Module: Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
⊢ ∃ φ U,
    ∃ (hU : IsOpen[inst✝⁷] U) (hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U) (h2φ :
      ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U),
      (e ≫ₕ e').EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)

Tactic:
refine ⟨fun b => (φ b).trans (φ' b), _, hU.inter hU', ?_, ?_,
      Setoid.trans (OpenPartialHomeomorph.EqOnSource.trans' heφ heφ') ⟨?_, ?_⟩⟩

Goal state after:
case refine_1
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑((fun b ↦ (φ b).trans (φ' b)) x)) (U ∩ U')

case refine_2
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
⊢ ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑((fun b ↦ (φ b).trans (φ' b)) x).symm) (U ∩ U')

case refine_3
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
⊢ (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯ ≫ₕ FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯).source =
    (FiberwiseLinear.openPartialHomeomorph (fun b ↦ (φ b).trans (φ' b)) ⋯ ⋯ ⋯).source

case refine_4
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : ℕ∞ω
e e' : OpenPartialHomeomorph (B × F) (B × F)
φ : B → F ≃L[𝕜] F
U : Set B
hU : IsOpen[inst✝⁷] U
hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x)) U
h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ x).symm) U
heφ : e.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯)
φ' : B → F ≃L[𝕜] F
U' : Set B
hU' : IsOpen[inst✝⁷] U'
hφ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x)) U'
h2φ' : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x ↦ ↑(φ' x).symm) U'
heφ' : e'.EqOnSource (FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯)
⊢ EqOn (↑(FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯ ≫ₕ FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯))
    (↑(FiberwiseLinear.openPartialHomeomorph (fun b ↦ (φ b).trans (φ' b)) ⋯ ⋯ ⋯))
    (FiberwiseLinear.openPartialHomeomorph φ hU ⋯ ⋯ ≫ₕ FiberwiseLinear.openPartialHomeomorph φ' hU' ⋯ ⋯).source