Support of a function

In this file we define Function.support f = {x | f x ≠ 0} and prove its basic properties. We also define Function.mulSupport f = {x | f x ≠ 1}.

def Function.support {α : Type u_1} {A : Type u_3} [Zero A] (f : α → A) : Set α

support of a function is the set of points x such that f x ≠ 0.

Equations

• Function.support f = {x : α | f x ≠ 0}

def Function.mulSupport {α : Type u_1} {M : Type u_5} [One M] (f : α → M) : Set α

mulSupport of a function is the set of points x such that f x ≠ 1.

Equations

• Function.mulSupport f = {x : α | f x ≠ 1}

theorem Function.support_eq_preimage {α : Type u_1} {M : Type u_5} [Zero M] (f : α → M) : Function.support f = f ⁻¹' {0}ᶜ

theorem Function.mulSupport_eq_preimage {α : Type u_1} {M : Type u_5} [One M] (f : α → M) : Function.mulSupport f = f ⁻¹' {1}ᶜ

theorem Function.nmem_support {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {x : α} : x ∉ Function.support f ↔ f x = 0

theorem Function.nmem_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {x : α} : x ∉ Function.mulSupport f ↔ f x = 1

theorem Function.compl_support {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} : (Function.support f)ᶜ = {x : α | f x = 0}

theorem Function.compl_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : α → M} : (Function.mulSupport f)ᶜ = {x : α | f x = 1}

@[simp]

theorem Function.mem_support {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {x : α} : x ∈ Function.support f ↔ f x ≠ 0

@[simp]

theorem Function.mem_mulSupport {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {x : α} : x ∈ Function.mulSupport f ↔ f x ≠ 1

@[simp]

theorem Function.support_subset_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {s : Set α} : Function.support f ⊆ s ↔ ∀ (x : α), f x ≠ 0 → x ∈ s

@[simp]

theorem Function.mulSupport_subset_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {s : Set α} : Function.mulSupport f ⊆ s ↔ ∀ (x : α), f x ≠ 1 → x ∈ s

theorem Function.support_subset_iff' {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {s : Set α} : Function.support f ⊆ s ↔ ∀ x ∉ s, f x = 0

theorem Function.mulSupport_subset_iff' {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {s : Set α} : Function.mulSupport f ⊆ s ↔ ∀ x ∉ s, f x = 1

theorem Function.support_eq_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {s : Set α} : Function.support f = s ↔ (∀ x ∈ s, f x ≠ 0) ∧ ∀ x ∉ s, f x = 0

theorem Function.mulSupport_eq_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {s : Set α} : Function.mulSupport f = s ↔ (∀ x ∈ s, f x ≠ 1) ∧ ∀ x ∉ s, f x = 1

theorem Function.support_extend_zero_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero N] {f : α → M} {g : α → N} : Function.support (Function.extend f g 0) ⊆ f '' Function.support g

theorem Function.mulSupport_extend_one_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One N] {f : α → M} {g : α → N} : Function.mulSupport (Function.extend f g 1) ⊆ f '' Function.mulSupport g

theorem Function.support_extend_zero {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero N] {f : α → M} {g : α → N} (hf : Function.Injective f) : Function.support (Function.extend f g 0) = f '' Function.support g

theorem Function.mulSupport_extend_one {α : Type u_1} {M : Type u_5} {N : Type u_6} [One N] {f : α → M} {g : α → N} (hf : Function.Injective f) : Function.mulSupport (Function.extend f g 1) = f '' Function.mulSupport g

theorem Function.support_disjoint_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {s : Set α} : Disjoint (Function.support f) s ↔ Set.EqOn f 0 s

theorem Function.mulSupport_disjoint_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {s : Set α} : Disjoint (Function.mulSupport f) s ↔ Set.EqOn f 1 s

theorem Function.disjoint_support_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {s : Set α} : Disjoint s (Function.support f) ↔ Set.EqOn f 0 s

theorem Function.disjoint_mulSupport_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {s : Set α} : Disjoint s (Function.mulSupport f) ↔ Set.EqOn f 1 s

@[simp]

theorem Function.support_eq_empty_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} : Function.support f = ∅ ↔ f = 0

@[simp]

theorem Function.mulSupport_eq_empty_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} : Function.mulSupport f = ∅ ↔ f = 1

@[simp]

theorem Function.support_nonempty_iff {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} : Set.Nonempty (Function.support f) ↔ f ≠ 0

@[simp]

theorem Function.mulSupport_nonempty_iff {α : Type u_1} {M : Type u_5} [One M] {f : α → M} : Set.Nonempty (Function.mulSupport f) ↔ f ≠ 1

theorem Function.range_subset_insert_image_support {α : Type u_1} {M : Type u_5} [Zero M] (f : α → M) : Set.range f ⊆ insert 0 (f '' Function.support f)

theorem Function.range_subset_insert_image_mulSupport {α : Type u_1} {M : Type u_5} [One M] (f : α → M) : Set.range f ⊆ insert 1 (f '' Function.mulSupport f)

theorem Function.range_eq_image_or_of_support_subset {α : Type u_1} {M : Type u_5} [Zero M] {f : α → M} {k : Set α} (h : Function.support f ⊆ k) : Set.range f = f '' k ∨ Set.range f = insert 0 (f '' k)

theorem Function.range_eq_image_or_of_mulSupport_subset {α : Type u_1} {M : Type u_5} [One M] {f : α → M} {k : Set α} (h : Function.mulSupport f ⊆ k) : Set.range f = f '' k ∨ Set.range f = insert 1 (f '' k)

@[simp]

theorem Function.support_zero' {α : Type u_1} {M : Type u_5} [Zero M] : Function.support 0 = ∅

@[simp]

theorem Function.mulSupport_one' {α : Type u_1} {M : Type u_5} [One M] : Function.mulSupport 1 = ∅

@[simp]

theorem Function.support_zero {α : Type u_1} {M : Type u_5} [Zero M] : (Function.support fun (x : α) => 0) = ∅

@[simp]

theorem Function.mulSupport_one {α : Type u_1} {M : Type u_5} [One M] : (Function.mulSupport fun (x : α) => 1) = ∅

theorem Function.support_const {α : Type u_1} {M : Type u_5} [Zero M] {c : M} (hc : c ≠ 0) : (Function.support fun (x : α) => c) = Set.univ

theorem Function.mulSupport_const {α : Type u_1} {M : Type u_5} [One M] {c : M} (hc : c ≠ 1) : (Function.mulSupport fun (x : α) => c) = Set.univ

theorem Function.support_binop_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} {P : Type u_7} [Zero M] [Zero N] [Zero P] (op : M → N → P) (op1 : op 0 0 = 0) (f : α → M) (g : α → N) : (Function.support fun (x : α) => op (f x) (g x)) ⊆ Function.support f ∪ Function.support g

theorem Function.mulSupport_binop_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} {P : Type u_7} [One M] [One N] [One P] (op : M → N → P) (op1 : op 1 1 = 1) (f : α → M) (g : α → N) : (Function.mulSupport fun (x : α) => op (f x) (g x)) ⊆ Function.mulSupport f ∪ Function.mulSupport g

theorem Function.support_comp_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : M → N} (hg : g 0 = 0) (f : α → M) : Function.support (g ∘ f) ⊆ Function.support f

theorem Function.mulSupport_comp_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : M → N} (hg : g 1 = 1) (f : α → M) : Function.mulSupport (g ∘ f) ⊆ Function.mulSupport f

theorem Function.support_subset_comp {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : M → N} (hg : ∀ {x : M}, g x = 0 → x = 0) (f : α → M) : Function.support f ⊆ Function.support (g ∘ f)

theorem Function.mulSupport_subset_comp {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : M → N} (hg : ∀ {x : M}, g x = 1 → x = 1) (f : α → M) : Function.mulSupport f ⊆ Function.mulSupport (g ∘ f)

theorem Function.support_comp_eq {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (g : M → N) (hg : ∀ {x : M}, g x = 0 ↔ x = 0) (f : α → M) : Function.support (g ∘ f) = Function.support f

theorem Function.mulSupport_comp_eq {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (g : M → N) (hg : ∀ {x : M}, g x = 1 ↔ x = 1) (f : α → M) : Function.mulSupport (g ∘ f) = Function.mulSupport f

theorem Function.support_comp_eq_of_range_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] {g : M → N} {f : α → M} (hg : ∀ {x : M}, x ∈ Set.range f → (g x = 0 ↔ x = 0)) : Function.support (g ∘ f) = Function.support f

theorem Function.mulSupport_comp_eq_of_range_subset {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] {g : M → N} {f : α → M} (hg : ∀ {x : M}, x ∈ Set.range f → (g x = 1 ↔ x = 1)) : Function.mulSupport (g ∘ f) = Function.mulSupport f

theorem Function.support_comp_eq_preimage {α : Type u_1} {β : Type u_2} {M : Type u_5} [Zero M] (g : β → M) (f : α → β) : Function.support (g ∘ f) = f ⁻¹' Function.support g

theorem Function.mulSupport_comp_eq_preimage {α : Type u_1} {β : Type u_2} {M : Type u_5} [One M] (g : β → M) (f : α → β) : Function.mulSupport (g ∘ f) = f ⁻¹' Function.mulSupport g

theorem Function.support_prod_mk {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (f : α → M) (g : α → N) : (Function.support fun (x : α) => (f x, g x)) = Function.support f ∪ Function.support g

theorem Function.mulSupport_prod_mk {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (f : α → M) (g : α → N) : (Function.mulSupport fun (x : α) => (f x, g x)) = Function.mulSupport f ∪ Function.mulSupport g

theorem Function.support_prod_mk' {α : Type u_1} {M : Type u_5} {N : Type u_6} [Zero M] [Zero N] (f : α → M × N) : Function.support f = (Function.support fun (x : α) => (f x).1) ∪ Function.support fun (x : α) => (f x).2

theorem Function.mulSupport_prod_mk' {α : Type u_1} {M : Type u_5} {N : Type u_6} [One M] [One N] (f : α → M × N) : Function.mulSupport f = (Function.mulSupport fun (x : α) => (f x).1) ∪ Function.mulSupport fun (x : α) => (f x).2

theorem Function.support_along_fiber_subset {α : Type u_1} {β : Type u_2} {M : Type u_5} [Zero M] (f : α × β → M) (a : α) : (Function.support fun (b : β) => f (a, b)) ⊆ Prod.snd '' Function.support f

theorem Function.mulSupport_along_fiber_subset {α : Type u_1} {β : Type u_2} {M : Type u_5} [One M] (f : α × β → M) (a : α) : (Function.mulSupport fun (b : β) => f (a, b)) ⊆ Prod.snd '' Function.mulSupport f

theorem Function.support_add {α : Type u_1} {M : Type u_5} [AddZeroClass M] (f : α → M) (g : α → M) : (Function.support fun (x : α) => f x + g x) ⊆ Function.support f ∪ Function.support g

theorem Function.mulSupport_mul {α : Type u_1} {M : Type u_5} [MulOneClass M] (f : α → M) (g : α → M) : (Function.mulSupport fun (x : α) => f x * g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g

theorem Function.support_nsmul {α : Type u_1} {M : Type u_5} [AddMonoid M] (f : α → M) (n : ℕ) : (Function.support fun (x : α) => n • f x) ⊆ Function.support f

theorem Function.mulSupport_pow {α : Type u_1} {M : Type u_5} [Monoid M] (f : α → M) (n : ℕ) : (Function.mulSupport fun (x : α) => f x ^ n) ⊆ Function.mulSupport f

@[simp]

theorem Function.support_neg {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : α → G) : (Function.support fun (x : α) => -f x) = Function.support f

@[simp]

theorem Function.mulSupport_inv {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : α → G) : (Function.mulSupport fun (x : α) => (f x)⁻¹) = Function.mulSupport f

@[simp]

theorem Function.support_neg' {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : α → G) : Function.support (-f) = Function.support f

@[simp]

theorem Function.mulSupport_inv' {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : α → G) : Function.mulSupport f⁻¹ = Function.mulSupport f

theorem Function.support_add_neg {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : α → G) (g : α → G) : (Function.support fun (x : α) => f x + -g x) ⊆ Function.support f ∪ Function.support g

theorem Function.mulSupport_mul_inv {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : α → G) (g : α → G) : (Function.mulSupport fun (x : α) => f x * (g x)⁻¹) ⊆ Function.mulSupport f ∪ Function.mulSupport g

theorem Function.support_sub {α : Type u_1} {G : Type u_10} [SubtractionMonoid G] (f : α → G) (g : α → G) : (Function.support fun (x : α) => f x - g x) ⊆ Function.support f ∪ Function.support g

theorem Function.mulSupport_div {α : Type u_1} {G : Type u_10} [DivisionMonoid G] (f : α → G) (g : α → G) : (Function.mulSupport fun (x : α) => f x / g x) ⊆ Function.mulSupport f ∪ Function.mulSupport g

@[simp]

theorem Function.support_one {α : Type u_1} (R : Type u_8) [Zero R] [One R] [NeZero 1] : Function.support 1 = Set.univ

@[simp]

theorem Function.mulSupport_zero {α : Type u_1} (R : Type u_8) [Zero R] [One R] [NeZero 1] : Function.mulSupport 0 = Set.univ

theorem Function.support_mul_subset_left {α : Type u_1} {M : Type u_5} [MulZeroClass M] (f : α → M) (g : α → M) : (Function.support fun (x : α) => f x * g x) ⊆ Function.support f

theorem Function.support_mul_subset_right {α : Type u_1} {M : Type u_5} [MulZeroClass M] (f : α → M) (g : α → M) : (Function.support fun (x : α) => f x * g x) ⊆ Function.support g

@[simp]

theorem Function.support_mul {α : Type u_1} {M : Type u_5} [MulZeroClass M] [NoZeroDivisors M] (f : α → M) (g : α → M) : (Function.support fun (x : α) => f x * g x) = Function.support f ∩ Function.support g

@[simp]

theorem Function.support_mul' {α : Type u_1} {M : Type u_5} [MulZeroClass M] [NoZeroDivisors M] (f : α → M) (g : α → M) : Function.support (f * g) = Function.support f ∩ Function.support g

@[simp]

theorem Function.support_pow {α : Type u_1} {M : Type u_5} [MonoidWithZero M] [NoZeroDivisors M] {n : ℕ} (f : α → M) (hn : n ≠ 0) : (Function.support fun (a : α) => f a ^ n) = Function.support f

@[simp]

theorem Function.support_pow' {α : Type u_1} {M : Type u_5} [MonoidWithZero M] [NoZeroDivisors M] {n : ℕ} (f : α → M) (hn : n ≠ 0) : Function.support (f ^ n) = Function.support f

@[simp]

theorem Function.support_inv {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : α → G₀) : (Function.support fun (a : α) => (f a)⁻¹) = Function.support f

@[simp]

theorem Function.support_inv' {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : α → G₀) : Function.support f⁻¹ = Function.support f

@[simp]

theorem Function.support_div {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : α → G₀) (g : α → G₀) : (Function.support fun (a : α) => f a / g a) = Function.support f ∩ Function.support g

@[simp]

theorem Function.support_div' {α : Type u_1} {G₀ : Type u_12} [GroupWithZero G₀] (f : α → G₀) (g : α → G₀) : Function.support (f / g) = Function.support f ∩ Function.support g

theorem Function.mulSupport_one_add {α : Type u_1} {R : Type u_8} [One R] [AddLeftCancelMonoid R] (f : α → R) : (Function.mulSupport fun (x : α) => 1 + f x) = Function.support f

theorem Function.mulSupport_one_add' {α : Type u_1} {R : Type u_8} [One R] [AddLeftCancelMonoid R] (f : α → R) : Function.mulSupport (1 + f) = Function.support f

theorem Function.mulSupport_add_one {α : Type u_1} {R : Type u_8} [One R] [AddRightCancelMonoid R] (f : α → R) : (Function.mulSupport fun (x : α) => f x + 1) = Function.support f

theorem Function.mulSupport_add_one' {α : Type u_1} {R : Type u_8} [One R] [AddRightCancelMonoid R] (f : α → R) : Function.mulSupport (f + 1) = Function.support f

theorem Function.mulSupport_one_sub' {α : Type u_1} {R : Type u_8} [One R] [AddGroup R] (f : α → R) : Function.mulSupport (1 - f) = Function.support f

theorem Function.mulSupport_one_sub {α : Type u_1} {R : Type u_8} [One R] [AddGroup R] (f : α → R) : (Function.mulSupport fun (x : α) => 1 - f x) = Function.support f

theorem Set.image_inter_support_eq {α : Type u_1} {β : Type u_2} {M : Type u_3} [Zero M] {f : α → M} {s : Set β} {g : β → α} : g '' s ∩ Function.support f = g '' (s ∩ Function.support (f ∘ g))

theorem Set.image_inter_mulSupport_eq {α : Type u_1} {β : Type u_2} {M : Type u_3} [One M] {f : α → M} {s : Set β} {g : β → α} : g '' s ∩ Function.mulSupport f = g '' (s ∩ Function.mulSupport (f ∘ g))

theorem Pi.support_single_subset {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} : Function.support (Pi.single a b) ⊆ {a}

theorem Pi.mulSupport_mulSingle_subset {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} : Function.mulSupport (Pi.mulSingle a b) ⊆ {a}

theorem Pi.support_single_zero {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} : Function.support (Pi.single a 0) = ∅

theorem Pi.mulSupport_mulSingle_one {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} : Function.mulSupport (Pi.mulSingle a 1) = ∅

@[simp]

theorem Pi.support_single_of_ne {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} (h : b ≠ 0) : Function.support (Pi.single a b) = {a}

@[simp]

theorem Pi.mulSupport_mulSingle_of_ne {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} (h : b ≠ 1) : Function.mulSupport (Pi.mulSingle a b) = {a}

theorem Pi.support_single {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {a : A} {b : B} [DecidableEq B] : Function.support (Pi.single a b) = if b = 0 then ∅ else {a}

theorem Pi.mulSupport_mulSingle {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {a : A} {b : B} [DecidableEq B] : Function.mulSupport (Pi.mulSingle a b) = if b = 1 then ∅ else {a}

theorem Pi.support_single_disjoint {A : Type u_1} {B : Type u_2} [DecidableEq A] [Zero B] {b : B} {b' : B} (hb : b ≠ 0) (hb' : b' ≠ 0) {i : A} {j : A} : Disjoint (Function.support (Pi.single i b)) (Function.support (Pi.single j b')) ↔ i ≠ j

theorem Pi.mulSupport_mulSingle_disjoint {A : Type u_1} {B : Type u_2} [DecidableEq A] [One B] {b : B} {b' : B} (hb : b ≠ 1) (hb' : b' ≠ 1) {i : A} {j : A} : Disjoint (Function.mulSupport (Pi.mulSingle i b)) (Function.mulSupport (Pi.mulSingle j b')) ↔ i ≠ j