Module: Mathlib.Geometry.Manifold.VectorBundle.Basic

Goal state before:
pp: case refine_1
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
s : (x : B) → E x
a : Set B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
ha : IsOpen[inst✝¹¹] a
ha' : a ⊆ e.baseSet
h : ContMDiffOn IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) a
x : B
hx : x ∈ a
this : ContMDiffAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) x
⊢ ContMDiffWithinAt IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) a x

pp: case refine_2
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
s : (x : B) → E x
a : Set B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
ha : IsOpen[inst✝¹¹] a
ha' : a ⊆ e.baseSet
h : ContMDiffOn IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) a
x : B
hx : x ∈ a
this : ContMDiffAt IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) x
⊢ ContMDiffWithinAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) a x

Tactic:
· exact ((e.contMDiffAt_section_iff (ha' hx)).mp this).contMDiffWithinAt

Goal state after:
case refine_2
n : ℕ∞ω
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁵ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁴ : NormedAddCommGroup EB
inst✝¹³ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹² : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹¹ : TopologicalSpace B
inst✝¹⁰ : ChartedSpace HB B
inst✝⁹ : (x : B) → AddCommMonoid (E x)
inst✝⁸ : (x : B) → Module 𝕜 (E x)
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → TopologicalSpace (E x)
inst✝³ : FiberBundle F E
inst✝² : VectorBundle 𝕜 F E
inst✝¹ : ContMDiffVectorBundle n F E IB
s : (x : B) → E x
a : Set B
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
ha : IsOpen[inst✝¹¹] a
ha' : a ⊆ e.baseSet
h : ContMDiffOn IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) a
x : B
hx : x ∈ a
this : ContMDiffAt IB 𝓘(𝕜, F) n (fun x ↦ (↑e { proj := x, snd := s x }).2) x
⊢ ContMDiffWithinAt IB (IB.prod 𝓘(𝕜, F)) n (fun x ↦ TotalSpace.mk' F x (s x)) a x