Module: Mathlib.Topology.VectorBundle.Hom

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip
h₃ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₄ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_continuousOn (h₃.mono ?_))).congr ?_

Goal state after:
case refine_1
𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip
h₃ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₄ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₂.baseSet ∩ e₂'.baseSet

case refine_2
𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip
h₃ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₄ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₁'.baseSet ∩ e₁.baseSet

case refine_3
𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))
h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip
h₃ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)
h₄ : ContinuousOn (fun b ↦ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) (e₂.baseSet ∩ e₂'.baseSet)
⊢ EqOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
    (fun x ↦
      (⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)) ∘ fun b ↦ ↑(Trivialization.coordChangeL 𝕜₂ e₂ e₂' b)) x ∘SL
        (⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip ∘ fun b ↦ ↑(Trivialization.coordChangeL 𝕜₁ e₁' e₁ b)) x)
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))