Theorem: Trivialization.continuousLinearEquivAt_prod
File: Mathlib/Topology/VectorBundle/Constructions.lean

Proof steps:
state_before: 𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
⊢ continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx =
    (continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)
state_after: case h.h
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v : E₁ x × E₂ x
⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) v =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) v
tactic: ext v : 2
state_before: case h.h
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v : E₁ x × E₂ x
⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) v =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) v
state_after: case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)
tactic: obtain ⟨v₁, v₂⟩ := v
state_before: case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx) (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)
state_after: case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (fun y =>
        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,
                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,
                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯,
                continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,
                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }
            { proj := x, snd := y }).2)
      (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)
tactic: rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod]
state_before: case h.h.mk
𝕜 : Type u_1
B : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : TopologicalSpace B
F₁ : Type u_3
inst✝¹⁵ : NormedAddCommGroup F₁
inst✝¹⁴ : NormedSpace 𝕜 F₁
E₁ : B → Type u_4
inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_5
inst✝¹² : NormedAddCommGroup F₂
inst✝¹¹ : NormedSpace 𝕜 F₂
E₂ : B → Type u_6
inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)
inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)
inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)
inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)
inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)
inst✝³ : FiberBundle F₁ E₁
inst✝² : FiberBundle F₂ E₂
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ TotalSpace.proj
inst✝¹ : Trivialization.IsLinear 𝕜 e₁
inst✝ : Trivialization.IsLinear 𝕜 e₂
x : B
hx : x ∈ (e₁.prod e₂).baseSet
v₁ : E₁ x
v₂ : E₂ x
⊢ (fun y =>
        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,
                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,
                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯,
                continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,
                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }
            { proj := x, snd := y }).2)
      (v₁, v₂) =
    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)
state_after: no goals
tactic: exact (congr_arg Prod.snd (prod_apply 𝕜 hx.1 hx.2 v₁ v₂) : _)