File: Mathlib/Topology/IndicatorConstPointwise.lean

full_name: tendsto_ite
code: lemma tendsto_ite {β : Type*} {p : ι → Prop} [DecidablePred p] {q : Prop} [Decidable q]
    {a b : β} {F G : Filter β}
    (haG : {a}ᶜ ∈ G) (hbF : {b}ᶜ ∈ F) (haF : principal {a} ≤ F) (hbG : principal {b} ≤ G) :
    Tendsto (fun i ↦ if p i then a else b) L (if q then F else G) ↔ ∀ᶠ i in L, p i ↔ q 
goal: Filter.Tendsto (fun i => if p i then a else b) L (if q then F else G) ↔ ∀ᶠ (i : ι) in L, p i ↔ q

full_name: tendsto_indicator_const_apply_iff_eventually'
code: lemma tendsto_indicator_const_apply_iff_eventually' (b : β)
    (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) (x : α) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b) x) L (𝓝 (A.indicator (fun (_ : α) ↦ b) x))
      ↔ ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
goal: Filter.Tendsto (fun i => (As i).indicator (fun x => b) x) L (𝓝 (A.indicator (fun x => b) x)) ↔
  ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

full_name: tendsto_indicator_const_iff_forall_eventually'
code: lemma tendsto_indicator_const_iff_forall_eventually'
    (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ ∀ x, ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

full_name: tendsto_indicator_const_apply_iff_eventually
code: /-- The indicator functions of `Asᵢ` evaluated at `x` tend to the indicator function of `A`
evaluated at `x` if and only if we eventually have the equivalence `x ∈ Asᵢ ↔ x ∈ A`. -/
@[simp] lemma tendsto_indicator_const_apply_iff_eventually [T1Space β] (b : β) [NeZero b]
    (x : α) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b) x) L (𝓝 (A.indicator (fun (_ : α) ↦ b) x))
      ↔ ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
goal: Filter.Tendsto (fun i => (As i).indicator (fun x => b) x) L (𝓝 (A.indicator (fun x => b) x)) ↔
  ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

full_name: tendsto_indicator_const_iff_forall_eventually
code: /-- The indicator functions of `Asᵢ` tend to the indicator function of `A` pointwise if and only if
for every `x`, we eventually have the equivalence `x ∈ Asᵢ ↔ x ∈ A`. -/
@[simp] lemma tendsto_indicator_const_iff_forall_eventually [T1Space β] (b : β) [NeZero b] :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ ∀ x, ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

full_name: tendsto_indicator_const_iff_tendsto_pi_pure'
code: lemma tendsto_indicator_const_iff_tendsto_pi_pure'
    (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ (Tendsto As L <| Filter.pi (pure <| · ∈ A)) 
goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  Filter.Tendsto As L (Filter.pi fun x => pure (x ∈ A))

full_name: tendsto_indicator_const_iff_tendsto_pi_pure
code: lemma tendsto_indicator_const_iff_tendsto_pi_pure [T1Space β] (b : β) [NeZero b] :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ (Tendsto As L <| Filter.pi (pure <| · ∈ A)) 
goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  Filter.Tendsto As L (Filter.pi fun x => pure (x ∈ A))