-- 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 : TotalSpace (F₁ →SL[σ] F₂) fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)
x : B
L : ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)
h_1 :
  Membership.mem (γ :=
    Set (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)) ⁻¹'
      (e₁.baseSet ∩ e₂.baseSet))
    (TotalSpace.mk (E := fun (x : B) ↦ ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) x L)
h₁ :
  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 L) ∈
    e₁.baseSet
h₂ :
  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 L) ∈
    e₂.baseSet
v : E₁ x
h :
  Eq.{u_7 + 1} (α := E₂ x)
    (((Trivialization.symmL 𝕜₂ (E := E₂) e₂ x : ContinuousLinearMap.{u_2, u_2, u_6, u_7} (RingHom.id 𝕜₂) F₂ (E₂ x)) :
          (a : F₂) → E₂ x)
        (((Trivialization.continuousLinearMapAt 𝕜₂ (E := E₂) e₂ x :
              ContinuousLinearMap.{u_2, u_2, u_7, u_6} (RingHom.id 𝕜₂) (E₂ x) F₂) :
            (a : E₂ x) → F₂)
          ((L : (a : E₁ x) → E₂ x)
              (((Trivialization.symmL 𝕜₁ (E := E₁) e₁ x :
                    ContinuousLinearMap.{u_1, u_1, u_4, u_5} (RingHom.id 𝕜₁) F₁ (E₁ x)) :
                  (a : F₁) → E₁ x)
                (((Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ x :
                      ContinuousLinearMap.{u_1, u_1, u_5, u_4} (RingHom.id 𝕜₁) (E₁ x) F₁) :
                    (a : E₁ x) → F₁)
                  v)) :
            E₂ x)) :
      E₂ x)
    ((L : (a : E₁ x) → E₂ x) v : E₂ x)
⊢ Eq.{u_7 + 1} (α := E₂ x)
    (((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)
              (comp.{u_1, u_1, u_2, u_5, u_4, u_6} (σ₁₂ := RingHom.id 𝕜₁) (M₂ := F₁)
                (comp (σ₁₂ := σ) (Trivialization.continuousLinearMapAt 𝕜₂ (E := E₂) e₂ x)
                  (L.comp (σ₁₂ := RingHom.id 𝕜₁) (Trivialization.symmL 𝕜₁ (E := E₁) e₁ x)))
                (Trivialization.continuousLinearMapAt 𝕜₁ (E := E₁) e₁ x)) :
            ContinuousLinearMap.{u_1, u_2, u_5, u_7} σ (E₁ x) (E₂ x)) :
          (a : E₁ x) → E₂ x)
        v :
      E₂ x)
    ((L : (a : E₁ x) → E₂ x) v : E₂ x)

Tactic:
dsimp only [comp_apply]

Premises used: ContinuousLinearMap.comp_apply