-- Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
theorem Trivialization.contMDiffOn_symm_trans :
    ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n
      (e.toPartialHomeomorph.symm ≫ₕ e'.toPartialHomeomorph) (e.target ∩ e'.target)

Proof state:
case mk
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
b : B
x : F
hb : (b, x) ∈ e.target ∩ e'.target
⊢ (↑(e.symm ≫ₕ e'.toPartialHomeomorph) (b, x)).1 = ((b, x).1, e.coordChange e' (b, x).1 (b, x).2).1

Tactic:
have : (e.toPartialHomeomorph.symm (b, x)).1 ∈ e'.baseSet := by
  simp_all only [Trivialization.mem_target, mfld_simps]

Result:
case mk
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
b : B
x : F
hb : (b, x) ∈ e.target ∩ e'.target
this : (↑e.symm (b, x)).proj ∈ e'.baseSet
⊢ (↑(e.symm ≫ₕ e'.toPartialHomeomorph) (b, x)).1 = ((b, x).1, e.coordChange e' (b, x).1 (b, x).2).1

Proof state:
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
⊢ ContMDiffOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) n (↑(e.symm ≫ₕ e'.toPartialHomeomorph)) (e.target ∩ e'.target)

Tactic:
refine (contMDiffOn_fst.prod_mk
  (contMDiffOn_fst.coordChange contMDiffOn_snd Hmaps.1 Hmaps.2)).congr ?_

Result:
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
⊢ ∀ y ∈ e.target ∩ e'.target, ↑(e.symm ≫ₕ e'.toPartialHomeomorph) y = (y.1, e.coordChange e' y.1 y.2)

Proof state:
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
b : B
x : F
hb : (b, x) ∈ e.target ∩ e'.target
⊢ (↑e.symm (b, x)).proj ∈ e'.baseSet

Tactic:
simp_all only [Trivialization.mem_target, mfld_simps]

Result:
no goals

Proof state:
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
⊢ ∀ y ∈ e.target ∩ e'.target, ↑(e.symm ≫ₕ e'.toPartialHomeomorph) y = (y.1, e.coordChange e' y.1 y.2)

Tactic:
rintro ⟨b, x⟩ hb

Result:
case mk
n : WithTop ℕ∞
𝕜 : Type u_1
B : Type u_2
F : Type u_4
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 : (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'
Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) e.baseSet ∧ MapsTo Prod.fst (e.target ∩ e'.target) e'.baseSet
b : B
x : F
hb : (b, x) ∈ e.target ∩ e'.target
⊢ ↑(e.symm ≫ₕ e'.toPartialHomeomorph) (b, x) = ((b, x).1, e.coordChange e' (b, x).1 (b, x).2)