-- 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 : 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_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 𝕜₂ (E := E₂) e₂
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂
rw :
  Eq (α := ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ b) (E₂ b))
    (cast (α :=
        ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ
          (E₁
            (TotalSpace.proj (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
              ((↑(PartialEquiv.symm (toPartialEquiv (continuousLinearMap σ (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
                  B × (F₁ →SL[σ] F₂) → TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ E₁ x →SL[σ] E₂ x)
                (b, L))))
          (E₂
            (TotalSpace.proj (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
              ((↑(PartialEquiv.symm (toPartialEquiv (continuousLinearMap σ (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
                  B × (F₁ →SL[σ] F₂) → TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ E₁ x →SL[σ] E₂ x)
                (b, L)))))
        ⋯
        (TotalSpace.snd (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
          ((↑(PartialEquiv.symm (toPartialEquiv (continuousLinearMap σ (E₁ := E₁) (E₂ := E₂) e₁ e₂))) :
              B × (F₁ →SL[σ] F₂) → TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ E₁ x →SL[σ] E₂ x)
            (b, L))) :
      ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ b) (E₂ b))
    (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 b)
        (L.comp (σ₁₂ := RingHom.id 𝕜₁) (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ b)) :
      ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ b) (E₂ b))
hb₁ :
  b ∈
    baseSet (F := F₁ →SL[σ] F₂) (Z :=
      TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
      (continuousLinearMap σ (E₁ := E₁) (E₂ := E₂) e₁ e₂)
⊢ Eq (α := ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ b) (E₂ b))
    (Pretrivialization.symm (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
      (continuousLinearMap σ (E₁ := E₁) (E₂ := E₂) e₁ e₂ :
        Pretrivialization (B := B) (F₁ →SL[σ] F₂) (Z :=
          TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))
          (TotalSpace.proj (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x))))
      b L)
    (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 b)
      (L.comp (σ₁₂ := RingHom.id 𝕜₁) (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ b)))

Tactic:
rw [symm_apply]

Premises used: Bundle.Pretrivialization.symm_apply