/-- Given trivializations `e₁`, `e₂` for vector bundles `E₁`, `E₂` over a base `B`,
`Pretrivialization.continuousLinearMap σ e₁ e₂` is the induced pretrivialization for the
continuous `σ`-semilinear maps from `E₁` to `E₂`. That is, the map which will later become a
trivialization, after the bundle of continuous semilinear maps is equipped with the right
topological vector bundle structure. -/
def continuousLinearMap :
    Pretrivialization (F₁ →SL[σ] F₂) (π (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂)) where
  toFun p := ⟨p.1, .comp (e₂.continuousLinearMapAt 𝕜₂ p.1) (p.2.comp (e₁.symmL 𝕜₁ p.1))⟩
  invFun p := ⟨p.1, .comp (e₂.symmL 𝕜₂ p.1) (p.2.comp (e₁.continuousLinearMapAt 𝕜₁ p.1))⟩
  source := Bundle.TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet)
  target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ Set.univ
  map_source' := fun ⟨x, L⟩ h => ⟨h, Set.mem_univ _⟩
  map_target' := fun ⟨x, f⟩ h => h.1
  left_inv' := fun ⟨x, L⟩ ⟨h₁, h₂⟩ => by
    simp only [TotalSpace.mk_inj]
    ext (v : E₁ x)
    dsimp only [comp_apply]
    rw [Trivialization.symmL_continuousLinearMapAt, Trivialization.symmL_continuousLinearMapAt]
    exacts [h₁, h₂]
  right_inv' := fun ⟨x, f⟩ ⟨⟨h₁, h₂⟩, _⟩ => by
    simp only [Prod.mk_inj_left]
    ext v
    

Proof state:
case h
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
x✝¹ : B × (F₁ →SL[σ] F₂)
x : B
f : F₁ →SL[σ] F₂
x✝ : (x, f) ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ
h₁ : (x, f).1 ∈ e₁.baseSet
h₂ : (x, f).1 ∈ e₂.baseSet
right✝ : (x, f).2 ∈ univ
v : F₁
⊢ (comp
        (Trivialization.continuousLinearMapAt 𝕜₂ e₂
          ((fun p =>
                { proj := p.1,
                  snd :=
                    comp (Trivialization.symmL 𝕜₂ e₂ p.1) (comp p.2 (Trivialization.continuousLinearMapAt 𝕜₁ e₁ p.1)) })
              (x, f)).proj)
        (comp
          (comp (Trivialization.symmL 𝕜₂ e₂ (x, f).1) (comp f (Trivialization.continuousLinearMapAt 𝕜₁ e₁ (x, f).1)))
          (Trivialization.symmL 𝕜₁ e₁
            ((fun p =>
                  { proj := p.1,
                    snd :=
                      comp (Trivialization.symmL 𝕜₂ e₂ p.1)
                        (comp p.2 (Trivialization.continuousLinearMapAt 𝕜₁ e₁ p.1)) })
                (x, f)).proj)))
      v =
    f v

Next tactic:
dsimp only [comp_apply]