Theorem: Trivialization.continuousOn_symm
File: Mathlib/Topology/FiberBundle/Trivialization.lean

Proof steps:
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
⊢ ContinuousOn (fun z => TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ univ)
state_after: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (fun z => TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ univ)
tactic: have : ∀ z ∈ e.baseSet ×ˢ (univ : Set F),
      TotalSpace.mk z.1 (e.symm z.1 z.2) = e.toPartialHomeomorph.symm z := by
    rintro x ⟨hx : x.1 ∈ e.baseSet, _⟩
    rw [e.mk_symm hx]
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (fun z => TotalSpace.mk' F z.1 (e.symm z.1 z.2)) (e.baseSet ×ˢ univ)
state_after: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) (e.baseSet ×ˢ univ)
tactic: refine ContinuousOn.congr ?_ this
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) (e.baseSet ×ˢ univ)
state_after: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) e.target
tactic: rw [← e.target_eq]
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
E : B → Type u_4
Z : Type u_5
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
proj : Z → B
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e' : Trivialization F TotalSpace.proj
x' : TotalSpace F E
b : B
y : E b
inst✝ : (x : B) → Zero (E x)
e : Trivialization F TotalSpace.proj
this : ∀ z ∈ e.baseSet ×ˢ univ, { proj := z.1, snd := e.symm z.1 z.2 } = ↑e.symm z
⊢ ContinuousOn (↑e.symm) e.target
state_after: no goals
tactic: exact e.toPartialHomeomorph.continuousOn_symm