File: Mathlib/Algebra/Order/Group/Indicator.lean

full_name: Function.support_sup
code: @[to_additive]
lemma mulSupport_sup [SemilatticeSup M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊔ g x) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.support fun x => f x ⊔ g x) ⊆ Function.support f ∪ Function.support g

full_name: Function.mulSupport_sup
code: @[to_additive]
lemma mulSupport_sup [SemilatticeSup M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊔ g x) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.mulSupport fun x => f x ⊔ g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g

full_name: Function.mulSupport_inf
code: @[to_additive]
lemma mulSupport_inf [SemilatticeInf M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊓ g x) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.mulSupport fun x => f x ⊓ g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g

full_name: Function.support_inf
code: @[to_additive]
lemma mulSupport_inf [SemilatticeInf M] (f g : α → M) :
    mulSupport (fun x ↦ f x ⊓ g x) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.support fun x => f x ⊓ g x) ⊆ Function.support f ∪ Function.support g

full_name: Function.support_max
code: @[to_additive]
lemma mulSupport_max [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ max (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.support fun x => max (f x) (g x)) ⊆ Function.support f ∪ Function.support g

full_name: Function.mulSupport_max
code: @[to_additive]
lemma mulSupport_max [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ max (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.mulSupport fun x => max (f x) (g x)) ⊆ Function.mulSupport f ∪ Function.mulSupport g

full_name: Function.support_min
code: @[to_additive]
lemma mulSupport_min [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ min (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.support fun x => min (f x) (g x)) ⊆ Function.support f ∪ Function.support g

full_name: Function.mulSupport_min
code: @[to_additive]
lemma mulSupport_min [LinearOrder M] (f g : α → M) :
    mulSupport (fun x ↦ min (f x) (g x)) ⊆ mulSupport f ∪ mulSupport g 
goal: (Function.mulSupport fun x => min (f x) (g x)) ⊆ Function.mulSupport f ∪ Function.mulSupport g

full_name: Function.support_iSup
code: @[to_additive]
lemma mulSupport_iSup [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨆ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
goal: (Function.support fun x => ⨆ i, f i x) ⊆ ⋃ i, Function.support (f i)

full_name: Function.mulSupport_iSup
code: @[to_additive]
lemma mulSupport_iSup [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨆ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
goal: (Function.mulSupport fun x => ⨆ i, f i x) ⊆ ⋃ i, Function.mulSupport (f i)

full_name: Function.mulSupport_iInf
code: @[to_additive]
lemma mulSupport_iInf [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨅ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
goal: (Function.mulSupport fun x => ⨅ i, f i x) ⊆ ⋃ i, Function.mulSupport (f i)

full_name: Function.support_iInf
code: @[to_additive]
lemma mulSupport_iInf [ConditionallyCompleteLattice M] [Nonempty ι] (f : ι → α → M) :
    mulSupport (fun x ↦ ⨅ i, f i x) ⊆ ⋃ i, mulSupport (f i) 
goal: (Function.support fun x => ⨅ i, f i x) ⊆ ⋃ i, Function.support (f i)

full_name: Set.mulIndicator_apply_le'
code: @[to_additive]
lemma mulIndicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
    mulIndicator s f a ≤ y 
goal: s.mulIndicator f a ≤ y

full_name: Set.indicator_apply_le'
code: @[to_additive]
lemma mulIndicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
    mulIndicator s f a ≤ y 
goal: s.indicator f a ≤ y

full_name: Set.mulIndicator_le'
code: @[to_additive]
lemma mulIndicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a, a ∉ s → 1 ≤ g a) :
    mulIndicator s f ≤ g 
goal: s.mulIndicator f ≤ g

full_name: Set.indicator_le'
code: @[to_additive]
lemma mulIndicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a, a ∉ s → 1 ≤ g a) :
    mulIndicator s f ≤ g 
goal: s.indicator f ≤ g

full_name: Set.le_indicator_apply
code: @[to_additive]
lemma le_mulIndicator_apply (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
    y ≤ mulIndicator s g a 
goal: y ≤ s.indicator g a

full_name: Set.le_mulIndicator_apply
code: @[to_additive]
lemma le_mulIndicator_apply (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
    y ≤ mulIndicator s g a 
goal: y ≤ s.mulIndicator g a

full_name: Set.le_mulIndicator
code: @[to_additive]
lemma le_mulIndicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
    f ≤ mulIndicator s g 
goal: f ≤ s.mulIndicator g

full_name: Set.le_indicator
code: @[to_additive]
lemma le_mulIndicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
    f ≤ mulIndicator s g 
goal: f ≤ s.indicator g

full_name: Set.one_le_mulIndicator_apply
code: @[to_additive indicator_apply_nonneg]
lemma one_le_mulIndicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mulIndicator s f a 
goal: 1 ≤ s.mulIndicator f a

full_name: Set.indicator_apply_nonneg
code: @[to_additive indicator_apply_nonneg]
lemma one_le_mulIndicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mulIndicator s f a 
goal: 0 ≤ s.indicator f a

full_name: Set.one_le_mulIndicator
code: @[to_additive indicator_nonneg]
lemma one_le_mulIndicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mulIndicator s f a 
goal: 1 ≤ s.mulIndicator f a

full_name: Set.indicator_nonneg
code: @[to_additive indicator_nonneg]
lemma one_le_mulIndicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mulIndicator s f a 
goal: 0 ≤ s.indicator f a

full_name: Set.mulIndicator_apply_le_one
code: @[to_additive]
lemma mulIndicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mulIndicator s f a ≤ 1 
goal: s.mulIndicator f a ≤ 1

full_name: Set.indicator_apply_nonpos
code: @[to_additive]
lemma mulIndicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mulIndicator s f a ≤ 1 
goal: s.indicator f a ≤ 0

full_name: Set.indicator_nonpos
code: @[to_additive]
lemma mulIndicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) : mulIndicator s f a ≤ 1 
goal: s.indicator f a ≤ 0

full_name: Set.mulIndicator_le_one
code: @[to_additive]
lemma mulIndicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) : mulIndicator s f a ≤ 1 
goal: s.mulIndicator f a ≤ 1

full_name: Set.indicator_le_indicator
code: @[to_additive]
lemma mulIndicator_le_mulIndicator (h : f a ≤ g a) : mulIndicator s f a ≤ mulIndicator s g a 
goal: s.indicator f a ≤ s.indicator g a

full_name: Set.mulIndicator_le_mulIndicator
code: @[to_additive]
lemma mulIndicator_le_mulIndicator (h : f a ≤ g a) : mulIndicator s f a ≤ mulIndicator s g a 
goal: s.mulIndicator f a ≤ s.mulIndicator g a

full_name: Set.indicator_le_indicator_of_subset
code: @[to_additive]
lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a) (a : α) :
    mulIndicator s f a ≤ mulIndicator t f a 
goal: s.indicator f a ≤ t.indicator f a

full_name: Set.mulIndicator_le_mulIndicator_of_subset
code: @[to_additive]
lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a) (a : α) :
    mulIndicator s f a ≤ mulIndicator t f a 
goal: s.mulIndicator f a ≤ t.mulIndicator f a

full_name: Set.mulIndicator_le_self'
code: @[to_additive]
lemma mulIndicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mulIndicator s f ≤ f 
goal: s.mulIndicator f ≤ f

full_name: Set.indicator_le_self'
code: @[to_additive]
lemma mulIndicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mulIndicator s f ≤ f 
goal: s.indicator f ≤ f

full_name: Set.indicator_le_indicator_nonneg
code: lemma indicator_le_indicator_nonneg (s : Set α) (f : α → M) :
    s.indicator f ≤ {a | 0 ≤ f a}.indicator f 
goal: s.indicator f ≤ {a | 0 ≤ f a}.indicator f

full_name: Set.indicator_nonpos_le_indicator
code: lemma indicator_nonpos_le_indicator (s : Set α) (f : α → M) :
    {a | f a ≤ 0}.indicator f ≤ s.indicator f 
goal: {a | f a ≤ 0}.indicator f ≤ s.indicator f

full_name: Set.indicator_iUnion_apply
code: @[to_additive]
lemma mulIndicator_iUnion_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋃ i, s i) f x = ⨆ i, mulIndicator (s i) f x 
goal: (⋃ i, s i).indicator f x = ⨆ i, (s i).indicator f x

full_name: Set.mulIndicator_iUnion_apply
code: @[to_additive]
lemma mulIndicator_iUnion_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋃ i, s i) f x = ⨆ i, mulIndicator (s i) f x 
goal: (⋃ i, s i).mulIndicator f x = ⨆ i, (s i).mulIndicator f x

full_name: Set.mulIndicator_iInter_apply
code: @[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x 
goal: (⋂ i, s i).mulIndicator f x = ⨅ i, (s i).mulIndicator f x

full_name: Set.indicator_iInter_apply
code: @[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
    mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x 
goal: (⋂ i, s i).indicator f x = ⨅ i, (s i).indicator f x

full_name: Set.indicator_le_self
code: @[to_additive]
lemma mulIndicator_le_self (s : Set α) (f : α → M) : mulIndicator s f ≤ f 
goal: s.indicator f ≤ f

full_name: Set.mulIndicator_le_self
code: @[to_additive]
lemma mulIndicator_le_self (s : Set α) (f : α → M) : mulIndicator s f ≤ f 
goal: s.mulIndicator f ≤ f

full_name: Set.indicator_apply_le
code: @[to_additive]
lemma mulIndicator_apply_le {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a) :
    mulIndicator s f a ≤ g a 
goal: s.indicator f a ≤ g a

full_name: Set.mulIndicator_apply_le
code: @[to_additive]
lemma mulIndicator_apply_le {a : α} {s : Set α} {f g : α → M} (hfg : a ∈ s → f a ≤ g a) :
    mulIndicator s f a ≤ g a 
goal: s.mulIndicator f a ≤ g a

full_name: Set.mulIndicator_le
code: @[to_additive]
lemma mulIndicator_le {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
    mulIndicator s f ≤ g 
goal: s.mulIndicator f ≤ g

full_name: Set.indicator_le
code: @[to_additive]
lemma mulIndicator_le {s : Set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
    mulIndicator s f ≤ g 
goal: s.indicator f ≤ g

full_name: Set.abs_indicator_symmDiff
code: @[to_additive]
lemma mabs_mulIndicator_symmDiff (s t : Set α) (f : α → M) (x : α) :
    |mulIndicator (s ∆ t) f x|ₘ = |mulIndicator s f x / mulIndicator t f x|ₘ 
goal: |(symmDiff s t).indicator f x| = |s.indicator f x - t.indicator f x|

full_name: Set.mabs_mulIndicator_symmDiff
code: @[to_additive]
lemma mabs_mulIndicator_symmDiff (s t : Set α) (f : α → M) (x : α) :
    |mulIndicator (s ∆ t) f x|ₘ = |mulIndicator s f x / mulIndicator t f x|ₘ 
goal: mabs ((symmDiff s t).mulIndicator f x) = mabs (s.mulIndicator f x / t.mulIndicator f x)