Theorem: ContinuousMap.piEquiv_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✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
⊢ LeftInverse (fun f i => (eval i).comp f) pi
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝ : (i : I) → C(A, X i)
⊢ (fun f i => (eval i).comp f) (pi x✝) = x✝
tactic: intro
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝ : (i : I) → C(A, X i)
⊢ (fun f i => (eval i).comp f) (pi x✝) = x✝
state_after: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝¹ : (i : I) → C(A, X i)
x✝ : I
a✝ : A
⊢ ((fun f i => (eval i).comp f) (pi 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✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝¹ : (i : I) → C(A, X i)
x✝ : I
a✝ : A
⊢ ((fun f i => (eval i).comp f) (pi x✝¹) x✝) a✝ = (x✝¹ x✝) a✝
state_after: no goals
tactic: simp [pi]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
⊢ Function.RightInverse (fun f i => (eval i).comp f) pi
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝ : C(A, (i : I) → X i)
⊢ pi ((fun f i => (eval i).comp f) x✝) = x✝
tactic: intro
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝ : C(A, (i : I) → X i)
⊢ pi ((fun f i => (eval i).comp f) x✝) = x✝
state_after: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝¹ : C(A, (i : I) → X i)
a✝ : A
x✝ : I
⊢ (pi ((fun f i => (eval i).comp f) x✝¹)) a✝ x✝ = x✝¹ a✝ x✝
tactic: ext
state_before: case h.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : TopologicalSpace δ
f g : C(α, β)
I : Type u_5
A : Type u_6
X : I → Type u_7
Y : I → Type u_8
inst✝² : TopologicalSpace A
inst✝¹ : (i : I) → TopologicalSpace (X i)
inst✝ : (i : I) → TopologicalSpace (Y i)
x✝¹ : C(A, (i : I) → X i)
a✝ : A
x✝ : I
⊢ (pi ((fun f i => (eval i).comp f) x✝¹)) a✝ x✝ = x✝¹ a✝ x✝
state_after: no goals
tactic: simp [pi]