case h.h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))
          (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))
      v =
    ↑(coordChangeL R e e' b) v
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
⊢ ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))
      (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)) =
    coordChangeL R e e' b
ext v
case h.h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))
          (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))
      v =
    (↑e' (totalSpaceMk b (Trivialization.symm e b v))).snd
case h.h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))
          (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))
      v =
    ↑(coordChangeL R e e' b) v
rw [coordChangeL_apply e e' hb]
no goals
case h.h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝¹⁰ : NontriviallyNormedField R
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module R (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace R F
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace (TotalSpace E)
inst✝³ : (x : B) → TopologicalSpace (E x)
inst✝² : FiberBundle F E
e e' : Trivialization F TotalSpace.proj
inst✝¹ : Trivialization.IsLinear R e
inst✝ : Trivialization.IsLinear R e'
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ ↑(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b ∈ e.baseSet)))
          (continuousLinearEquivAt R e' b (_ : b ∈ e'.baseSet)))
      v =
    (↑e' (totalSpaceMk b (Trivialization.symm e b v))).snd
rfl