/-- Make a `SmoothVectorBundle` from a `SmoothVectorPrebundle`. -/
theorem smoothVectorBundle : @SmoothVectorBundle
    _ _ F E _ _ _ _ _ _ IB _ _ _ _ _ _ a.totalSpaceTopology _ a.toFiberBundle a.toVectorBundle :=
  letI := a.totalSpaceTopology; letI := a.toFiberBundle; letI := a.toVectorBundle
  { smoothOn_coordChangeL := by
      rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
      refine' (a.smoothOn_smoothCoordChange he he').congr _
      intro b hb
      ext v
      rw [a.smoothCoordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
        Trivialization.coordChangeL_apply]
      

Proof state:
case mk.intro.intro.mk.intro.intro.h
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a
this✝ : FiberBundle F E := toFiberBundle a
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ (toFiberPrebundle a).pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (↑(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')
        { proj := b,
          snd :=
            Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b
              v }).2 =
    (↑e' { proj := b, snd := Pretrivialization.symm e b v }).2
case mk.intro.intro.mk.intro.intro.h.hb
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a
this✝ : FiberBundle F E := toFiberBundle a
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ (toFiberPrebundle a).pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ b ∈
    (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩
      (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet

Next tactic:
exacts [rfl, hb]