File: Mathlib/Order/Filter/IndicatorFunction.lean

full_name: mulIndicator_eventuallyEq
code: @[to_additive]
theorem mulIndicator_eventuallyEq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
    mulIndicator s f =ᶠ[l] mulIndicator t g 
goal: s.mulIndicator f =ᶠ[l] t.mulIndicator g

full_name: indicator_eventuallyEq
code: @[to_additive]
theorem mulIndicator_eventuallyEq (hf : f =ᶠ[l ⊓ 𝓟 s] g) (hs : s =ᶠ[l] t) :
    mulIndicator s f =ᶠ[l] mulIndicator t g 
goal: s.indicator f =ᶠ[l] t.indicator g

full_name: mulIndicator_union_eventuallyEq
code: @[to_additive]
theorem mulIndicator_union_eventuallyEq (h : ∀ᶠ a in l, a ∉ s ∩ t) :
    mulIndicator (s ∪ t) f =ᶠ[l] mulIndicator s f * mulIndicator t f 
goal: (s ∪ t).mulIndicator f =ᶠ[l] s.mulIndicator f * t.mulIndicator f

full_name: indicator_union_eventuallyEq
code: @[to_additive]
theorem mulIndicator_union_eventuallyEq (h : ∀ᶠ a in l, a ∉ s ∩ t) :
    mulIndicator (s ∪ t) f =ᶠ[l] mulIndicator s f * mulIndicator t f 
goal: (s ∪ t).indicator f =ᶠ[l] s.indicator f + t.indicator f

full_name: mulIndicator_eventuallyLE_mulIndicator
code: @[to_additive]
theorem mulIndicator_eventuallyLE_mulIndicator (h : f ≤ᶠ[l ⊓ 𝓟 s] g) :
    mulIndicator s f ≤ᶠ[l] mulIndicator s g 
goal: s.mulIndicator f ≤ᶠ[l] s.mulIndicator g

full_name: indicator_eventuallyLE_indicator
code: @[to_additive]
theorem mulIndicator_eventuallyLE_mulIndicator (h : f ≤ᶠ[l ⊓ 𝓟 s] g) :
    mulIndicator s f ≤ᶠ[l] mulIndicator s g 
goal: s.indicator f ≤ᶠ[l] s.indicator g

full_name: Monotone.indicator_eventuallyEq_iUnion
code: @[to_additive]
theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Monotone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a 
goal: (fun i => (s i).indicator f a) =ᶠ[Filter.atTop] fun x => (⋃ i, s i).indicator f a

full_name: Monotone.mulIndicator_eventuallyEq_iUnion
code: @[to_additive]
theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Monotone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a 
goal: (fun i => (s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (⋃ i, s i).mulIndicator f a

full_name: Monotone.tendsto_indicator
code: @[to_additive]
theorem Monotone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋃ i, s i) f a) 
goal: Filter.Tendsto (fun i => (s i).indicator f a) Filter.atTop (pure ((⋃ i, s i).indicator f a))

full_name: Monotone.tendsto_mulIndicator
code: @[to_additive]
theorem Monotone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋃ i, s i) f a) 
goal: Filter.Tendsto (fun i => (s i).mulIndicator f a) Filter.atTop (pure ((⋃ i, s i).mulIndicator f a))

full_name: Antitone.mulIndicator_eventuallyEq_iInter
code: @[to_additive]
theorem Antitone.mulIndicator_eventuallyEq_iInter {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Antitone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋂ i, s i) f a 
goal: (fun i => (s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (⋂ i, s i).mulIndicator f a

full_name: Antitone.indicator_eventuallyEq_iInter
code: @[to_additive]
theorem Antitone.mulIndicator_eventuallyEq_iInter {ι} [Preorder ι] [One β] (s : ι → Set α)
    (hs : Antitone s) (f : α → β) (a : α) :
    (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋂ i, s i) f a 
goal: (fun i => (s i).indicator f a) =ᶠ[Filter.atTop] fun x => (⋂ i, s i).indicator f a

full_name: Antitone.tendsto_mulIndicator
code: @[to_additive]
theorem Antitone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋂ i, s i) f a) 
goal: Filter.Tendsto (fun i => (s i).mulIndicator f a) Filter.atTop (pure ((⋂ i, s i).mulIndicator f a))

full_name: Antitone.tendsto_indicator
code: @[to_additive]
theorem Antitone.tendsto_mulIndicator {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Antitone s)
    (f : α → β) (a : α) :
    Tendsto (fun i => mulIndicator (s i) f a) atTop (pure <| mulIndicator (⋂ i, s i) f a) 
goal: Filter.Tendsto (fun i => (s i).indicator f a) Filter.atTop (pure ((⋂ i, s i).indicator f a))

full_name: mulIndicator_biUnion_finset_eventuallyEq
code: @[to_additive]
theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
      fun _ ↦ mulIndicator (iUnion s) f a 
goal: (fun n => (⋃ i ∈ n, s i).mulIndicator f a) =ᶠ[Filter.atTop] fun x => (Set.iUnion s).mulIndicator f a

full_name: indicator_biUnion_finset_eventuallyEq
code: @[to_additive]
theorem mulIndicator_biUnion_finset_eventuallyEq {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) =ᶠ[atTop]
      fun _ ↦ mulIndicator (iUnion s) f a 
goal: (fun n => (⋃ i ∈ n, s i).indicator f a) =ᶠ[Filter.atTop] fun x => (Set.iUnion s).indicator f a

full_name: tendsto_mulIndicator_biUnion_finset
code: @[to_additive]
theorem tendsto_mulIndicator_biUnion_finset {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    Tendsto (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) atTop
      (pure <| mulIndicator (iUnion s) f a) 
goal: Filter.Tendsto (fun n => (⋃ i ∈ n, s i).mulIndicator f a) Filter.atTop (pure ((Set.iUnion s).mulIndicator f a))

full_name: tendsto_indicator_biUnion_finset
code: @[to_additive]
theorem tendsto_mulIndicator_biUnion_finset {ι} [One β] (s : ι → Set α) (f : α → β) (a : α) :
    Tendsto (fun n : Finset ι => mulIndicator (⋃ i ∈ n, s i) f a) atTop
      (pure <| mulIndicator (iUnion s) f a) 
goal: Filter.Tendsto (fun n => (⋃ i ∈ n, s i).indicator f a) Filter.atTop (pure ((Set.iUnion s).indicator f a))

full_name: Filter.EventuallyEq.support
code: @[to_additive]
protected theorem Filter.EventuallyEq.mulSupport [One β] {f g : α → β} {l : Filter α}
    (h : f =ᶠ[l] g) :
    Function.mulSupport f =ᶠ[l] Function.mulSupport g 
goal: Function.support f =ᶠ[l] Function.support g

full_name: Filter.EventuallyEq.mulSupport
code: @[to_additive]
protected theorem Filter.EventuallyEq.mulSupport [One β] {f g : α → β} {l : Filter α}
    (h : f =ᶠ[l] g) :
    Function.mulSupport f =ᶠ[l] Function.mulSupport g 
goal: Function.mulSupport f =ᶠ[l] Function.mulSupport g

full_name: Filter.EventuallyEq.mulIndicator
code: @[to_additive]
protected theorem Filter.EventuallyEq.mulIndicator [One β] {l : Filter α} {f g : α → β} {s : Set α}
    (hfg : f =ᶠ[l] g) : s.mulIndicator f =ᶠ[l] s.mulIndicator g 
goal: s.mulIndicator f =ᶠ[l] s.mulIndicator g

full_name: Filter.EventuallyEq.indicator
code: @[to_additive]
protected theorem Filter.EventuallyEq.mulIndicator [One β] {l : Filter α} {f g : α → β} {s : Set α}
    (hfg : f =ᶠ[l] g) : s.mulIndicator f =ᶠ[l] s.mulIndicator g 
goal: s.indicator f =ᶠ[l] s.indicator g

full_name: Filter.EventuallyEq.indicator_zero
code: @[to_additive]
theorem Filter.EventuallyEq.mulIndicator_one [One β] {l : Filter α} {f : α → β} {s : Set α}
    (hf : f =ᶠ[l] 1) : s.mulIndicator f =ᶠ[l] 1 
goal: s.indicator f =ᶠ[l] 0

full_name: Filter.EventuallyEq.mulIndicator_one
code: @[to_additive]
theorem Filter.EventuallyEq.mulIndicator_one [One β] {l : Filter α} {f : α → β} {s : Set α}
    (hf : f =ᶠ[l] 1) : s.mulIndicator f =ᶠ[l] 1 
goal: s.mulIndicator f =ᶠ[l] 1

full_name: Filter.EventuallyEq.of_mulIndicator
code: @[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator [One β] {l : Filter α} {f : α → β}
    (hf : ∀ᶠ x in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f) :
    s =ᶠ[l] t 
goal: s =ᶠ[l] t

full_name: Filter.EventuallyEq.of_indicator
code: @[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator [One β] {l : Filter α} {f : α → β}
    (hf : ∀ᶠ x in l, f x ≠ 1) {s t : Set α} (h : s.mulIndicator f =ᶠ[l] t.mulIndicator f) :
    s =ᶠ[l] t 
goal: s =ᶠ[l] t

full_name: Filter.EventuallyEq.of_indicator_const
code: @[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator_const [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} (h : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator fun _ ↦ c) : s =ᶠ[l] t 
goal: s =ᶠ[l] t

full_name: Filter.EventuallyEq.of_mulIndicator_const
code: @[to_additive]
theorem Filter.EventuallyEq.of_mulIndicator_const [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} (h : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator fun _ ↦ c) : s =ᶠ[l] t 
goal: s =ᶠ[l] t

full_name: Filter.mulIndicator_const_eventuallyEq
code: @[to_additive]
theorem Filter.mulIndicator_const_eventuallyEq [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator (fun _ ↦ c) ↔ s =ᶠ[l] t 
goal: ((s.mulIndicator fun x => c) =ᶠ[l] t.mulIndicator fun x => c) ↔ s =ᶠ[l] t

full_name: Filter.indicator_const_eventuallyEq
code: @[to_additive]
theorem Filter.mulIndicator_const_eventuallyEq [One β] {l : Filter α} {c : β} (hc : c ≠ 1)
    {s t : Set α} : s.mulIndicator (fun _ ↦ c) =ᶠ[l] t.mulIndicator (fun _ ↦ c) ↔ s =ᶠ[l] t 
goal: ((s.indicator fun x => c) =ᶠ[l] t.indicator fun x => c) ↔ s =ᶠ[l] t