Theorem: IsOpen.ite'
File: Mathlib/Topology/ContinuousOn.lean

tactic: classical
  simp only [isOpen_iff_continuous_mem, Set.ite] at *
  convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
  rename_i x
  by_cases hx : x ∈ t <;> simp [hx]
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
s s' t : Set α
hs : IsOpen s
hs' : IsOpen s'
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
⊢ IsOpen (t.ite s s')
state_after: no goals

tactic: simp only [isOpen_iff_continuous_mem, Set.ite] at *
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
s s' t : Set α
hs : IsOpen s
hs' : IsOpen s'
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
⊢ IsOpen (t.ite s s')
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
⊢ Continuous fun x => x ∈ s ∩ t ∪ s' \ t

tactic: convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
⊢ Continuous fun x => x ∈ s ∩ t ∪ s' \ t
state_after: case h.e'_5.h.a
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
x✝ : α
⊢ x✝ ∈ s ∩ t ∪ s' \ t ↔ t.piecewise (fun x => x ∈ s) (fun x => x ∈ s') x✝

tactic: rename_i x
state_before: case h.e'_5.h.a
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
x✝ : α
⊢ x✝ ∈ s ∩ t ∪ s' \ t ↔ t.piecewise (fun x => x ∈ s) (fun x => x ∈ s') x✝
state_after: case h.e'_5.h.a
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
x : α
⊢ x ∈ s ∩ t ∪ s' \ t ↔ t.piecewise (fun x => x ∈ s) (fun x => x ∈ s') x

tactic: by_cases hx : x ∈ t <;> simp [hx]
state_before: case h.e'_5.h.a
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
s s' t : Set α
ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s'
hs : Continuous fun x => x ∈ s
hs' : Continuous fun x => x ∈ s'
x : α
⊢ x ∈ s ∩ t ∪ s' \ t ↔ t.piecewise (fun x => x ∈ s) (fun x => x ∈ s') x
state_after: no goals