-- Module: Mathlib.Topology.VectorBundle.Hom
Goal:
𝕜₁ : Type u_1
inst : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst_1 : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst_2 : NormedAddCommGroup F₁
inst_3 : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst_4 : (x : B) → AddCommGroup (E₁ x)
inst_5 : (x : B) → Module.{u_1, u_5} 𝕜₁ (E₁ x)
inst_6 : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst_7 : NormedAddCommGroup F₂
inst_8 : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst_9 : (x : B) → AddCommGroup (E₂ x)
inst_10 : (x : B) → Module.{u_2, u_7} 𝕜₂ (E₂ x)
inst_11 : TopologicalSpace (TotalSpace F₂ E₂)
inst_12 : TopologicalSpace B
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_13 : (x : B) → TopologicalSpace (E₁ x)
inst_14 : FiberBundle F₁ E₁
inst_15 : (x : B) → TopologicalSpace (E₂ x)
inst_16 : FiberBundle F₂ E₂
inst_17 : Trivialization.IsLinear 𝕜₁ (E := E₁) e₁
inst_18 : Trivialization.IsLinear 𝕜₁ (E := E₁) e₁'
inst_19 : Trivialization.IsLinear 𝕜₂ (E := E₂) e₂
inst_20 : Trivialization.IsLinear 𝕜₂ (E := E₂) e₂'
x_1 : B × (F₁ →SL[σ] F₂)
x : B
f : F₁ →SL[σ] F₂
h : (x, f) ∈ SProd.sprod (β := Set (F₁ →SL[σ] F₂)) (e₁.baseSet ∩ e₂.baseSet) univ
h₁ : (x, f).1 ∈ e₁.baseSet
h₂ : (x, f).1 ∈ e₂.baseSet
right : Membership.mem (γ := Set (F₁ →SL[σ] F₂)) univ (x, f).2
simp :
  Trivialization.continuousLinearMapAt 𝕜₂ (E := E₂) e₂ x ∘SL
      comp (σ₂₃ := σ)
        (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
          (Trivialization.symmL 𝕜₂ (E := E₂) e₂ x)
          (f.comp (σ₁₂ := RingHom.id 𝕜₁) (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ x)))
        (Trivialization.symmL 𝕜₁ (E := E₁) e₁ x) =
    f
⊢ (TotalSpace.proj (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
        (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
          (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
            (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
            ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁) (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1)))),
      comp (M₂ :=
        E₂
          (TotalSpace.proj (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦
            ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
              (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
                (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
                ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁)
                  (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1))))))
        (Trivialization.continuousLinearMapAt 𝕜₂ (E := E₂) e₂
          (TotalSpace.proj (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦
            ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
              (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
                (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
                ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁)
                  (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1))))))
        (comp (σ₂₃ := σ) (M₂ :=
          E₁
            (TotalSpace.proj (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦
              ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
              (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
                (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
                  (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
                  ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁)
                    (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1))))))
          (TotalSpace.snd (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦
            ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
            (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
              (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
                (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
                ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁)
                  (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1)))))
          (Trivialization.symmL 𝕜₁ (E := E₁) e₁
            (TotalSpace.proj (F := F₁ →SL[σ] F₂) (E := fun (x : B) ↦
              ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
              (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) (x, f).1
                (comp.{u_1, u_2, u_2, u_5, u_6, u_7} (R₂ := 𝕜₂) (σ₁₂ := σ) (σ₂₃ := RingHom.id 𝕜₂) (M₂ := F₂)
                  (Trivialization.symmL 𝕜₂ (E := E₂) e₂ (x, f).1)
                  ((x, f).2.comp (σ₁₂ := RingHom.id 𝕜₁)
                    (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ (x, f).1)))))))) =
    (x, f)

Tactic:
simp only [Prod.mk_right_inj]

Premises used: Prod.mk_right_inj