File: Mathlib/Algebra/GroupWithZero/Indicator.lean

full_name: Set.indicator_mul
code: lemma indicator_mul (s : Set ι) (f g : ι → M₀) :
    indicator s (fun i ↦ f i * g i) = fun i ↦ indicator s f i * indicator s g i 
goal: (s.indicator fun i => f i * g i) = fun i => s.indicator f i * s.indicator g i

full_name: Set.indicator_mul_left
code: lemma indicator_mul_left (s : Set ι) (f g : ι → M₀) :
    indicator s (fun j ↦ f j * g j) i = indicator s f i * g i 
goal: s.indicator (fun j => f j * g j) i = s.indicator f i * g i

full_name: Set.indicator_mul_right
code: lemma indicator_mul_right (s : Set ι) (f g : ι → M₀) :
    indicator s (fun j ↦ f j * g j) i = f i * indicator s g i 
goal: s.indicator (fun j => f j * g j) i = f i * s.indicator g i

full_name: Set.inter_indicator_mul
code: lemma inter_indicator_mul (f g : ι → M₀) (i : ι) :
    (s ∩ t).indicator (fun j ↦ f j * g j) i = s.indicator f i * t.indicator g i 
goal: (s ∩ t).indicator (fun j => f j * g j) i = s.indicator f i * t.indicator g i

full_name: Set.inter_indicator_one
code: lemma inter_indicator_one : (s ∩ t).indicator (1 : ι → M₀) = s.indicator 1 * t.indicator 1 
goal: (s ∩ t).indicator 1 = s.indicator 1 * t.indicator 1

full_name: Set.indicator_prod_one
code: lemma indicator_prod_one {t : Set κ} {j : κ} :
    (s ×ˢ t).indicator (1 : ι × κ → M₀) (i, j) = s.indicator 1 i * t.indicator 1 j 
goal: (s ×ˢ t).indicator 1 (i, j) = s.indicator 1 i * t.indicator 1 j

full_name: Set.indicator_eq_zero_iff_not_mem
code: lemma indicator_eq_zero_iff_not_mem : indicator s 1 i = (0 : M₀) ↔ i ∉ s 
goal: s.indicator 1 i = 0 ↔ i ∉ s

full_name: Set.indicator_eq_one_iff_mem
code: lemma indicator_eq_one_iff_mem : indicator s 1 i = (1 : M₀) ↔ i ∈ s 
goal: s.indicator 1 i = 1 ↔ i ∈ s

full_name: Set.indicator_one_inj
code: lemma indicator_one_inj (h : indicator s (1 : ι → M₀) = indicator t 1) : s = t 
goal: s = t

full_name: Function.support_one
code: @[simp] lemma support_one : support (1 : ι → R) = univ 
goal: Function.support 1 = Set.univ

full_name: Function.mulSupport_zero
code: @[simp] lemma mulSupport_zero : mulSupport (0 : ι → R) = univ 
goal: Function.mulSupport 0 = Set.univ

full_name: Function.support_mul_subset_left
code: lemma support_mul_subset_left (f g : ι → M₀) : support (fun x ↦ f x * g x) ⊆ support f 
goal: (Function.support fun x => f x * g x) ⊆ Function.support f

full_name: Function.support_mul_subset_right
code: lemma support_mul_subset_right (f g : ι → M₀) : support (fun x ↦ f x * g x) ⊆ support g 
goal: (Function.support fun x => f x * g x) ⊆ Function.support g

full_name: Function.support_mul
code: @[simp] lemma support_mul (f g : ι → M₀) : support (fun x ↦ f x * g x) = support f ∩ support g 
goal: (Function.support fun x => f x * g x) = Function.support f ∩ Function.support g

full_name: Function.support_mul'
code: @[simp] lemma support_mul' (f g : ι → M₀) : support (f * g) = support f ∩ support g 
goal: Function.support (f * g) = Function.support f ∩ Function.support g

full_name: Function.support_pow
code: @[simp] lemma support_pow (f : ι → M₀) (hn : n ≠ 0) : support (fun a ↦ f a ^ n) = support f 
goal: (Function.support fun a => f a ^ n) = Function.support f

full_name: Function.support_pow'
code: @[simp] lemma support_pow' (f : ι → M₀) (hn : n ≠ 0) : support (f ^ n) = support f 
goal: Function.support (f ^ n) = Function.support f

full_name: Function.support_inv
code: @[simp] lemma support_inv (f : ι → G₀) : support (fun a ↦ (f a)⁻¹) = support f 
goal: (Function.support fun a => (f a)⁻¹) = Function.support f

full_name: Function.support_inv'
code: @[simp] lemma support_inv' (f : ι → G₀) : support f⁻¹ = support f 
goal: Function.support f⁻¹ = Function.support f

full_name: Function.support_div
code: @[simp] lemma support_div (f g : ι → G₀) : support (fun a ↦ f a / g a) = support f ∩ support g 
goal: (Function.support fun a => f a / g a) = Function.support f ∩ Function.support g

full_name: Function.support_div'
code: @[simp] lemma support_div' (f g : ι → G₀) : support (f / g) = support f ∩ support g 
goal: Function.support (f / g) = Function.support f ∩ Function.support g

full_name: Function.mulSupport_one_add
code: lemma mulSupport_one_add [AddLeftCancelMonoid R] (f : ι → R) :
    mulSupport (fun x ↦ 1 + f x) = support f 
goal: (Function.mulSupport fun x => 1 + f x) = Function.support f

full_name: Function.mulSupport_one_add'
code: lemma mulSupport_one_add' [AddLeftCancelMonoid R] (f : ι → R) : mulSupport (1 + f) = support f 
goal: Function.mulSupport (1 + f) = Function.support f

full_name: Function.mulSupport_add_one
code: lemma mulSupport_add_one [AddRightCancelMonoid R] (f : ι → R) :
    mulSupport (fun x ↦ f x + 1) = support f 
goal: (Function.mulSupport fun x => f x + 1) = Function.support f

full_name: Function.mulSupport_add_one'
code: lemma mulSupport_add_one' [AddRightCancelMonoid R] (f : ι → R) : mulSupport (f + 1) = support f 
goal: Function.mulSupport (f + 1) = Function.support f

full_name: Function.mulSupport_one_sub'
code: lemma mulSupport_one_sub' [AddGroup R] (f : ι → R) : mulSupport (1 - f) = support f 
goal: Function.mulSupport (1 - f) = Function.support f

full_name: Function.mulSupport_one_sub
code: lemma mulSupport_one_sub [AddGroup R] (f : ι → R) :
    mulSupport (fun x ↦ 1 - f x) = support f 
goal: (Function.mulSupport fun x => 1 - f x) = Function.support f