Theorem: ContinuousMap.sigmaEquiv_apply
File: Mathlib/Topology/ContinuousFunction/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
⊢ LeftInverse (fun f i => f.comp (sigmaMk i)) sigma
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : (i : I) → C(X i, A)
⊢ (fun f i => f.comp (sigmaMk i)) (sigma x✝) = x✝
tactic: intro
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : (i : I) → C(X i, A)
⊢ (fun f i => f.comp (sigmaMk i)) (sigma x✝) = x✝
state_after: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝¹ : (i : I) → C(X i, A)
x✝ : I
a✝ : X x✝
⊢ ((fun f i => f.comp (sigmaMk i)) (sigma x✝¹) x✝) a✝ = (x✝¹ x✝) a✝
tactic: ext
state_before: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝¹ : (i : I) → C(X i, A)
x✝ : I
a✝ : X x✝
⊢ ((fun f i => f.comp (sigmaMk i)) (sigma x✝¹) x✝) a✝ = (x✝¹ x✝) a✝
state_after: no goals
tactic: simp
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
⊢ Function.RightInverse (fun f i => f.comp (sigmaMk i)) sigma
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : C((i : I) × X i, A)
⊢ sigma ((fun f i => f.comp (sigmaMk i)) x✝) = x✝
tactic: intro
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : C((i : I) × X i, A)
⊢ sigma ((fun f i => f.comp (sigmaMk i)) x✝) = x✝
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : C((i : I) × X i, A)
a✝ : (i : I) × X i
⊢ (sigma ((fun f i => f.comp (sigmaMk i)) x✝)) a✝ = x✝ a✝
tactic: ext
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
inst✝² : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
inst✝¹ : TopologicalSpace A
inst✝ : (i : I) → TopologicalSpace (X i)
x✝ : C((i : I) × X i, A)
a✝ : (i : I) × X i
⊢ (sigma ((fun f i => f.comp (sigmaMk i)) x✝)) a✝ = x✝ a✝
state_after: no goals
tactic: simp