-- 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 e₁' : Trivialization (B := B) F₁ (Z := TotalSpace F₁ E₁) (TotalSpace.proj (E := E₁))
e₂ 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 := E₁) e₁'
inst_18 : Trivialization.IsLinear 𝕜 (E := E₂) e₂
inst_19 : Trivialization.IsLinear 𝕜 (E := E₂) e₂'
x : B
f : F₁ [⋀^ι]→L[𝕜] F₂
h₁ : (x, f).1 ∈ e₁.baseSet
h₂ : (x, f).1 ∈ e₂.baseSet
simp :
  ContinuousLinearMap.compContinuousAlternatingMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) e₂ x)
      (ContinuousAlternatingMap.compContinuousLinearMap
        (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
          (Trivialization.symmL 𝕜 (E := E₂) e₂ x)
          (f.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ x)))
        (Trivialization.symmL 𝕜 (E := E₁) e₁ x)) =
    f
⊢ (TotalSpace.proj (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
        ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
        (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) (x, f).1
          (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
            (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
            ((x, f).2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1)))),
      ContinuousLinearMap.compContinuousAlternatingMap (N :=
        E₂
          (TotalSpace.proj (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
            ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) (x, f).1
              (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
                (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
                ((x, f).2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1))))))
        (Trivialization.continuousLinearMapAt 𝕜 (E := E₂) e₂
          (TotalSpace.proj (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
            ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) (x, f).1
              (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
                (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
                ((x, f).2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1))))))
        (ContinuousAlternatingMap.compContinuousLinearMap (M :=
          E₁
            (TotalSpace.proj (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
              ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
              (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
                (x, f).1
                (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
                  (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
                  ((x, f).2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1))))))
          (TotalSpace.snd (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
            ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι) (x, f).1
              (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
                (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
                ((x, f).2.compContinuousLinearMap (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1)))))
          (Trivialization.symmL 𝕜 (E := E₁) e₁
            (TotalSpace.proj (F := F₁ [⋀^ι]→L[𝕜] F₂) (E := fun (x : B) ↦
              ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
              (TotalSpace.mk (E := fun (x : B) ↦ ContinuousAlternatingMap.{u_1, u_5, u_7, u_2} 𝕜 (E₁ x) (E₂ x) ι)
                (x, f).1
                (ContinuousLinearMap.compContinuousAlternatingMap.{u_1, u_5, u_6, u_7, u_2} (N := F₂)
                  (Trivialization.symmL 𝕜 (E := E₂) e₂ (x, f).1)
                  ((x, f).2.compContinuousLinearMap
                    (Trivialization.continuousLinearMapAt 𝕜 (E := E₁) e₁ (x, f).1)))))))) =
    (x, f)

Tactic:
simp only [Prod.mk_right_inj]

Premises used: Prod.mk_right_inj