Theorem: VectorPrebundle.toVectorBundle
File: Mathlib/Topology/VectorBundle/Basic.lean

Proof steps:
state_before: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
⊢ ∀ (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear R e
state_after: case mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
tactic: rintro _ ⟨e, he, rfl⟩
state_before: case mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
state_after: no goals
tactic: apply linear_trivializationOfMemPretrivializationAtlas
state_before: R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],
    ContinuousOn (fun b => ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet)
state_after: case mk.intro.intro.mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ ContinuousOn
    (fun b =>
      ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b))
    ((a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet)
tactic: rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩
state_before: case mk.intro.intro.mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
⊢ ContinuousOn
    (fun b =>
      ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b))
    ((a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet)
state_after: case mk.intro.intro.mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
        (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b) =
    a.coordChange he he' b
tactic: refine (a.continuousOn_coordChange he he').congr fun b hb ↦ ?_
state_before: case mk.intro.intro.mk.intro.intro
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
        (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b) =
    a.coordChange he he' b
state_after: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
tactic: ext v
state_before: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
state_after: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he)
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
tactic: haveI h₁ := a.linear_trivializationOfMemPretrivializationAtlas he
state_before: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he)
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
state_after: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he')
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
tactic: haveI h₂ := a.linear_trivializationOfMemPretrivializationAtlas he'
state_before: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.trivializationOfMemPretrivializationAtlas he')
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
state_after: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
tactic: rw [trivializationOfMemPretrivializationAtlas] at h₁ h₂
state_before: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ ↑(Trivialization.coordChangeL R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
          (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') b)
      v =
    (a.coordChange he he' b) v
state_after: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ (↑(a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
        { proj := b, snd := (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).symm b v }).2 =
    (↑e' { proj := b, snd := e.symm b v }).2

case mk.intro.intro.mk.intro.intro.h.hb
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ b ∈
    (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet
tactic: rw [a.coordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,
        Trivialization.coordChangeL_apply]
state_before: case mk.intro.intro.mk.intro.intro.h
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ (↑(a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
        { proj := b, snd := (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).symm b v }).2 =
    (↑e' { proj := b, snd := e.symm b v }).2

case mk.intro.intro.mk.intro.intro.h.hb
R : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
inst✝⁶ : NontriviallyNormedField R
inst✝⁵ : (x : B) → AddCommMonoid (E x)
inst✝⁴ : (x : B) → Module R (E x)
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace R F
inst✝¹ : TopologicalSpace B
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle R F E
this✝ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this : FiberBundle F E := a.toFiberBundle
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
h₁ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he)
h₂ : Trivialization.IsLinear R (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he')
⊢ b ∈
    (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).baseSet ∩
      (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he').baseSet
state_after: no goals
tactic: exacts [rfl, hb]