ring_theory.subsemiring.basic

@[class]

structure subsemiring_class (S : Type u_1) (R : out_param (Type u)) [non_assoc_semiring R] [set_like S R] :

Type

to_submonoid_class : submonoid_class S R
add_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a + b ∈ s
zero_mem : ∀ (s : S), 0 ∈ s

subsemiring_class S R states that S is a type of subsets s ⊆ R that are both a multiplicative and an additive submonoid.

Instances

source

@[protected, instance]

def subsemiring_class.add_submonoid_class (S : Type u_1) (R : out_param (Type u)) [non_assoc_semiring R] [set_like S R] [h : subsemiring_class S R] :

add_submonoid_class S R

Equations

subsemiring_class.add_submonoid_class S R = {to_add_mem_class := {add_mem := _}, zero_mem := _}

source

theorem coe_nat_mem {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) (n : ℕ) :

↑n ∈ s

source

@[protected, instance]

def subsemiring_class.to_non_assoc_semiring {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) :

non_assoc_semiring ↥s

A subsemiring of a non_assoc_semiring inherits a non_assoc_semiring structure

Equations

subsemiring_class.to_non_assoc_semiring s = function.injective.non_assoc_semiring coe _ _ _ _ _ _

source

@[protected, instance]

def subsemiring_class.nontrivial {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) [nontrivial R] :

nontrivial ↥s

source

@[protected, instance]

def subsemiring_class.no_zero_divisors {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) [no_zero_divisors R] :

no_zero_divisors ↥s

source

def subsemiring_class.subtype {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) :

↥s →+* R

The natural ring hom from a subsemiring of semiring R to R.

Equations

subsemiring_class.subtype s = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

source

@[simp]

theorem subsemiring_class.coe_subtype {R : Type u} {S : Type v} [non_assoc_semiring R] [set_like S R] [hSR : subsemiring_class S R] (s : S) :

⇑(subsemiring_class.subtype s) = coe

source

@[protected, instance]

def subsemiring_class.to_semiring {S : Type v} (s : S) {R : Type u_1} [semiring R] [set_like S R] [subsemiring_class S R] :

semiring ↥s

A subsemiring of a semiring is a semiring.

Equations

subsemiring_class.to_semiring s = function.injective.semiring coe _ _ _ _ _ _ _

source

@[simp, norm_cast]

theorem subsemiring_class.coe_pow {S : Type v} (s : S) {R : Type u_1} [semiring R] [set_like S R] [subsemiring_class S R] (x : ↥s) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

source

@[protected, instance]

def subsemiring_class.to_comm_semiring {S : Type v} (s : S) {R : Type u_1} [comm_semiring R] [set_like S R] [subsemiring_class S R] :

comm_semiring ↥s

A subsemiring of a comm_semiring is a comm_semiring.

Equations

subsemiring_class.to_comm_semiring s = function.injective.comm_semiring coe _ _ _ _ _ _ _

source

@[protected, instance]

def subsemiring_class.to_ordered_semiring {S : Type v} (s : S) {R : Type u_1} [ordered_semiring R] [set_like S R] [subsemiring_class S R] :

ordered_semiring ↥s

A subsemiring of an ordered_semiring is an ordered_semiring.

Equations

subsemiring_class.to_ordered_semiring s = function.injective.ordered_semiring coe _ _ _ _ _ _ _

source

@[protected, instance]

def subsemiring_class.to_ordered_comm_semiring {S : Type v} (s : S) {R : Type u_1} [ordered_comm_semiring R] [set_like S R] [subsemiring_class S R] :

ordered_comm_semiring ↥s

A subsemiring of an ordered_comm_semiring is an ordered_comm_semiring.

Equations

subsemiring_class.to_ordered_comm_semiring s = function.injective.ordered_comm_semiring coe _ _ _ _ _ _ _

source

@[protected, instance]

def subsemiring_class.to_linear_ordered_semiring {S : Type v} (s : S) {R : Type u_1} [linear_ordered_semiring R] [set_like S R] [subsemiring_class S R] :

linear_ordered_semiring ↥s

A subsemiring of a linear_ordered_semiring is a linear_ordered_semiring.

Equations

subsemiring_class.to_linear_ordered_semiring s = function.injective.linear_ordered_semiring coe _ _ _ _ _ _ _

source

def subsemiring.to_add_submonoid {R : Type u} [non_assoc_semiring R] (self : subsemiring R) :

add_submonoid R

Reinterpret a subsemiring as an add_submonoid.

source

def subsemiring.to_submonoid {R : Type u} [non_assoc_semiring R] (self : subsemiring R) :

submonoid R

Reinterpret a subsemiring as a submonoid.

source

structure subsemiring (R : Type u) [non_assoc_semiring R] :

Type u

carrier : set R
mul_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier
one_mem' : 1 ∈ self.carrier
add_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
zero_mem' : 0 ∈ self.carrier

A subsemiring of a semiring R is a subset s that is both a multiplicative and an additive submonoid.

source

@[protected, instance]

def subsemiring.set_like {R : Type u} [non_assoc_semiring R] :

set_like (subsemiring R) R

Equations

subsemiring.set_like = {coe := subsemiring.carrier _inst_1, coe_injective' := _}

source

@[protected, instance]

def subsemiring.subsemiring_class {R : Type u} [non_assoc_semiring R] :

subsemiring_class (subsemiring R) R

Equations

subsemiring.subsemiring_class = {to_submonoid_class := {to_mul_mem_class := {mul_mem := _}, one_mem := _}, add_mem := _, zero_mem := _}

source

@[simp]

theorem subsemiring.mem_carrier {R : Type u} [non_assoc_semiring R] {s : subsemiring R} {x : R} :

x ∈ s.carrier ↔ x ∈ s

source

@[ext]

theorem subsemiring.ext {R : Type u} [non_assoc_semiring R] {S T : subsemiring R} (h : ∀ (x : R), x ∈ S ↔ x ∈ T) :

S = T

Two subsemirings are equal if they have the same elements.

source

@[protected]

def subsemiring.copy {R : Type u} [non_assoc_semiring R] (S : subsemiring R) (s : set R) (hs : s = ↑S) :

subsemiring R

Copy of a subsemiring with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

S.copy s hs = {carrier := s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[simp]

theorem subsemiring.coe_copy {R : Type u} [non_assoc_semiring R] (S : subsemiring R) (s : set R) (hs : s = ↑S) :

↑(S.copy s hs) = s

source

theorem subsemiring.copy_eq {R : Type u} [non_assoc_semiring R] (S : subsemiring R) (s : set R) (hs : s = ↑S) :

S.copy s hs = S

source

theorem subsemiring.to_submonoid_injective {R : Type u} [non_assoc_semiring R] :

function.injective subsemiring.to_submonoid

source

theorem subsemiring.to_submonoid_strict_mono {R : Type u} [non_assoc_semiring R] :

strict_mono subsemiring.to_submonoid

source

theorem subsemiring.to_submonoid_mono {R : Type u} [non_assoc_semiring R] :

monotone subsemiring.to_submonoid

source

theorem subsemiring.to_add_submonoid_injective {R : Type u} [non_assoc_semiring R] :

function.injective subsemiring.to_add_submonoid

source

theorem subsemiring.to_add_submonoid_strict_mono {R : Type u} [non_assoc_semiring R] :

strict_mono subsemiring.to_add_submonoid

source

theorem subsemiring.to_add_submonoid_mono {R : Type u} [non_assoc_semiring R] :

monotone subsemiring.to_add_submonoid

source

@[protected]

def subsemiring.mk' {R : Type u} [non_assoc_semiring R] (s : set R) (sm : submonoid R) (hm : ↑sm = s) (sa : add_submonoid R) (ha : ↑sa = s) :

subsemiring R

Construct a subsemiring R from a set s, a submonoid sm, and an additive submonoid sa such that x ∈ s ↔ x ∈ sm ↔ x ∈ sa.

Equations

subsemiring.mk' s sm hm sa ha = {carrier := s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[simp]

theorem subsemiring.coe_mk' {R : Type u} [non_assoc_semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :

↑(subsemiring.mk' s sm hm sa ha) = s

source

@[simp]

theorem subsemiring.mem_mk' {R : Type u} [non_assoc_semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) {x : R} :

x ∈ subsemiring.mk' s sm hm sa ha ↔ x ∈ s

source

@[simp]

theorem subsemiring.mk'_to_submonoid {R : Type u} [non_assoc_semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :

(subsemiring.mk' s sm hm sa ha).to_submonoid = sm

source

@[simp]

theorem subsemiring.mk'_to_add_submonoid {R : Type u} [non_assoc_semiring R] {s : set R} {sm : submonoid R} (hm : ↑sm = s) {sa : add_submonoid R} (ha : ↑sa = s) :

(subsemiring.mk' s sm hm sa ha).to_add_submonoid = sa

source

@[protected]

theorem subsemiring.one_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

1 ∈ s

A subsemiring contains the semiring's 1.

source

@[protected]

theorem subsemiring.zero_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

0 ∈ s

A subsemiring contains the semiring's 0.

source

@[protected]

theorem subsemiring.mul_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) {x y : R} :

x ∈ s → y ∈ s → x * y ∈ s

A subsemiring is closed under multiplication.

source

@[protected]

theorem subsemiring.add_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) {x y : R} :

x ∈ s → y ∈ s → x + y ∈ s

A subsemiring is closed under addition.

source

theorem subsemiring.list_prod_mem {R : Type u_1} [semiring R] (s : subsemiring R) {l : list R} :

(∀ (x : R), x ∈ l → x ∈ s) → l.prod ∈ s

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

source

@[protected]

theorem subsemiring.list_sum_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) {l : list R} :

(∀ (x : R), x ∈ l → x ∈ s) → l.sum ∈ s

Sum of a list of elements in a subsemiring is in the subsemiring.

source

@[protected]

theorem subsemiring.multiset_prod_mem {R : Type u_1} [comm_semiring R] (s : subsemiring R) (m : multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → m.prod ∈ s

Product of a multiset of elements in a subsemiring of a comm_semiring is in the subsemiring.

source

@[protected]

theorem subsemiring.multiset_sum_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) (m : multiset R) :

(∀ (a : R), a ∈ m → a ∈ s) → m.sum ∈ s

Sum of a multiset of elements in a subsemiring of a semiring is in the add_subsemiring.

source

@[protected]

theorem subsemiring.prod_mem {R : Type u_1} [comm_semiring R] (s : subsemiring R) {ι : Type u_2} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

∏ (i : ι) in t, f i ∈ s

Product of elements of a subsemiring of a comm_semiring indexed by a finset is in the subsemiring.

source

@[protected]

theorem subsemiring.sum_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) {ι : Type u_1} {t : finset ι} {f : ι → R} (h : ∀ (c : ι), c ∈ t → f c ∈ s) :

∑ (i : ι) in t, f i ∈ s

Sum of elements in an subsemiring of an semiring indexed by a finset is in the add_subsemiring.

source

@[protected, instance]

def subsemiring.to_non_assoc_semiring {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

non_assoc_semiring ↥s

A subsemiring of a non_assoc_semiring inherits a non_assoc_semiring structure

Equations

s.to_non_assoc_semiring = {add := add_comm_monoid.add s.to_add_submonoid.to_add_comm_monoid, add_assoc := _, zero := add_comm_monoid.zero s.to_add_submonoid.to_add_comm_monoid, zero_add := _, add_zero := _, nsmul := add_comm_monoid.nsmul s.to_add_submonoid.to_add_comm_monoid, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := mul_one_class.mul s.to_submonoid.to_mul_one_class, left_distrib := _, right_distrib := _, zero_mul := _, mul_zero := _, one := mul_one_class.one s.to_submonoid.to_mul_one_class, one_mul := _, mul_one := _}

source

@[simp, norm_cast]

theorem subsemiring.coe_one {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

↑1 = 1

source

@[simp, norm_cast]

theorem subsemiring.coe_zero {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

↑0 = 0

source

@[simp, norm_cast]

theorem subsemiring.coe_add {R : Type u} [non_assoc_semiring R] (s : subsemiring R) (x y : ↥s) :

↑(x + y) = ↑x + ↑y

source

@[simp, norm_cast]

theorem subsemiring.coe_mul {R : Type u} [non_assoc_semiring R] (s : subsemiring R) (x y : ↥s) :

↑x * y = (↑x) * ↑y

source

@[protected, instance]

def subsemiring.nontrivial {R : Type u} [non_assoc_semiring R] (s : subsemiring R) [nontrivial R] :

nontrivial ↥s

source

@[protected]

theorem subsemiring.pow_mem {R : Type u_1} [semiring R] (s : subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :

x ^ n ∈ s

source

@[protected, instance]

def subsemiring.no_zero_divisors {R : Type u} [non_assoc_semiring R] (s : subsemiring R) [no_zero_divisors R] :

no_zero_divisors ↥s

source

@[protected, instance]

def subsemiring.to_semiring {R : Type u_1} [semiring R] (s : subsemiring R) :

semiring ↥s

A subsemiring of a semiring is a semiring.

Equations

s.to_semiring = {add := non_assoc_semiring.add s.to_non_assoc_semiring, add_assoc := _, zero := non_assoc_semiring.zero s.to_non_assoc_semiring, zero_add := _, add_zero := _, nsmul := non_assoc_semiring.nsmul s.to_non_assoc_semiring, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := non_assoc_semiring.mul s.to_non_assoc_semiring, left_distrib := _, right_distrib := _, zero_mul := _, mul_zero := _, mul_assoc := _, one := non_assoc_semiring.one s.to_non_assoc_semiring, one_mul := _, mul_one := _, npow := monoid.npow s.to_submonoid.to_monoid, npow_zero' := _, npow_succ' := _}

source

@[simp, norm_cast]

theorem subsemiring.coe_pow {R : Type u_1} [semiring R] (s : subsemiring R) (x : ↥s) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

source

@[protected, instance]

def subsemiring.to_comm_semiring {R : Type u_1} [comm_semiring R] (s : subsemiring R) :

comm_semiring ↥s

A subsemiring of a comm_semiring is a comm_semiring.

Equations

s.to_comm_semiring = {add := semiring.add s.to_semiring, add_assoc := _, zero := semiring.zero s.to_semiring, zero_add := _, add_zero := _, nsmul := semiring.nsmul s.to_semiring, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := semiring.mul s.to_semiring, left_distrib := _, right_distrib := _, zero_mul := _, mul_zero := _, mul_assoc := _, one := semiring.one s.to_semiring, one_mul := _, mul_one := _, npow := semiring.npow s.to_semiring, npow_zero' := _, npow_succ' := _, mul_comm := _}

source

def subsemiring.subtype {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

↥s →+* R

The natural ring hom from a subsemiring of semiring R to R.

Equations

s.subtype = {to_fun := coe coe_to_lift, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

source

@[simp]

theorem subsemiring.coe_subtype {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

⇑(s.subtype) = coe

source

@[protected, instance]

def subsemiring.to_ordered_semiring {R : Type u_1} [ordered_semiring R] (s : subsemiring R) :

ordered_semiring ↥s

A subsemiring of an ordered_semiring is an ordered_semiring.

Equations

s.to_ordered_semiring = function.injective.ordered_semiring coe _ _ _ _ _ _ _

source

@[protected, instance]

def subsemiring.to_ordered_comm_semiring {R : Type u_1} [ordered_comm_semiring R] (s : subsemiring R) :

ordered_comm_semiring ↥s

A subsemiring of an ordered_comm_semiring is an ordered_comm_semiring.

Equations

s.to_ordered_comm_semiring = function.injective.ordered_comm_semiring coe _ _ _ _ _ _ _

source

@[protected, instance]

def subsemiring.to_linear_ordered_semiring {R : Type u_1} [linear_ordered_semiring R] (s : subsemiring R) :

linear_ordered_semiring ↥s

A subsemiring of a linear_ordered_semiring is a linear_ordered_semiring.

Equations

s.to_linear_ordered_semiring = function.injective.linear_ordered_semiring coe _ _ _ _ _ _ _

source

@[protected]

theorem subsemiring.nsmul_mem {R : Type u} [non_assoc_semiring R] (s : subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) :

n • x ∈ s

source

@[simp]

theorem subsemiring.mem_to_submonoid {R : Type u} [non_assoc_semiring R] {s : subsemiring R} {x : R} :

x ∈ s.to_submonoid ↔ x ∈ s

source

@[simp]

theorem subsemiring.coe_to_submonoid {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

↑(s.to_submonoid) = ↑s

source

@[simp]

theorem subsemiring.mem_to_add_submonoid {R : Type u} [non_assoc_semiring R] {s : subsemiring R} {x : R} :

x ∈ s.to_add_submonoid ↔ x ∈ s

source

@[simp]

theorem subsemiring.coe_to_add_submonoid {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

↑(s.to_add_submonoid) = ↑s

source

@[protected, instance]

def subsemiring.has_top {R : Type u} [non_assoc_semiring R] :

has_top (subsemiring R)

The subsemiring R of the semiring R.

Equations

subsemiring.has_top = {top := {carrier := ⊤.carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}}

source

@[simp]

theorem subsemiring.mem_top {R : Type u} [non_assoc_semiring R] (x : R) :

x ∈ ⊤

source

@[simp]

theorem subsemiring.coe_top {R : Type u} [non_assoc_semiring R] :

↑⊤ = set.univ

source

def subsemiring.comap {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : subsemiring S) :

subsemiring R

The preimage of a subsemiring along a ring homomorphism is a subsemiring.

Equations

subsemiring.comap f s = {carrier := ⇑f ⁻¹' ↑s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[simp]

theorem subsemiring.coe_comap {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring S) (f : R →+* S) :

↑(subsemiring.comap f s) = ⇑f ⁻¹' ↑s

source

@[simp]

theorem subsemiring.mem_comap {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {s : subsemiring S} {f : R →+* S} {x : R} :

x ∈ subsemiring.comap f s ↔ ⇑f x ∈ s

source

theorem subsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (s : subsemiring T) (g : S →+* T) (f : R →+* S) :

subsemiring.comap f (subsemiring.comap g s) = subsemiring.comap (g.comp f) s

source

def subsemiring.map {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : subsemiring R) :

subsemiring S

The image of a subsemiring along a ring homomorphism is a subsemiring.

Equations

subsemiring.map f s = {carrier := ⇑f '' ↑s, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[simp]

theorem subsemiring.coe_map {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : subsemiring R) :

↑(subsemiring.map f s) = ⇑f '' ↑s

source

@[simp]

theorem subsemiring.mem_map {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f : R →+* S} {s : subsemiring R} {y : S} :

y ∈ subsemiring.map f s ↔ ∃ (x : R) (H : x ∈ s), ⇑f x = y

source

@[simp]

theorem subsemiring.map_id {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

subsemiring.map (ring_hom.id R) s = s

source

theorem subsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (s : subsemiring R) (g : S →+* T) (f : R →+* S) :

subsemiring.map g (subsemiring.map f s) = subsemiring.map (g.comp f) s

source

theorem subsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f : R →+* S} {s : subsemiring R} {t : subsemiring S} :

subsemiring.map f s ≤ t ↔ s ≤ subsemiring.comap f t

source

theorem subsemiring.gc_map_comap {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

galois_connection (subsemiring.map f) (subsemiring.comap f)

source

noncomputable def subsemiring.equiv_map_of_injective {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (f : R →+* S) (hf : function.injective ⇑f) :

↥s ≃+* ↥(subsemiring.map f s)

A subsemiring is isomorphic to its image under an injective function

Equations

s.equiv_map_of_injective f hf = {to_fun := (equiv.set.image ⇑f ↑s hf).to_fun, inv_fun := (equiv.set.image ⇑f ↑s hf).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem subsemiring.coe_equiv_map_of_injective_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (f : R →+* S) (hf : function.injective ⇑f) (x : ↥s) :

↑(⇑(s.equiv_map_of_injective f hf) x) = ⇑f ↑x

source

def ring_hom.srange {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

subsemiring S

The range of a ring homomorphism is a subsemiring. See Note [range copy pattern].

Equations

f.srange = (subsemiring.map f ⊤).copy (set.range ⇑f) _

source

@[simp]

theorem ring_hom.coe_srange {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

↑(f.srange) = set.range ⇑f

source

@[simp]

theorem ring_hom.mem_srange {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f : R →+* S} {y : S} :

y ∈ f.srange ↔ ∃ (x : R), ⇑f x = y

source

theorem ring_hom.srange_eq_map {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

f.srange = subsemiring.map f ⊤

source

theorem ring_hom.mem_srange_self {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (x : R) :

⇑f x ∈ f.srange

source

theorem ring_hom.map_srange {R : Type u} {S : Type v} {T : Type w} [non_assoc_semiring R] [non_assoc_semiring S] [non_assoc_semiring T] (g : S →+* T) (f : R →+* S) :

subsemiring.map g f.srange = (g.comp f).srange

source

@[protected, instance]

def ring_hom.fintype_srange {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] [fintype R] [decidable_eq S] (f : R →+* S) :

fintype ↥(f.srange)

The range of a morphism of semirings is a fintype, if the domain is a fintype. Note: this instance can form a diamond with subtype.fintype in the presence of fintype S.

Equations

f.fintype_srange = set.fintype_range ⇑f

source

@[protected, instance]

def subsemiring.has_bot {R : Type u} [non_assoc_semiring R] :

has_bot (subsemiring R)

Equations

subsemiring.has_bot = {bot := (nat.cast_ring_hom R).srange}

source

@[protected, instance]

def subsemiring.inhabited {R : Type u} [non_assoc_semiring R] :

inhabited (subsemiring R)

Equations

subsemiring.inhabited = {default := ⊥}

source

theorem subsemiring.coe_bot {R : Type u} [non_assoc_semiring R] :

↑⊥ = set.range coe

source

theorem subsemiring.mem_bot {R : Type u} [non_assoc_semiring R] {x : R} :

x ∈ ⊥ ↔ ∃ (n : ℕ), ↑n = x

source

@[protected, instance]

def subsemiring.has_inf {R : Type u} [non_assoc_semiring R] :

has_inf (subsemiring R)

The inf of two subsemirings is their intersection.

Equations

subsemiring.has_inf = {inf := λ (s t : subsemiring R), {carrier := ↑s ∩ ↑t, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}}

source

@[simp]

theorem subsemiring.coe_inf {R : Type u} [non_assoc_semiring R] (p p' : subsemiring R) :

↑(p ⊓ p') = ↑p ∩ ↑p'

source

@[simp]

theorem subsemiring.mem_inf {R : Type u} [non_assoc_semiring R] {p p' : subsemiring R} {x : R} :

x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

source

@[protected, instance]

def subsemiring.has_Inf {R : Type u} [non_assoc_semiring R] :

has_Inf (subsemiring R)

Equations

subsemiring.has_Inf = {Inf := λ (s : set (subsemiring R)), subsemiring.mk' (⋂ (t : subsemiring R) (H : t ∈ s), ↑t) (⨅ (t : subsemiring R) (H : t ∈ s), t.to_submonoid) _ (⨅ (t : subsemiring R) (H : t ∈ s), t.to_add_submonoid) _}

source

@[simp, norm_cast]

theorem subsemiring.coe_Inf {R : Type u} [non_assoc_semiring R] (S : set (subsemiring R)) :

↑(Inf S) = ⋂ (s : subsemiring R) (H : s ∈ S), ↑s

source

theorem subsemiring.mem_Inf {R : Type u} [non_assoc_semiring R] {S : set (subsemiring R)} {x : R} :

x ∈ Inf S ↔ ∀ (p : subsemiring R), p ∈ S → x ∈ p

source

@[simp]

theorem subsemiring.Inf_to_submonoid {R : Type u} [non_assoc_semiring R] (s : set (subsemiring R)) :

(Inf s).to_submonoid = ⨅ (t : subsemiring R) (H : t ∈ s), t.to_submonoid

source

@[simp]

theorem subsemiring.Inf_to_add_submonoid {R : Type u} [non_assoc_semiring R] (s : set (subsemiring R)) :

(Inf s).to_add_submonoid = ⨅ (t : subsemiring R) (H : t ∈ s), t.to_add_submonoid

source

@[protected, instance]

def subsemiring.complete_lattice {R : Type u} [non_assoc_semiring R] :

complete_lattice (subsemiring R)

Subsemirings of a semiring form a complete lattice.

Equations

subsemiring.complete_lattice = {sup := complete_lattice.sup (complete_lattice_of_Inf (subsemiring R) subsemiring.complete_lattice._proof_1), le := complete_lattice.le (complete_lattice_of_Inf (subsemiring R) subsemiring.complete_lattice._proof_1), lt := complete_lattice.lt (complete_lattice_of_Inf (subsemiring R) subsemiring.complete_lattice._proof_1), le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := has_inf.inf subsemiring.has_inf, inf_le_left := _, inf_le_right := _, le_inf := _, Sup := complete_lattice.Sup (complete_lattice_of_Inf (subsemiring R) subsemiring.complete_lattice._proof_1), le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf (complete_lattice_of_Inf (subsemiring R) subsemiring.complete_lattice._proof_1), Inf_le := _, le_Inf := _, top := ⊤, bot := ⊥, le_top := _, bot_le := _}

source

theorem subsemiring.eq_top_iff' {R : Type u} [non_assoc_semiring R] (A : subsemiring R) :

A = ⊤ ↔ ∀ (x : R), x ∈ A

source

def subsemiring.center (R : Type u_1) [semiring R] :

subsemiring R

The center of a semiring R is the set of elements that commute with everything in R

Equations

subsemiring.center R = {carrier := set.center R (distrib.to_has_mul R), mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

theorem subsemiring.coe_center (R : Type u_1) [semiring R] :

↑(subsemiring.center R) = set.center R

source

@[simp]

theorem subsemiring.center_to_submonoid (R : Type u_1) [semiring R] :

(subsemiring.center R).to_submonoid = submonoid.center R

source

theorem subsemiring.mem_center_iff {R : Type u_1} [semiring R] {z : R} :

z ∈ subsemiring.center R ↔ ∀ (g : R), g * z = z * g

source

@[protected, instance]

def subsemiring.decidable_mem_center {R : Type u_1} [semiring R] [decidable_eq R] [fintype R] :

decidable_pred (λ (_x : R), _x ∈ subsemiring.center R)

Equations

subsemiring.decidable_mem_center = λ (_x : R), decidable_of_iff' (∀ (g : R), g * _x = _x * g) subsemiring.mem_center_iff

source

@[simp]

theorem subsemiring.center_eq_top (R : Type u_1) [comm_semiring R] :

subsemiring.center R = ⊤

source

@[protected, instance]

def subsemiring.center.comm_semiring {R : Type u_1} [semiring R] :

comm_semiring ↥(subsemiring.center R)

The center is commutative.

Equations

subsemiring.center.comm_semiring = {add := semiring.add (subsemiring.center R).to_semiring, add_assoc := _, zero := semiring.zero (subsemiring.center R).to_semiring, zero_add := _, add_zero := _, nsmul := semiring.nsmul (subsemiring.center R).to_semiring, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := comm_monoid.mul submonoid.center.comm_monoid, left_distrib := _, right_distrib := _, zero_mul := _, mul_zero := _, mul_assoc := _, one := comm_monoid.one submonoid.center.comm_monoid, one_mul := _, mul_one := _, npow := comm_monoid.npow submonoid.center.comm_monoid, npow_zero' := _, npow_succ' := _, mul_comm := _}

source

def subsemiring.centralizer {R : Type u_1} [semiring R] (s : set R) :

subsemiring R

The centralizer of a set as subsemiring.

Equations

subsemiring.centralizer s = {carrier := s.centralizer (distrib.to_has_mul R), mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[simp, norm_cast]

theorem subsemiring.coe_centralizer {R : Type u_1} [semiring R] (s : set R) :

↑(subsemiring.centralizer s) = s.centralizer

source

theorem subsemiring.centralizer_to_submonoid {R : Type u_1} [semiring R] (s : set R) :

(subsemiring.centralizer s).to_submonoid = submonoid.centralizer s

source

theorem subsemiring.mem_centralizer_iff {R : Type u_1} [semiring R] {s : set R} {z : R} :

z ∈ subsemiring.centralizer s ↔ ∀ (g : R), g ∈ s → g * z = z * g

source

theorem subsemiring.centralizer_le {R : Type u_1} [semiring R] (s t : set R) (h : s ⊆ t) :

subsemiring.centralizer t ≤ subsemiring.centralizer s

source

@[simp]

theorem subsemiring.centralizer_univ {R : Type u_1} [semiring R] :

subsemiring.centralizer set.univ = subsemiring.center R

source

def subsemiring.closure {R : Type u} [non_assoc_semiring R] (s : set R) :

subsemiring R

The subsemiring generated by a set.

Equations

subsemiring.closure s = Inf {S : subsemiring R | s ⊆ ↑S}

source

theorem subsemiring.mem_closure {R : Type u} [non_assoc_semiring R] {x : R} {s : set R} :

x ∈ subsemiring.closure s ↔ ∀ (S : subsemiring R), s ⊆ ↑S → x ∈ S

source

@[simp]

theorem subsemiring.subset_closure {R : Type u} [non_assoc_semiring R] {s : set R} :

s ⊆ ↑(subsemiring.closure s)

The subsemiring generated by a set includes the set.

source

theorem subsemiring.not_mem_of_not_mem_closure {R : Type u} [non_assoc_semiring R] {s : set R} {P : R} (hP : P ∉ subsemiring.closure s) :

P ∉ s

source

@[simp]

theorem subsemiring.closure_le {R : Type u} [non_assoc_semiring R] {s : set R} {t : subsemiring R} :

subsemiring.closure s ≤ t ↔ s ⊆ ↑t

A subsemiring S includes closure s if and only if it includes s.

source

theorem subsemiring.closure_mono {R : Type u} [non_assoc_semiring R] ⦃s t : set R⦄ (h : s ⊆ t) :

subsemiring.closure s ≤ subsemiring.closure t

Subsemiring closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

source

theorem subsemiring.closure_eq_of_le {R : Type u} [non_assoc_semiring R] {s : set R} {t : subsemiring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ subsemiring.closure s) :

subsemiring.closure s = t

source

theorem subsemiring.mem_map_equiv {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f : R ≃+* S} {K : subsemiring R} {x : S} :

x ∈ subsemiring.map ↑f K ↔ ⇑(f.symm) x ∈ K

source

theorem subsemiring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) (K : subsemiring R) :

subsemiring.map ↑f K = subsemiring.comap ↑(f.symm) K

source

theorem subsemiring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R ≃+* S) (K : subsemiring S) :

subsemiring.comap ↑f K = subsemiring.map ↑(f.symm) K

source

def submonoid.subsemiring_closure {R : Type u} [non_assoc_semiring R] (M : submonoid R) :

subsemiring R

The additive closure of a submonoid is a subsemiring.

Equations

M.subsemiring_closure = {carrier := (add_submonoid.closure ↑M).carrier, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

theorem submonoid.subsemiring_closure_coe {R : Type u} [non_assoc_semiring R] (M : submonoid R) :

↑(M.subsemiring_closure) = ↑(add_submonoid.closure ↑M)

source

theorem submonoid.subsemiring_closure_to_add_submonoid {R : Type u} [non_assoc_semiring R] (M : submonoid R) :

M.subsemiring_closure.to_add_submonoid = add_submonoid.closure ↑M

source

theorem submonoid.subsemiring_closure_eq_closure {R : Type u} [non_assoc_semiring R] (M : submonoid R) :

M.subsemiring_closure = subsemiring.closure ↑M

The subsemiring generated by a multiplicative submonoid coincides with the subsemiring.closure of the submonoid itself .

source

@[simp]

theorem subsemiring.closure_submonoid_closure {R : Type u} [non_assoc_semiring R] (s : set R) :

subsemiring.closure ↑(submonoid.closure s) = subsemiring.closure s

source

theorem subsemiring.coe_closure_eq {R : Type u} [non_assoc_semiring R] (s : set R) :

↑(subsemiring.closure s) = ↑(add_submonoid.closure ↑(submonoid.closure s))

The elements of the subsemiring closure of M are exactly the elements of the additive closure of a multiplicative submonoid M.

source

theorem subsemiring.mem_closure_iff {R : Type u} [non_assoc_semiring R] {s : set R} {x : R} :

x ∈ subsemiring.closure s ↔ x ∈ add_submonoid.closure ↑(submonoid.closure s)

source

@[simp]

theorem subsemiring.closure_add_submonoid_closure {R : Type u} [non_assoc_semiring R] {s : set R} :

subsemiring.closure ↑(add_submonoid.closure s) = subsemiring.closure s

source

theorem subsemiring.closure_induction {R : Type u} [non_assoc_semiring R] {s : set R} {p : R → Prop} {x : R} (h : x ∈ subsemiring.closure s) (Hs : ∀ (x : R), x ∈ s → p x) (H0 : p 0) (H1 : p 1) (Hadd : ∀ (x y : R), p x → p y → p (x + y)) (Hmul : ∀ (x y : R), p x → p y → p (x * y)) :

p x

An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

source

theorem subsemiring.closure_induction₂ {R : Type u} [non_assoc_semiring R] {s : set R} {p : R → R → Prop} {x y : R} (hx : x ∈ subsemiring.closure s) (hy : y ∈ subsemiring.closure s) (Hs : ∀ (x : R), x ∈ s → ∀ (y : R), y ∈ s → p x y) (H0_left : ∀ (x : R), p 0 x) (H0_right : ∀ (x : R), p x 0) (H1_left : ∀ (x : R), p 1 x) (H1_right : ∀ (x : R), p x 1) (Hadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y) (Hadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)) (Hmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y) (Hmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)) :

p x y

An induction principle for closure membership for predicates with two arguments.

source

theorem subsemiring.mem_closure_iff_exists_list {R : Type u_1} [semiring R] {s : set R} {x : R} :

x ∈ subsemiring.closure s ↔ ∃ (L : list (list R)), (∀ (t : list R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s) ∧ (list.map list.prod L).sum = x

source

@[protected]

def subsemiring.gi (R : Type u) [non_assoc_semiring R] :

galois_insertion subsemiring.closure coe

closure forms a Galois insertion with the coercion to set.

Equations

subsemiring.gi R = {choice := λ (s : set R) (_x : ↑(subsemiring.closure s) ≤ s), subsemiring.closure s, gc := _, le_l_u := _, choice_eq := _}

source

theorem subsemiring.closure_eq {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

subsemiring.closure ↑s = s

Closure of a subsemiring S equals S.

source

@[simp]

theorem subsemiring.closure_empty {R : Type u} [non_assoc_semiring R] :

subsemiring.closure ∅ = ⊥

source

@[simp]

theorem subsemiring.closure_univ {R : Type u} [non_assoc_semiring R] :

subsemiring.closure set.univ = ⊤

source

theorem subsemiring.closure_union {R : Type u} [non_assoc_semiring R] (s t : set R) :

subsemiring.closure (s ∪ t) = subsemiring.closure s ⊔ subsemiring.closure t

source

theorem subsemiring.closure_Union {R : Type u} [non_assoc_semiring R] {ι : Sort u_1} (s : ι → set R) :

subsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), subsemiring.closure (s i)

source

theorem subsemiring.closure_sUnion {R : Type u} [non_assoc_semiring R] (s : set (set R)) :

subsemiring.closure (⋃₀s) = ⨆ (t : set R) (H : t ∈ s), subsemiring.closure t

source

theorem subsemiring.map_sup {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s t : subsemiring R) (f : R →+* S) :

subsemiring.map f (s ⊔ t) = subsemiring.map f s ⊔ subsemiring.map f t

source

theorem subsemiring.map_supr {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subsemiring R) :

subsemiring.map f (supr s) = ⨆ (i : ι), subsemiring.map f (s i)

source

theorem subsemiring.comap_inf {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s t : subsemiring S) (f : R →+* S) :

subsemiring.comap f (s ⊓ t) = subsemiring.comap f s ⊓ subsemiring.comap f t

source

theorem subsemiring.comap_infi {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → subsemiring S) :

subsemiring.comap f (infi s) = ⨅ (i : ι), subsemiring.comap f (s i)

source

@[simp]

theorem subsemiring.map_bot {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

subsemiring.map f ⊥ = ⊥

source

@[simp]

theorem subsemiring.comap_top {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

subsemiring.comap f ⊤ = ⊤

source

def subsemiring.prod {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (t : subsemiring S) :

subsemiring (R × S)

Given subsemirings s, t of semirings R, S respectively, s.prod t is s × t as a subsemiring of R × S.

Equations

s.prod t = {carrier := ↑s ×ˢ ↑t, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

@[norm_cast]

theorem subsemiring.coe_prod {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (t : subsemiring S) :

↑(s.prod t) = ↑s ×ˢ ↑t

source

theorem subsemiring.mem_prod {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {s : subsemiring R} {t : subsemiring S} {p : R × S} :

p ∈ s.prod t ↔ p.fst ∈ s ∧ p.snd ∈ t

source

theorem subsemiring.prod_mono {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] ⦃s₁ s₂ : subsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ t₂ : subsemiring S⦄ (ht : t₁ ≤ t₂) :

s₁.prod t₁ ≤ s₂.prod t₂

source

theorem subsemiring.prod_mono_right {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) :

monotone (λ (t : subsemiring S), s.prod t)

source

theorem subsemiring.prod_mono_left {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (t : subsemiring S) :

monotone (λ (s : subsemiring R), s.prod t)

source

theorem subsemiring.prod_top {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) :

s.prod ⊤ = subsemiring.comap (ring_hom.fst R S) s

source

theorem subsemiring.top_prod {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring S) :

⊤.prod s = subsemiring.comap (ring_hom.snd R S) s

source

@[simp]

theorem subsemiring.top_prod_top {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] :

⊤.prod ⊤ = ⊤

source

def subsemiring.prod_equiv {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (t : subsemiring S) :

↥(s.prod t) ≃+* ↥s × ↥t

Product of subsemirings is isomorphic to their product as monoids.

Equations

s.prod_equiv t = {to_fun := (equiv.set.prod ↑s ↑t).to_fun, inv_fun := (equiv.set.prod ↑s ↑t).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

theorem subsemiring.mem_supr_of_directed {R : Type u} [non_assoc_semiring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subsemiring R} (hS : directed has_le.le S) {x : R} :

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

source

theorem subsemiring.coe_supr_of_directed {R : Type u} [non_assoc_semiring R] {ι : Sort u_1} [hι : nonempty ι] {S : ι → subsemiring R} (hS : directed has_le.le S) :

(↑⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

source

theorem subsemiring.mem_Sup_of_directed_on {R : Type u} [non_assoc_semiring R] {S : set (subsemiring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) {x : R} :

x ∈ Sup S ↔ ∃ (s : subsemiring R) (H : s ∈ S), x ∈ s

source

theorem subsemiring.coe_Sup_of_directed_on {R : Type u} [non_assoc_semiring R] {S : set (subsemiring R)} (Sne : S.nonempty) (hS : directed_on has_le.le S) :

↑(Sup S) = ⋃ (s : subsemiring R) (H : s ∈ S), ↑s

source

def ring_hom.srestrict {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : subsemiring R) :

↥s →+* S

Restriction of a ring homomorphism to a subsemiring of the domain.

Equations

f.srestrict s = f.comp s.subtype

source

@[simp]

theorem ring_hom.srestrict_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {s : subsemiring R} (f : R →+* S) (x : ↥s) :

⇑(f.srestrict s) x = ⇑f ↑x

source

def ring_hom.cod_srestrict {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : subsemiring S) (h : ∀ (x : R), ⇑f x ∈ s) :

R →+* ↥s

Restriction of a ring homomorphism to a subsemiring of the codomain.

Equations

f.cod_srestrict s h = {to_fun := λ (n : R), ⟨⇑f n, _⟩, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

source

def ring_hom.srange_restrict {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

R →+* ↥(f.srange)

Restriction of a ring homomorphism to its range interpreted as a subsemiring.

This is the bundled version of set.range_factorization.

Equations

f.srange_restrict = f.cod_srestrict f.srange _

source

@[simp]

theorem ring_hom.coe_srange_restrict {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (x : R) :

↑(⇑(f.srange_restrict) x) = ⇑f x

source

theorem ring_hom.srange_restrict_surjective {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) :

function.surjective ⇑(f.srange_restrict)

source

theorem ring_hom.srange_top_iff_surjective {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f : R →+* S} :

f.srange = ⊤ ↔ function.surjective ⇑f

source

theorem ring_hom.srange_top_of_surjective {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (hf : function.surjective ⇑f) :

f.srange = ⊤

The range of a surjective ring homomorphism is the whole of the codomain.

source

def ring_hom.eq_slocus {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f g : R →+* S) :

subsemiring R

The subsemiring of elements x : R such that f x = g x

Equations

f.eq_slocus g = {carrier := {x : R | ⇑f x = ⇑g x}, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _}

source

theorem ring_hom.eq_on_sclosure {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f g : R →+* S} {s : set R} (h : set.eq_on ⇑f ⇑g s) :

set.eq_on ⇑f ⇑g ↑(subsemiring.closure s)

If two ring homomorphisms are equal on a set, then they are equal on its subsemiring closure.

source

theorem ring_hom.eq_of_eq_on_stop {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {f g : R →+* S} (h : set.eq_on ⇑f ⇑g ↑⊤) :

f = g

source

theorem ring_hom.eq_of_eq_on_sdense {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {s : set R} (hs : subsemiring.closure s = ⊤) {f g : R →+* S} (h : set.eq_on ⇑f ⇑g s) :

f = g

source

theorem ring_hom.sclosure_preimage_le {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : set S) :

subsemiring.closure (⇑f ⁻¹' s) ≤ subsemiring.comap f (subsemiring.closure s)

source

theorem ring_hom.map_sclosure {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (f : R →+* S) (s : set R) :

subsemiring.map f (subsemiring.closure s) = subsemiring.closure (⇑f '' s)

The image under a ring homomorphism of the subsemiring generated by a set equals the subsemiring generated by the image of the set.

source

def subsemiring.inclusion {R : Type u} [non_assoc_semiring R] {S T : subsemiring R} (h : S ≤ T) :

↥S →* ↥T

The ring homomorphism associated to an inclusion of subsemirings.

Equations

subsemiring.inclusion h = ↑(S.subtype.cod_srestrict T _)

source

@[simp]

theorem subsemiring.srange_subtype {R : Type u} [non_assoc_semiring R] (s : subsemiring R) :

s.subtype.srange = s

source

@[simp]

theorem subsemiring.range_fst {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] :

(ring_hom.fst R S).srange = ⊤

source

@[simp]

theorem subsemiring.range_snd {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] :

(ring_hom.snd R S).srange = ⊤

source

@[simp]

theorem subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (s : subsemiring R) (t : subsemiring S) :

s.prod ⊥ ⊔ ⊥.prod t = s.prod t

source

def ring_equiv.subsemiring_congr {R : Type u} [non_assoc_semiring R] {s t : subsemiring R} (h : s = t) :

↥s ≃+* ↥t

Makes the identity isomorphism from a proof two subsemirings of a multiplicative monoid are equal.

Equations

ring_equiv.subsemiring_congr h = {to_fun := (equiv.set_congr _).to_fun, inv_fun := (equiv.set_congr _).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

def ring_equiv.sof_left_inverse {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) :

R ≃+* ↥(f.srange)

Restrict a ring homomorphism with a left inverse to a ring isomorphism to its ring_hom.srange.

Equations

ring_equiv.sof_left_inverse h = {to_fun := λ (x : R), ⇑(f.srange_restrict) x, inv_fun := λ (x : ↥(f.srange)), (g ∘ ⇑(f.srange.subtype)) x, left_inv := h, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.sof_left_inverse_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : R) :

↑(⇑(ring_equiv.sof_left_inverse h) x) = ⇑f x

source

@[simp]

theorem ring_equiv.sof_left_inverse_symm_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] {g : S → R} {f : R →+* S} (h : function.left_inverse g ⇑f) (x : ↥(f.srange)) :

⇑((ring_equiv.sof_left_inverse h).symm) x = g ↑x

source

def ring_equiv.subsemiring_map {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (s : subsemiring R) :

↥s ≃+* ↥(subsemiring.map e.to_ring_hom s)

Given an equivalence e : R ≃+* S of semirings and a subsemiring s of R, subsemiring_map e s is the induced equivalence between s and s.map e

Equations

e.subsemiring_map s = {to_fun := (e.to_add_equiv.add_submonoid_map s.to_add_submonoid).to_fun, inv_fun := (e.to_add_equiv.add_submonoid_map s.to_add_submonoid).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

source

@[simp]

theorem ring_equiv.subsemiring_map_symm_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (s : subsemiring R) (ᾰ : ↥(add_submonoid.map e.to_add_equiv.to_add_monoid_hom s.to_add_submonoid)) :

⇑((e.subsemiring_map s).symm) ᾰ = (e.to_add_equiv.add_submonoid_map s.to_add_submonoid).inv_fun ᾰ

source

@[simp]

theorem ring_equiv.subsemiring_map_apply {R : Type u} {S : Type v} [non_assoc_semiring R] [non_assoc_semiring S] (e : R ≃+* S) (s : subsemiring R) (ᾰ : ↥(s.to_add_submonoid)) :

⇑(e.subsemiring_map s) ᾰ = (e.to_add_equiv.add_submonoid_map s.to_add_submonoid).to_fun ᾰ

source

@[protected, instance]

def subsemiring.has_scalar {R' : Type u_1} {α : Type u_2} [non_assoc_semiring R'] [has_scalar R' α] (S : subsemiring R') :

has_scalar ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.has_scalar = S.to_submonoid.has_scalar

source

theorem subsemiring.smul_def {R' : Type u_1} {α : Type u_2} [non_assoc_semiring R'] [has_scalar R' α] {S : subsemiring R'} (g : ↥S) (m : α) :

g • m = ↑g • m

source

@[protected, instance]

def subsemiring.smul_comm_class_left {R' : Type u_1} {α : Type u_2} {β : Type u_3} [non_assoc_semiring R'] [has_scalar R' β] [has_scalar α β] [smul_comm_class R' α β] (S : subsemiring R') :

smul_comm_class ↥S α β

source

@[protected, instance]

def subsemiring.smul_comm_class_right {R' : Type u_1} {α : Type u_2} {β : Type u_3} [non_assoc_semiring R'] [has_scalar α β] [has_scalar R' β] [smul_comm_class α R' β] (S : subsemiring R') :

smul_comm_class α ↥S β

source

@[protected, instance]

def subsemiring.is_scalar_tower {R' : Type u_1} {α : Type u_2} {β : Type u_3} [non_assoc_semiring R'] [has_scalar α β] [has_scalar R' α] [has_scalar R' β] [is_scalar_tower R' α β] (S : subsemiring R') :

is_scalar_tower ↥S α β

Note that this provides is_scalar_tower S R R which is needed by smul_mul_assoc.

source

@[protected, instance]

def subsemiring.has_faithful_scalar {R' : Type u_1} {α : Type u_2} [non_assoc_semiring R'] [has_scalar R' α] [has_faithful_scalar R' α] (S : subsemiring R') :

has_faithful_scalar ↥S α

source

@[protected, instance]

def subsemiring.smul_with_zero {R' : Type u_1} {α : Type u_2} [non_assoc_semiring R'] [has_zero α] [smul_with_zero R' α] (S : subsemiring R') :

smul_with_zero ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.smul_with_zero = smul_with_zero.comp_hom α S.subtype.to_monoid_with_zero_hom.to_zero_hom

source

@[protected, instance]

def subsemiring.mul_action {R' : Type u_1} {α : Type u_2} [semiring R'] [mul_action R' α] (S : subsemiring R') :

mul_action ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.mul_action = S.to_submonoid.mul_action

source

@[protected, instance]

def subsemiring.distrib_mul_action {R' : Type u_1} {α : Type u_2} [semiring R'] [add_monoid α] [distrib_mul_action R' α] (S : subsemiring R') :

distrib_mul_action ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.distrib_mul_action = S.to_submonoid.distrib_mul_action

source

@[protected, instance]

def subsemiring.mul_distrib_mul_action {R' : Type u_1} {α : Type u_2} [semiring R'] [monoid α] [mul_distrib_mul_action R' α] (S : subsemiring R') :

mul_distrib_mul_action ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.mul_distrib_mul_action = S.to_submonoid.mul_distrib_mul_action

source

@[protected, instance]

def subsemiring.mul_action_with_zero {R' : Type u_1} {α : Type u_2} [semiring R'] [has_zero α] [mul_action_with_zero R' α] (S : subsemiring R') :

mul_action_with_zero ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.mul_action_with_zero = mul_action_with_zero.comp_hom α S.subtype.to_monoid_with_zero_hom

source

@[protected, instance]

def subsemiring.module {R' : Type u_1} {α : Type u_2} [semiring R'] [add_comm_monoid α] [module R' α] (S : subsemiring R') :

module ↥S α

The action by a subsemiring is the action by the underlying semiring.

Equations

S.module = {to_distrib_mul_action := {to_mul_action := {to_has_scalar := {smul := has_scalar.smul S.has_scalar}, one_smul := _, mul_smul := _}, smul_add := _, smul_zero := _}, add_smul := _, zero_smul := _}

source

def subsemiring.closure_comm_semiring_of_comm {R' : Type u_1} [semiring R'] {s : set R'} (hcomm : ∀ (a : R'), a ∈ s → ∀ (b : R'), b ∈ s → a * b = b * a) :

comm_semiring ↥(subsemiring.closure s)

If all the elements of a set s commute, then closure s is a commutative monoid.

Equations