/-- The continuous `σ`-semilinear maps between two topological vector bundles form a
`VectorPrebundle` (this is an auxiliary construction for the
`VectorBundle` instance, in which the pretrivializations are collated but no topology
on the total space is yet provided). -/
def Bundle.ContinuousLinearMap.vectorPrebundle :
    VectorPrebundle 𝕜₂ (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) where
  pretrivializationAtlas :=
    {e | ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
      (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
        e = Pretrivialization.continuousLinearMap σ e₁ e₂}
  pretrivialization_linear' := by
    rintro _ ⟨e₁, he₁, e₂, he₂, rfl⟩
    infer_instance
  pretrivializationAt x :=
    Pretrivialization.continuousLinearMap σ (trivializationAt F₁ E₁ x) (trivializationAt F₂ E₂ x)
  mem_base_pretrivializationAt x :=
    ⟨mem_baseSet_trivializationAt F₁ E₁ x, mem_baseSet_trivializationAt F₂ E₂ x⟩
  pretrivialization_mem_atlas x :=
    ⟨trivializationAt F₁ E₁ x, trivializationAt F₂ E₂ x, inferInstance, inferInstance, rfl⟩
  exists_coordChange := by
    rintro _ ⟨e₁, e₂, he₁, he₂, rfl⟩ _ ⟨e₁', e₂', he₁', he₂', rfl⟩
    exact ⟨continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂',
      continuousOn_continuousLinearMapCoordChange,
      continuousLinearMapCoordChange_apply σ e₁ e₁' e₂ e₂'⟩
  totalSpaceMk_inducing := by
    intro b
    let L₁ : E₁ b ≃L[𝕜₁] F₁ :=
      (trivializationAt F₁ E₁ b).continuousLinearEquivAt 𝕜₁ b
        (mem_baseSet_trivializationAt _ _ _)
    let L₂ : E₂ b ≃L[𝕜₂] F₂ :=
      (trivializationAt F₂ E₂ b).continuousLinearEquivAt 𝕜₂ b
        (mem_baseSet_trivializationAt _ _ _)
    let φ : (E₁ b →SL[σ] E₂ b) ≃L[𝕜₂] F₁ →SL[σ] F₂ := L₁.arrowCongrSL L₂
    have : Inducing fun x => (b, φ x) := inducing_const_prod.mpr φ.toHomeomorph.inducing
    convert this
    ext f
    dsimp [Pretrivialization.continuousLinearMap_apply]
    

Proof state:
case h.e'_5.h.h.e'_4.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 : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₂ E₂
inst✝² : VectorBundle 𝕜₂ F₂ E₂
inst✝¹ : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝ : ∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)
b : B
L₁ : E₁ b ≃L[𝕜₁] F₁ :=
  Trivialization.continuousLinearEquivAt 𝕜₁ (trivializationAt F₁ E₁ b) b (_ : b ∈ (trivializationAt F₁ E₁ b).baseSet)
L₂ : E₂ b ≃L[𝕜₂] F₂ :=
  Trivialization.continuousLinearEquivAt 𝕜₂ (trivializationAt F₂ E₂ b) b (_ : b ∈ (trivializationAt F₂ E₂ b).baseSet)
φ : (E₁ b →SL[σ] E₂ b) ≃L[𝕜₂] F₁ →SL[σ] F₂ := ContinuousLinearEquiv.arrowCongrSL L₁ L₂
this : Inducing fun x => (b, φ x)
x✝ : Bundle.ContinuousLinearMap σ E₁ E₂ b
f : F₁
⊢ (Trivialization.linearMapAt 𝕜₂ (trivializationAt F₂ E₂ b) b)
      (x✝ (Trivialization.symm (trivializationAt F₁ E₁ b) b f)) =
    (↑(trivializationAt F₂ E₂ b) { proj := b, snd := x✝ (Trivialization.symm (trivializationAt F₁ E₁ b) b f) }).2

Next tactic:
rw [Trivialization.linearMapAt_def_of_mem _ (mem_baseSet_trivializationAt _ _ _)]