_17 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ [⋀^ι]→L[𝕜] F₂
rw :
  Eq (α := ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι)
    (cast (α :=
        ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜
          (E₁
            (TotalSpace.proj (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
              ((↑(PartialEquiv.symm (toPartialEquiv (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
                  B × F₁ [⋀^ι]→L[𝕜] F₂ → TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)
                (b, L))))
          (E₂
            (TotalSpace.proj (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
              ((↑(PartialEquiv.symm (toPartialEquiv (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
                  B × F₁ [⋀^ι]→L[𝕜] F₂ → TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)
                (b, L))))
          ι)
        ⋯
        (TotalSpace.snd (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
          ((↑(PartialEquiv.symm (toPartialEquiv (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
              B × F₁ [⋀^ι]→L[𝕜] F₂ → TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ E₁ x [⋀^ι]→L[𝕜] E₂ x)
            (b, L))) :
      ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι)
    (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
        (Trivialization.symmL 𝕜 (E := E₂) e₂ b)
        (L.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ b)) :
      ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι)
hb₁ :
  b ∈
    baseSet (F := F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
      TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂)
⊢ Eq (α := ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ b) (E₂ b) ι)
    (Pretrivialization.symm (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
      (continuousAlternatingMap 𝕜 ι (E₁ := E₁) (E₂ := E₂) e₁ e₂ :
        Pretrivialization (B := B) (F₁ [⋀^ι]→L[𝕜] F₂) (Z :=
          TotalSpace (F₁ [⋀^ι]→L[𝕜] F₂) fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
          (TotalSpace.proj (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)))
      b L)
    (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
      (Trivialization.symmL 𝕜 (E := E₂) e₂ b)
      (L.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ b)))

Tactic:
rw [Pretrivialization.symm_apply]

Premises used: Bundle.Pretrivialization.symm_apply