Theorem: continuousOn_piecewise_ite'
File: Mathlib/Topology/ContinuousOn.lean

tactic: apply ContinuousOn.piecewise
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ContinuousOn (t.piecewise f f') (t.ite s s')
state_after: case ht
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ∀ a ∈ t.ite s s' ∩ frontier t, f a = f' a

case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure t)

case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ContinuousOn f' (t.ite s s' ∩ closure tᶜ)

tactic: rwa [ite_inter_of_inter_eq _ H]
state_before: case ht
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ∀ a ∈ t.ite s s' ∩ frontier t, f a = f' a
state_after: no goals

tactic: rwa [ite_inter_closure_eq_of_inter_frontier_eq H]
state_before: case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ContinuousOn f (t.ite s s' ∩ closure t)
state_after: no goals

tactic: rwa [ite_inter_closure_compl_eq_of_inter_frontier_eq H]
state_before: case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝⁴ : (i : ι) → TopologicalSpace (π i)
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
s s' t : Set α
f f' : α → β
inst✝ : (x : α) → Decidable (x ∈ t)
h : ContinuousOn f (s ∩ closure t)
h' : ContinuousOn f' (s' ∩ closure tᶜ)
H : s ∩ frontier t = s' ∩ frontier t
Heq : EqOn f f' (s ∩ frontier t)
⊢ ContinuousOn f' (t.ite s s' ∩ closure tᶜ)
state_after: no goals