Theorem: mulIndicator_cthickening_eventually_eq_mulIndicator_closure
File: Mathlib/Topology/MetricSpace/ThickenedIndicator.lean

tactic: by_cases x_mem_closure : x ∈ closure E
state_before: α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: case pos
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x

case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x

tactic: filter_upwards [univ_mem] with δ _
state_before: case pos
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
δ : ℝ
a✝ : δ ∈ univ
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x

tactic: have obs : x ∈ cthickening δ E := closure_subset_cthickening δ E x_mem_closure
state_before: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
δ : ℝ
a✝ : δ ∈ univ
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
δ : ℝ
a✝ : δ ∈ univ
obs : x ∈ cthickening δ E
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x

tactic: rw [mulIndicator_of_mem obs f, mulIndicator_of_mem x_mem_closure f]
state_before: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure E
δ : ℝ
a✝ : δ ∈ univ
obs : x ∈ cthickening δ E
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: no goals

tactic: filter_upwards [eventually_not_mem_cthickening_of_infEdist_pos x_mem_closure] with δ hδ
state_before: case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0, (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure E
δ : ℝ
hδ : x ∉ cthickening δ E
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x

tactic: simp only [hδ, not_false_eq_true, mulIndicator_of_not_mem, x_mem_closure]
state_before: case h
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure E
δ : ℝ
hδ : x ∉ cthickening δ E
⊢ (cthickening δ E).mulIndicator f x = (closure E).mulIndicator f x
state_after: no goals