-- Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
theorem ContMDiffWithinAt.coordChange
    (hf : ContMDiffWithinAt IM IB n f s x) (hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x)
    (he : f x ∈ e.baseSet) (he' : f x ∈ e'.baseSet) :
    ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ e.coordChange e' (f y) (g y)) s x

Proof state:
case h
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
s : Set M
f : M → B
g : M → F
x : M
hf : ContMDiffWithinAt IM IB n f s x
hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x
he : f x ∈ e.baseSet
he' : f x ∈ e'.baseSet
this : e.baseSet ∩ e'.baseSet ∈ 𝓝 (f x)
y : M
hy : y ∈ f ⁻¹' (e.baseSet ∩ e'.baseSet)
⊢ e.coordChange e' (f y) (g y) = ↑(Trivialization.coordChangeL 𝕜 e e' (f y)) (g y)

Tactic:
exact (Trivialization.coordChangeL_apply' e e' hy (g y)).symm

Result:
no goals

Proof state:
case refine_1
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
s : Set M
f : M → B
g : M → F
x : M
hf : ContMDiffWithinAt IM IB n f s x
hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x
he : f x ∈ e.baseSet
he' : f x ∈ e'.baseSet
⊢ (fun y => e.coordChange e' (f y) (g y)) =ᶠ[𝓝[s] x] fun x => ↑(Trivialization.coordChangeL 𝕜 e e' (f x)) (g x)

Tactic:
have : e.baseSet ∩ e'.baseSet ∈ 𝓝 (f x) :=
 (e.open_baseSet.inter e'.open_baseSet).mem_nhds ⟨he, he'⟩

Result:
case refine_1
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝²¹ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝²⁰ : NormedAddCommGroup EB
inst✝¹⁹ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹⁸ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹⁷ : TopologicalSpace B
inst✝¹⁶ : ChartedSpace HB B
EM : Type u_9
inst✝¹⁵ : NormedAddCommGroup EM
inst✝¹⁴ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝¹³ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝¹² : TopologicalSpace M
inst✝¹¹ : ChartedSpace HM M
inst✝¹⁰ : (x : B) → AddCommMonoid (E x)
inst✝⁹ : (x : B) → Module 𝕜 (E x)
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : TopologicalSpace (TotalSpace F E)
inst✝⁵ : (x : B) → TopologicalSpace (E x)
inst✝⁴ : FiberBundle F E
inst✝³ : VectorBundle 𝕜 F E
inst✝² : ContMDiffVectorBundle n F E IB
e e' : Trivialization F TotalSpace.proj
inst✝¹ : MemTrivializationAtlas e
inst✝ : MemTrivializationAtlas e'
s : Set M
f : M → B
g : M → F
x : M
hf : ContMDiffWithinAt IM IB n f s x
hg : ContMDiffWithinAt IM 𝓘(𝕜, F) n g s x
he : f x ∈ e.baseSet
he' : f x ∈ e'.baseSet
this : e.baseSet ∩ e'.baseSet ∈ 𝓝 (f x)
⊢ (fun y => e.coordChange e' (f y) (g y)) =ᶠ[𝓝[s] x] fun x => ↑(Trivialization.coordChangeL 𝕜 e e' (f x)) (g x)