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group_theory.submonoid.membership

Submonoids: membership criteria

In this file we prove various facts about membership in a submonoid:

list_prod_mem, multiset_prod_mem, prod_mem: if each element of a collection belongs to a multiplicative submonoid, then so does their product;
list_sum_mem, multiset_sum_mem, sum_mem: if each element of a collection belongs to an additive submonoid, then so does their sum;
pow_mem, nsmul_mem: if x ∈ S where S is a multiplicative (resp., additive) submonoid and n is a natural number, then x^n (resp., n •ℕ x) belongs to S;
mem_supr_of_directed, coe_supr_of_directed, mem_Sup_of_directed_on, coe_Sup_of_directed_on: the supremum of a directed collection of submonoid is their union.
sup_eq_range, mem_sup: supremum of two submonoids S, T of a commutative monoid is the set of products;
closure_singleton_eq, mem_closure_singleton: the multiplicative (resp., additive) closure of {x} consists of powers (resp., natural multiples) of x.

Tags

submonoid, submonoids

theorem submonoid.list_prod_mem {M : Type u_1} [monoid M] (S : submonoid M) {l : list M} (a : ∀ (x : M), x ∈ l → x ∈ S) :

Product of a list of elements in a submonoid is in the submonoid.

theorem add_submonoid.list_sum_mem {M : Type u_1} [add_monoid M] (S : add_submonoid M) {l : list M} (a : ∀ (x : M), x ∈ l → x ∈ S) :

Sum of a list of elements in an add_submonoid is in the add_submonoid.

theorem submonoid.multiset_prod_mem {M : Type u_1} [comm_monoid M] (S : submonoid M) (m : multiset M) (a : ∀ (a : M), a ∈ m → a ∈ S) :

Product of a multiset of elements in a submonoid of a comm_monoid is in the submonoid.

theorem add_submonoid.multiset_sum_mem {M : Type u_1} [add_comm_monoid M] (S : add_submonoid M) (m : multiset M) (a : ∀ (a : M), a ∈ m → a ∈ S) :

Sum of a multiset of elements in an add_submonoid of an add_comm_monoid is in the add_submonoid.

theorem add_submonoid.sum_mem {M : Type u_1} [add_comm_monoid M] (S : add_submonoid M) {ι : Type u_2} {t : finset ι} {f : ι → M} (h : ∀ (c : ι), c ∈ t → f c ∈ S) :

∑ (c : ι) in t, f c ∈ S

Sum of elements in an add_submonoid of an add_comm_monoid indexed by a finset is in the add_submonoid.

theorem submonoid.prod_mem {M : Type u_1} [comm_monoid M] (S : submonoid M) {ι : Type u_2} {t : finset ι} {f : ι → M} (h : ∀ (c : ι), c ∈ t → f c ∈ S) :

∏ (c : ι) in t, f c ∈ S

Product of elements of a submonoid of a comm_monoid indexed by a finset is in the submonoid.

theorem submonoid.pow_mem {M : Type u_1} [monoid M] (S : submonoid M) {x : M} (hx : x ∈ S) (n : ℕ) :

x ^ n ∈ S

theorem add_submonoid.mem_supr_of_directed {M : Type u_1} [add_monoid M] {ι : Sort u_2} [hι : nonempty ι] {S : ι → add_submonoid M} (hS : directed has_le.le S) {x : M} :

(x ∈ ⨆ (i : ι), S i) ↔ ∃ (i : ι), x ∈ S i

theorem submonoid.mem_supr_of_directed {M : Type u_1} [monoid M] {ι : Sort u_2} [hι : nonempty ι] {S : ι → submonoid M} (hS : directed has_le.le S) {x : M} :

(x ∈ ⨆ (i : ι), S i) ↔ ∃ (i : ι), x ∈ S i

theorem add_submonoid.coe_supr_of_directed {M : Type u_1} [add_monoid M] {ι : Sort u_2} [nonempty ι] {S : ι → add_submonoid M} (hS : directed has_le.le S) :

(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem submonoid.coe_supr_of_directed {M : Type u_1} [monoid M] {ι : Sort u_2} [nonempty ι] {S : ι → submonoid M} (hS : directed has_le.le S) :

(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem submonoid.mem_Sup_of_directed_on {M : Type u_1} [monoid M] {S : set (submonoid M)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : M} :

x ∈ Sup S ↔ ∃ (s : submonoid M) (H : s ∈ S), x ∈ s

theorem add_submonoid.mem_Sup_of_directed_on {M : Type u_1} [add_monoid M] {S : set (add_submonoid M)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : M} :

x ∈ Sup S ↔ ∃ (s : add_submonoid M) (H : s ∈ S), x ∈ s

theorem add_submonoid.coe_Sup_of_directed_on {M : Type u_1} [add_monoid M] {S : set (add_submonoid M)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :

↑(Sup S) = ⋃ (s : add_submonoid M) (H : s ∈ S), ↑s

theorem submonoid.coe_Sup_of_directed_on {M : Type u_1} [monoid M] {S : set (submonoid M)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :

↑(Sup S) = ⋃ (s : submonoid M) (H : s ∈ S), ↑s

theorem submonoid.mem_sup_left {M : Type u_1} [monoid M] {S T : submonoid M} {x : M} (a : x ∈ S) :

x ∈ S ⊔ T

theorem add_submonoid.mem_sup_left {M : Type u_1} [add_monoid M] {S T : add_submonoid M} {x : M} (a : x ∈ S) :

x ∈ S ⊔ T

theorem submonoid.mem_sup_right {M : Type u_1} [monoid M] {S T : submonoid M} {x : M} (a : x ∈ T) :

x ∈ S ⊔ T

theorem add_submonoid.mem_sup_right {M : Type u_1} [add_monoid M] {S T : add_submonoid M} {x : M} (a : x ∈ T) :

x ∈ S ⊔ T

theorem submonoid.mem_supr_of_mem {M : Type u_1} [monoid M] {ι : Type u_2} {S : ι → submonoid M} (i : ι) {x : M} (a : x ∈ S i) :

theorem add_submonoid.mem_supr_of_mem {M : Type u_1} [add_monoid M] {ι : Type u_2} {S : ι → add_submonoid M} (i : ι) {x : M} (a : x ∈ S i) :

theorem add_submonoid.mem_Sup_of_mem {M : Type u_1} [add_monoid M] {S : set (add_submonoid M)} {s : add_submonoid M} (hs : s ∈ S) {x : M} (a : x ∈ s) :

theorem submonoid.mem_Sup_of_mem {M : Type u_1} [monoid M] {S : set (submonoid M)} {s : submonoid M} (hs : s ∈ S) {x : M} (a : x ∈ s) :

theorem free_monoid.closure_range_of {α : Type u_3} :

submonoid.closure (set.range free_monoid.of) = ⊤

theorem free_add_monoid.closure_range_of {α : Type u_3} :

add_submonoid.closure (set.range free_add_monoid.of) = ⊤

theorem submonoid.closure_singleton_eq {M : Type u_1} [monoid M] (x : M) :

submonoid.closure {x} = (⇑(powers_hom M) x).mrange

theorem submonoid.mem_closure_singleton {M : Type u_1} [monoid M] {x y : M} :

y ∈ submonoid.closure {x} ↔ ∃ (n : ℕ), x ^ n = y

The submonoid generated by an element of a monoid equals the set of natural number powers of the element.

theorem submonoid.mem_closure_singleton_self {M : Type u_1} [monoid M] {y : M} :

y ∈ submonoid.closure {y}

theorem submonoid.closure_eq_mrange {M : Type u_1} [monoid M] (s : set M) :

submonoid.closure s = (⇑free_monoid.lift coe).mrange

theorem add_submonoid.closure_eq_mrange {M : Type u_1} [add_monoid M] (s : set M) :

add_submonoid.closure s = (⇑free_add_monoid.lift coe).mrange

theorem submonoid.exists_list_of_mem_closure {M : Type u_1} [monoid M] {s : set M} {x : M} (hx : x ∈ submonoid.closure s) :

∃ (l : list M) (hl : ∀ (y : M), y ∈ l → y ∈ s), l.prod = x

theorem add_submonoid.exists_list_of_mem_closure {M : Type u_1} [add_monoid M] {s : set M} {x : M} (hx : x ∈ add_submonoid.closure s) :

∃ (l : list M) (hl : ∀ (y : M), y ∈ l → y ∈ s), l.sum = x

def submonoid.powers {N : Type u_3} [monoid N] (n : N) :

The submonoid generated by an element.

Equations

submonoid.powers n = (⇑(powers_hom N) n).mrange.copy (set.range (pow n)) _

@[simp]

theorem submonoid.mem_powers {N : Type u_3} [monoid N] (n : N) :

n ∈ submonoid.powers n

theorem submonoid.powers_eq_closure {N : Type u_3} [monoid N] (n : N) :

submonoid.powers n = submonoid.closure {n}

theorem submonoid.powers_subset {N : Type u_3} [monoid N] {n : N} {P : submonoid N} (h : n ∈ P) :

submonoid.powers n ≤ P

theorem add_submonoid.sup_eq_range {N : Type u_3} [add_comm_monoid N] (s t : add_submonoid N) :

s ⊔ t = (s.subtype.coprod t.subtype).mrange

theorem submonoid.sup_eq_range {N : Type u_3} [comm_monoid N] (s t : submonoid N) :

s ⊔ t = (s.subtype.coprod t.subtype).mrange

theorem submonoid.mem_sup {N : Type u_3} [comm_monoid N] {s t : submonoid N} {x : N} :

x ∈ s ⊔ t ↔ ∃ (y : N) (H : y ∈ s) (z : N) (H : z ∈ t), y * z = x

theorem add_submonoid.mem_sup {N : Type u_3} [add_comm_monoid N] {s t : add_submonoid N} {x : N} :

x ∈ s ⊔ t ↔ ∃ (y : N) (H : y ∈ s) (z : N) (H : z ∈ t), y + z = x

theorem add_submonoid.nsmul_mem {A : Type u_2} [add_monoid A] (S : add_submonoid A) {x : A} (hx : x ∈ S) (n : ℕ) :

n •ℕ x ∈ S

theorem add_submonoid.closure_singleton_eq {A : Type u_2} [add_monoid A] (x : A) :

add_submonoid.closure {x} = (⇑(multiples_hom A) x).mrange

theorem add_submonoid.mem_closure_singleton {A : Type u_2} [add_monoid A] {x y : A} :

y ∈ add_submonoid.closure {x} ↔ ∃ (n : ℕ), n •ℕ x = y

The add_submonoid generated by an element of an add_monoid equals the set of natural number multiples of the element.

def add_submonoid.multiples {A : Type u_2} [add_monoid A] (x : A) :

add_submonoid A

The additive submonoid generated by an element.

Equations

add_submonoid.multiples x = (⇑(multiples_hom A) x).mrange.copy (set.range (λ (i : ℕ), i •ℕ x)) _

@[simp]

theorem add_submonoid.mem_multiples {A : Type u_2} [add_monoid A] (x : A) :

x ∈ add_submonoid.multiples x

theorem add_submonoid.multiples_eq_closure {A : Type u_2} [add_monoid A] (x : A) :

add_submonoid.multiples x = add_submonoid.closure {x}

theorem add_submonoid.multiples_subset {A : Type u_2} [add_monoid A] {x : A} {P : add_submonoid A} (h : x ∈ P) :

add_submonoid.multiples x ≤ P