Theorem: continuousOn_iff_continuous_restrict
File: Mathlib/Topology/ContinuousOn.lean

tactic: rw [ContinuousOn, continuous_iff_continuousAt]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ ContinuousOn f s ↔ Continuous (s.restrict f)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) ↔ ∀ (x : ↑s), ContinuousAt (s.restrict f) x

tactic: constructor
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) ↔ ∀ (x : ↑s), ContinuousAt (s.restrict f) x
state_after: case mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) → ∀ (x : ↑s), ContinuousAt (s.restrict f) x

case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ (x : ↑s), ContinuousAt (s.restrict f) x) → ∀ x ∈ s, ContinuousWithinAt f s x

tactic: intro h x xs
state_before: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ (x : ↑s), ContinuousAt (s.restrict f) x) → ∀ x ∈ s, ContinuousWithinAt f s x
state_after: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
h : ∀ (x : ↑s), ContinuousAt (s.restrict f) x
x : α
xs : x ∈ s
⊢ ContinuousWithinAt f s x

tactic: exact (continuousWithinAt_iff_continuousAt_restrict f xs).mpr (h ⟨x, xs⟩)
state_before: case mpr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
h : ∀ (x : ↑s), ContinuousAt (s.restrict f) x
x : α
xs : x ∈ s
⊢ ContinuousWithinAt f s x
state_after: no goals

tactic: rintro h ⟨x, xs⟩
state_before: case mp
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
⊢ (∀ x ∈ s, ContinuousWithinAt f s x) → ∀ (x : ↑s), ContinuousAt (s.restrict f) x
state_after: case mp.mk
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
h : ∀ x ∈ s, ContinuousWithinAt f s x
x : α
xs : x ∈ s
⊢ ContinuousAt (s.restrict f) ⟨x, xs⟩

tactic: exact (continuousWithinAt_iff_continuousAt_restrict f xs).mp (h x xs)
state_before: case mp.mk
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
s : Set α
h : ∀ x ∈ s, ContinuousWithinAt f s x
x : α
xs : x ∈ s
⊢ ContinuousAt (s.restrict f) ⟨x, xs⟩
state_after: no goals