# Documentation

Mathlib.RingTheory.Subsemiring.Basic

# Bundled subsemirings #

We define bundled subsemirings and some standard constructions: CompleteLattice structure, Subtype and inclusion ring homomorphisms, subsemiring map, comap and range (rangeS) of a RingHom etc.

class AddSubmonoidWithOneClass (S : Type u_1) (R : Type u_2) [inst : ] [inst : SetLike S R] extends , :

AddSubmonoidWithOneClass S R says S is a type of subsets s ≤ R≤ R that contain 0, 1, and are closed under (+)

Instances
theorem natCast_mem {S : Type u_1} {R : Type u_2} [inst : ] [inst : SetLike S R] (s : S) [inst : ] (n : ) :
n s
instance AddSubmonoidWithOneClass.toAddMonoidWithOne {S : Type u_1} {R : Type u_2} [inst : ] [inst : SetLike S R] (s : S) [inst : ] :
AddMonoidWithOne { x // x s }
Equations
• = let src := ; AddMonoidWithOne.mk
class SubsemiringClass (S : Type u_1) (R : Type u) [inst : ] [inst : SetLike S R] extends , :

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

Instances
instance SubsemiringClass.addSubmonoidWithOneClass (S : Type u_1) (R : Type u) [inst : ] [inst : SetLike S R] [h : ] :
Equations
theorem coe_nat_mem {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) (n : ) :
n s
instance SubsemiringClass.toNonAssocSemiring {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) :
NonAssocSemiring { x // x s }

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.nontrivial {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) [inst : ] :
Nontrivial { x // x s }
Equations
instance SubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) [inst : ] :
NoZeroDivisors { x // x s }
Equations
def SubsemiringClass.subtype {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) :
{ x // x s } →+* R

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem SubsemiringClass.coe_subtype {R : Type u} {S : Type v} [inst : ] [inst : SetLike S R] [hSR : ] (s : S) :
= Subtype.val
instance SubsemiringClass.toSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :
Semiring { x // x s }

A subsemiring of a Semiring is a Semiring.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem SubsemiringClass.coe_pow {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] (x : { x // x s }) (n : ) :
↑(x ^ n) = x ^ n
instance SubsemiringClass.toCommSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :
CommSemiring { x // x s }

A subsemiring of a CommSemiring is a CommSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :
OrderedSemiring { x // x s }

A subsemiring of an OrderedSemiring is an OrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toStrictOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :

A subsemiring of an StrictOrderedSemiring is an StrictOrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :

A subsemiring of an OrderedCommSemiring is an OrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toStrictOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :

A subsemiring of an StrictOrderedCommSemiring is an StrictOrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toLinearOrderedSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :

A subsemiring of a LinearOrderedSemiring is a LinearOrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance SubsemiringClass.toLinearOrderedCommSemiring {S : Type v} (s : S) {R : Type u_1} [inst : ] [inst : SetLike S R] [inst : ] :

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
structure Subsemiring (R : Type u) [inst : ] extends :
• The sum of two elements of an additive subsemigroup belongs to the subsemigroup.

add_mem' : ∀ {a b : R}, a toSubmonoid.toSubsemigroup.carrierb toSubmonoid.toSubsemigroup.carriera + b toSubmonoid.toSubsemigroup.carrier
• An additive submonoid contains 0.

zero_mem' : 0 toSubmonoid.toSubsemigroup.carrier

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

Instances For
abbrev Subsemiring.toAddSubmonoid {R : Type u} [inst : ] (self : ) :

Reinterpret a Subsemiring as an AddSubmonoid.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.instSetLikeSubsemiring {R : Type u} [inst : ] :
SetLike () R
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.mem_toSubmonoid {R : Type u} [inst : ] {s : } {x : R} :
x s.toSubmonoid x s
theorem Subsemiring.mem_carrier {R : Type u} [inst : ] {s : } {x : R} :
x s.toSubmonoid.toSubsemigroup.carrier x s
theorem Subsemiring.ext {R : Type u} [inst : ] {S : } {T : } (h : ∀ (x : R), x S x T) :
S = T

Two subsemirings are equal if they have the same elements.

def Subsemiring.copy {R : Type u} [inst : ] (S : ) (s : Set R) (hs : s = S) :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_copy {R : Type u} [inst : ] (S : ) (s : Set R) (hs : s = S) :
↑(Subsemiring.copy S s hs) = s
theorem Subsemiring.copy_eq {R : Type u} [inst : ] (S : ) (s : Set R) (hs : s = S) :
theorem Subsemiring.toSubmonoid_injective {R : Type u} [inst : ] :
Function.Injective Subsemiring.toSubmonoid
theorem Subsemiring.toSubmonoid_strictMono {R : Type u} [inst : ] :
StrictMono Subsemiring.toSubmonoid
theorem Subsemiring.toSubmonoid_mono {R : Type u} [inst : ] :
Monotone Subsemiring.toSubmonoid
theorem Subsemiring.toAddSubmonoid_injective {R : Type u} [inst : ] :
theorem Subsemiring.toAddSubmonoid_strictMono {R : Type u} [inst : ] :
theorem Subsemiring.toAddSubmonoid_mono {R : Type u} [inst : ] :
def Subsemiring.mk' {R : Type u} [inst : ] (s : Set R) (sm : ) (hm : sm = s) (sa : ) (ha : sa = s) :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_mk' {R : Type u} [inst : ] {s : Set R} {sm : } (hm : sm = s) {sa : } (ha : sa = s) :
↑(Subsemiring.mk' s sm hm sa ha) = s
@[simp]
theorem Subsemiring.mem_mk' {R : Type u} [inst : ] {s : Set R} {sm : } (hm : sm = s) {sa : } (ha : sa = s) {x : R} :
x Subsemiring.mk' s sm hm sa ha x s
@[simp]
theorem Subsemiring.mk'_toSubmonoid {R : Type u} [inst : ] {s : Set R} {sm : } (hm : sm = s) {sa : } (ha : sa = s) :
(Subsemiring.mk' s sm hm sa ha).toSubmonoid = sm
@[simp]
theorem Subsemiring.mk'_toAddSubmonoid {R : Type u} [inst : ] {s : Set R} {sm : } (hm : sm = s) {sa : } (ha : sa = s) :
theorem Subsemiring.one_mem {R : Type u} [inst : ] (s : ) :
1 s

A subsemiring contains the semiring's 1.

theorem Subsemiring.zero_mem {R : Type u} [inst : ] (s : ) :
0 s

A subsemiring contains the semiring's 0.

theorem Subsemiring.mul_mem {R : Type u} [inst : ] (s : ) {x : R} {y : R} :
x sy sx * y s

A subsemiring is closed under multiplication.

theorem Subsemiring.add_mem {R : Type u} [inst : ] (s : ) {x : R} {y : R} :
x sy sx + y s

A subsemiring is closed under addition.

theorem Subsemiring.list_prod_mem {R : Type u_1} [inst : ] (s : ) {l : List R} :
(∀ (x : R), x lx s) → s

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

theorem Subsemiring.list_sum_mem {R : Type u} [inst : ] (s : ) {l : List R} :
(∀ (x : R), x lx s) → s

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

theorem Subsemiring.multiset_prod_mem {R : Type u_1} [inst : ] (s : ) (m : ) :
(∀ (a : R), a ma s) →

Product of a multiset of elements in a Subsemiring of a CommSemiring is in the Subsemiring.

theorem Subsemiring.multiset_sum_mem {R : Type u} [inst : ] (s : ) (m : ) :
(∀ (a : R), a ma s) →

Sum of a multiset of elements in a Subsemiring of a Semiring is in the add_subsemiring.

theorem Subsemiring.prod_mem {R : Type u_1} [inst : ] (s : ) {ι : Type u_2} {t : } {f : ιR} (h : ∀ (c : ι), c tf c s) :
(Finset.prod t fun i => f i) s

Product of elements of a subsemiring of a CommSemiring indexed by a Finset is in the subsemiring.

theorem Subsemiring.sum_mem {R : Type u} [inst : ] (s : ) {ι : Type u_1} {t : } {f : ιR} (h : ∀ (c : ι), c tf c s) :
(Finset.sum t fun i => f i) s

Sum of elements in an Subsemiring of an Semiring indexed by a Finset is in the add_subsemiring.

instance Subsemiring.toNonAssocSemiring {R : Type u} [inst : ] (s : ) :
NonAssocSemiring { x // x s }

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

Equations
@[simp]
theorem Subsemiring.coe_one {R : Type u} [inst : ] (s : ) :
1 = 1
@[simp]
theorem Subsemiring.coe_zero {R : Type u} [inst : ] (s : ) :
0 = 0
@[simp]
theorem Subsemiring.coe_add {R : Type u} [inst : ] (s : ) (x : { x // x s }) (y : { x // x s }) :
↑(x + y) = x + y
@[simp]
theorem Subsemiring.coe_mul {R : Type u} [inst : ] (s : ) (x : { x // x s }) (y : { x // x s }) :
↑(x * y) = x * y
instance Subsemiring.nontrivial {R : Type u} [inst : ] (s : ) [inst : ] :
Nontrivial { x // x s }
Equations
theorem Subsemiring.pow_mem {R : Type u_1} [inst : ] (s : ) {x : R} (hx : x s) (n : ) :
x ^ n s
instance Subsemiring.noZeroDivisors {R : Type u} [inst : ] (s : ) [inst : ] :
NoZeroDivisors { x // x s }
Equations
instance Subsemiring.toSemiring {R : Type u_1} [inst : ] (s : ) :
Semiring { x // x s }

A subsemiring of a Semiring is a Semiring.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_pow {R : Type u_1} [inst : ] (s : ) (x : { x // x s }) (n : ) :
↑(x ^ n) = x ^ n
instance Subsemiring.toCommSemiring {R : Type u_1} [inst : ] (s : ) :
CommSemiring { x // x s }

A subsemiring of a CommSemiring is a CommSemiring.

Equations
def Subsemiring.subtype {R : Type u} [inst : ] (s : ) :
{ x // x s } →+* R

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_subtype {R : Type u} [inst : ] (s : ) :
↑() = Subtype.val
instance Subsemiring.toOrderedSemiring {R : Type u_1} [inst : ] (s : ) :
OrderedSemiring { x // x s }

A subsemiring of an OrderedSemiring is an OrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.toStrictOrderedSemiring {R : Type u_1} [inst : ] (s : ) :

A subsemiring of a StrictOrderedSemiring is a StrictOrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.toOrderedCommSemiring {R : Type u_1} [inst : ] (s : ) :

A subsemiring of an OrderedCommSemiring is an OrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.toStrictOrderedCommSemiring {R : Type u_1} [inst : ] (s : ) :

A subsemiring of a StrictOrderedCommSemiring is a StrictOrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.toLinearOrderedSemiring {R : Type u_1} [inst : ] (s : ) :

A subsemiring of a LinearOrderedSemiring is a LinearOrderedSemiring.

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.toLinearOrderedCommSemiring {R : Type u_1} [inst : ] (s : ) :

A subsemiring of a LinearOrderedCommSemiring is a LinearOrderedCommSemiring.

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.nsmul_mem {R : Type u} [inst : ] (s : ) {x : R} (hx : x s) (n : ) :
n x s
@[simp]
theorem Subsemiring.coe_toSubmonoid {R : Type u} [inst : ] (s : ) :
s.toSubmonoid = s
@[simp]
theorem Subsemiring.coe_carrier_toSubmonoid {R : Type u} [inst : ] (s : ) :
s.toSubmonoid.toSubsemigroup.carrier = s
theorem Subsemiring.mem_toAddSubmonoid {R : Type u} [inst : ] {s : } {x : R} :
x s
theorem Subsemiring.coe_toAddSubmonoid {R : Type u} [inst : ] (s : ) :
= s
instance Subsemiring.instTopSubsemiring {R : Type u} [inst : ] :
Top ()

The subsemiring R of the semiring R.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.mem_top {R : Type u} [inst : ] (x : R) :
@[simp]
theorem Subsemiring.coe_top {R : Type u} [inst : ] :
= Set.univ
@[simp]
theorem Subsemiring.topEquiv_apply {R : Type u} [inst : ] (r : { x // x }) :
Subsemiring.topEquiv r = r
@[simp]
theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [inst : ] (r : R) :
↑(↑(RingEquiv.symm Subsemiring.topEquiv) r) = r
def Subsemiring.topEquiv {R : Type u} [inst : ] :
{ x // x } ≃+* R

The ring equiv between the top element of Subsemiring R and R.

Equations
• One or more equations did not get rendered due to their size.
def Subsemiring.comap {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (s : ) :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (f : R →+* S) :
↑() = f ⁻¹' s
@[simp]
theorem Subsemiring.mem_comap {R : Type u} {S : Type v} [inst : ] [inst : ] {s : } {f : R →+* S} {x : R} :
x f x s
theorem Subsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [inst : ] [inst : ] [inst : ] (s : ) (g : S →+* T) (f : R →+* S) :
=
def Subsemiring.map {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (s : ) :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_map {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (s : ) :
↑() = f '' s
@[simp]
theorem Subsemiring.mem_map {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} {s : } {y : S} :
y x, x s f x = y
@[simp]
theorem Subsemiring.map_id {R : Type u} [inst : ] (s : ) :
= s
theorem Subsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [inst : ] [inst : ] [inst : ] (s : ) (g : S →+* T) (f : R →+* S) :
theorem Subsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} {s : } {t : } :
t s
theorem Subsemiring.gc_map_comap {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
noncomputable def Subsemiring.equivMapOfInjective {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (f : R →+* S) (hf : ) :
{ x // x s } ≃+* { x // x }

A subsemiring is isomorphic to its image under an injective function

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (f : R →+* S) (hf : ) (x : { x // x s }) :
↑(↑() x) = f x
def RingHom.rangeS {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :

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

Equations
@[simp]
theorem RingHom.coe_rangeS {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
↑() =
@[simp]
theorem RingHom.mem_rangeS {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} {y : S} :
x, f x = y
theorem RingHom.rangeS_eq_map {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
theorem RingHom.mem_rangeS_self {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (x : R) :
f x
theorem RingHom.map_rangeS {R : Type u} {S : Type v} {T : Type w} [inst : ] [inst : ] [inst : ] (g : S →+* T) (f : R →+* S) :
=
instance RingHom.fintypeRangeS {R : Type u} {S : Type v} [inst : ] [inst : ] [inst : ] [inst : ] (f : R →+* S) :
Fintype { x // }

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
instance Subsemiring.instBotSubsemiring {R : Type u} [inst : ] :
Bot ()
Equations
• Subsemiring.instBotSubsemiring = { bot := }
instance Subsemiring.instInhabitedSubsemiring {R : Type u} [inst : ] :
Equations
• Subsemiring.instInhabitedSubsemiring = { default := }
theorem Subsemiring.coe_bot {R : Type u} [inst : ] :
= Set.range Nat.cast
theorem Subsemiring.mem_bot {R : Type u} [inst : ] {x : R} :
x n, n = x
instance Subsemiring.instInfSubsemiring {R : Type u} [inst : ] :
Inf ()

The inf of two subsemirings is their intersection.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_inf {R : Type u} [inst : ] (p : ) (p' : ) :
↑(p p') = p p'
@[simp]
theorem Subsemiring.mem_inf {R : Type u} [inst : ] {p : } {p' : } {x : R} :
x p p' x p x p'
instance Subsemiring.instInfSetSubsemiring {R : Type u} [inst : ] :
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_infₛ {R : Type u} [inst : ] (S : Set ()) :
↑(infₛ S) = Set.interᵢ fun s => Set.interᵢ fun h => s
theorem Subsemiring.mem_infₛ {R : Type u} [inst : ] {S : Set ()} {x : R} :
x infₛ S ∀ (p : ), p Sx p
@[simp]
theorem Subsemiring.infₛ_toSubmonoid {R : Type u} [inst : ] (s : Set ()) :
(infₛ s).toSubmonoid = t, h, t.toSubmonoid
@[simp]
theorem Subsemiring.infₛ_toAddSubmonoid {R : Type u} [inst : ] (s : Set ()) :
= t,

Subsemirings of a semiring form a complete lattice.

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.eq_top_iff' {R : Type u} [inst : ] (A : ) :
A = ∀ (x : R), x A
def Subsemiring.center (R : Type u_1) [inst : ] :

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

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.coe_center (R : Type u_1) [inst : ] :
@[simp]
theorem Subsemiring.center_toSubmonoid (R : Type u_1) [inst : ] :
().toSubmonoid =
theorem Subsemiring.mem_center_iff {R : Type u_1} [inst : ] {z : R} :
∀ (g : R), g * z = z * g
instance Subsemiring.decidableMemCenter {R : Type u_1} [inst : ] [inst : ] [inst : ] :
DecidablePred fun x =>
Equations
@[simp]
theorem Subsemiring.center_eq_top (R : Type u_1) [inst : ] :
instance Subsemiring.commSemiring {R : Type u_1} [inst : ] :
CommSemiring { x // }

The center is commutative.

Equations
• One or more equations did not get rendered due to their size.
def Subsemiring.centralizer {R : Type u_1} [inst : ] (s : Set R) :

The centralizer of a set as subsemiring.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Subsemiring.coe_centralizer {R : Type u_1} [inst : ] (s : Set R) :
theorem Subsemiring.centralizer_toSubmonoid {R : Type u_1} [inst : ] (s : Set R) :
().toSubmonoid =
theorem Subsemiring.mem_centralizer_iff {R : Type u_1} [inst : ] {s : Set R} {z : R} :
∀ (g : R), g sg * z = z * g
theorem Subsemiring.centralizer_le {R : Type u_1} [inst : ] (s : Set R) (t : Set R) (h : s t) :
@[simp]
theorem Subsemiring.centralizer_univ {R : Type u_1} [inst : ] :
def Subsemiring.closure {R : Type u} [inst : ] (s : Set R) :

The Subsemiring generated by a set.

Equations
theorem Subsemiring.mem_closure {R : Type u} [inst : ] {x : R} {s : Set R} :
∀ (S : ), s Sx S
@[simp]
theorem Subsemiring.subset_closure {R : Type u} [inst : ] {s : Set R} :
s ↑()

The subsemiring generated by a set includes the set.

theorem Subsemiring.not_mem_of_not_mem_closure {R : Type u} [inst : ] {s : Set R} {P : R} (hP : ) :
¬P s
@[simp]
theorem Subsemiring.closure_le {R : Type u} [inst : ] {s : Set R} {t : } :
s t

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

theorem Subsemiring.closure_mono {R : Type u} [inst : ] ⦃s : Set R ⦃t : Set R (h : s t) :

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

theorem Subsemiring.closure_eq_of_le {R : Type u} [inst : ] {s : Set R} {t : } (h₁ : s t) (h₂ : ) :
theorem Subsemiring.mem_map_equiv {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R ≃+* S} {K : } {x : S} :
x Subsemiring.map (f) K ↑() x K
theorem Subsemiring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R ≃+* S) (K : ) :
theorem Subsemiring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R ≃+* S) (K : ) :
def Submonoid.subsemiringClosure {R : Type u} [inst : ] (M : ) :

The additive closure of a submonoid is a subsemiring.

Equations
• One or more equations did not get rendered due to their size.
theorem Submonoid.subsemiringClosure_coe {R : Type u} [inst : ] (M : ) :
theorem Submonoid.subsemiringClosure_eq_closure {R : Type u} [inst : ] (M : ) :

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

@[simp]
theorem Subsemiring.closure_submonoid_closure {R : Type u} [inst : ] (s : Set R) :
theorem Subsemiring.coe_closure_eq {R : Type u} [inst : ] (s : Set R) :

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

theorem Subsemiring.mem_closure_iff {R : Type u} [inst : ] {s : Set R} {x : R} :
@[simp]
theorem Subsemiring.closure_addSubmonoid_closure {R : Type u} [inst : ] {s : Set R} :
theorem Subsemiring.closure_induction {R : Type u} [inst : ] {s : Set R} {p : RProp} {x : R} (h : ) (Hs : (x : R) → x sp x) (H0 : p 0) (H1 : p 1) (Hadd : (x y : R) → p xp yp (x + y)) (Hmul : (x y : R) → p xp yp (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.

theorem Subsemiring.closure_induction₂ {R : Type u} [inst : ] {s : Set R} {p : RRProp} {x : R} {y : R} (hx : ) (hy : ) (Hs : (x : R) → x s(y : R) → y sp 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 yp x₂ yp (x₁ + x₂) y) (Hadd_right : (x y₁ y₂ : R) → p x y₁p x y₂p x (y₁ + y₂)) (Hmul_left : (x₁ x₂ y : R) → p x₁ yp x₂ yp (x₁ * x₂) y) (Hmul_right : (x y₁ y₂ : R) → p x y₁p x y₂p x (y₁ * y₂)) :
p x y

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

theorem Subsemiring.mem_closure_iff_exists_list {R : Type u_1} [inst : ] {s : Set R} {x : R} :
L, (∀ (t : List R), t L∀ (y : R), y ty s) List.sum (List.map List.prod L) = x
def Subsemiring.gi (R : Type u) [inst : ] :
GaloisInsertion Subsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.closure_eq {R : Type u} [inst : ] (s : ) :

Closure of a subsemiring S equals S.

@[simp]
theorem Subsemiring.closure_empty {R : Type u} [inst : ] :
@[simp]
theorem Subsemiring.closure_univ {R : Type u} [inst : ] :
theorem Subsemiring.closure_union {R : Type u} [inst : ] (s : Set R) (t : Set R) :
theorem Subsemiring.closure_unionᵢ {R : Type u} [inst : ] {ι : Sort u_1} (s : ιSet R) :
theorem Subsemiring.closure_unionₛ {R : Type u} [inst : ] (s : Set (Set R)) :
= t, h,
theorem Subsemiring.map_sup {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) (f : R →+* S) :
theorem Subsemiring.map_supᵢ {R : Type u} {S : Type v} [inst : ] [inst : ] {ι : Sort u_1} (f : R →+* S) (s : ι) :
theorem Subsemiring.comap_inf {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) (f : R →+* S) :
theorem Subsemiring.comap_infᵢ {R : Type u} {S : Type v} [inst : ] [inst : ] {ι : Sort u_1} (f : R →+* S) (s : ι) :
= i, Subsemiring.comap f (s i)
@[simp]
theorem Subsemiring.map_bot {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
@[simp]
theorem Subsemiring.comap_top {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
def Subsemiring.prod {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) :

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

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.coe_prod {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) :
↑() = s ×ˢ t
theorem Subsemiring.mem_prod {R : Type u} {S : Type v} [inst : ] [inst : ] {s : } {t : } {p : R × S} :
p p.fst s p.snd t
theorem Subsemiring.prod_mono {R : Type u} {S : Type v} [inst : ] [inst : ] ⦃s₁ : ⦃s₂ : (hs : s₁ s₂) ⦃t₁ : ⦃t₂ : (ht : t₁ t₂) :
theorem Subsemiring.prod_mono_right {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) :
Monotone fun t =>
theorem Subsemiring.prod_mono_left {R : Type u} {S : Type v} [inst : ] [inst : ] (t : ) :
Monotone fun s =>
theorem Subsemiring.prod_top {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) :
theorem Subsemiring.top_prod {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) :
@[simp]
theorem Subsemiring.top_prod_top {R : Type u} {S : Type v} [inst : ] [inst : ] :
def Subsemiring.prodEquiv {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) :
{ x // x } ≃+* { x // x s } × { x // x t }

Product of subsemirings is isomorphic to their product as monoids.

Equations
• One or more equations did not get rendered due to their size.
theorem Subsemiring.mem_supᵢ_of_directed {R : Type u} [inst : ] {ι : Sort u_1} [hι : ] {S : ι} (hS : Directed (fun x x_1 => x x_1) S) {x : R} :
(x i, S i) i, x S i
theorem Subsemiring.coe_supᵢ_of_directed {R : Type u} [inst : ] {ι : Sort u_1} [hι : ] {S : ι} (hS : Directed (fun x x_1 => x x_1) S) :
↑(i, S i) = Set.unionᵢ fun i => ↑(S i)
theorem Subsemiring.mem_supₛ_of_directedOn {R : Type u} [inst : ] {S : Set ()} (Sne : ) (hS : DirectedOn (fun x x_1 => x x_1) S) {x : R} :
x supₛ S s, s S x s
theorem Subsemiring.coe_supₛ_of_directedOn {R : Type u} [inst : ] {S : Set ()} (Sne : ) (hS : DirectedOn (fun x x_1 => x x_1) S) :
↑(supₛ S) = Set.unionᵢ fun s => Set.unionᵢ fun h => s
def RingHom.domRestrict {R : Type u} {S : Type v} [inst : ] [inst : ] {σR : Type u_1} [inst : SetLike σR R] [inst : ] (f : R →+* S) (s : σR) :
{ x // x s } →+* S

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

Equations
@[simp]
theorem RingHom.restrict_apply {R : Type u} {S : Type v} [inst : ] [inst : ] {σR : Type u_1} [inst : SetLike σR R] [inst : ] (f : R →+* S) {s : σR} (x : { x // x s }) :
↑() x = f x
def RingHom.codRestrict {R : Type u} {S : Type v} [inst : ] [inst : ] {σS : Type u_1} [inst : SetLike σS S] [inst : ] (f : R →+* S) (s : σS) (h : ∀ (x : R), f x s) :
R →+* { x // x s }

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

Equations
• One or more equations did not get rendered due to their size.
def RingHom.restrict {R : Type u} {S : Type v} [inst : ] [inst : ] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : ] [inst : ] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x s'f x s) :
{ x // x s' } →+* { x // x s }

The ring homomorphism from the preimage of s to s.

Equations
@[simp]
theorem RingHom.coe_restrict_apply {R : Type u} {S : Type v} [inst : ] [inst : ] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : ] [inst : ] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x s'f x s) (x : { x // x s' }) :
↑(↑(RingHom.restrict f s' s h) x) = f x
@[simp]
theorem RingHom.comp_restrict {R : Type u} {S : Type v} [inst : ] [inst : ] {σR : Type u_1} {σS : Type u_2} [inst : SetLike σR R] [inst : SetLike σS S] [inst : ] [inst : ] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ (x : R), x s'f x s) :
def RingHom.rangeSRestrict {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
R →+* { x // }

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

This is the bundled version of Set.rangeFactorization.

Equations
@[simp]
theorem RingHom.coe_rangeSRestrict {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (x : R) :
↑(↑() x) = f x
theorem RingHom.rangeSRestrict_surjective {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
theorem RingHom.rangeS_top_iff_surjective {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} :
theorem RingHom.rangeS_top_of_surjective {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (hf : ) :

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

def RingHom.eqLocusS {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (g : R →+* S) :

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

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem RingHom.eqLocusS_same {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) :
theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} {g : R →+* S} {s : Set R} (h : Set.EqOn (f) (g) s) :
Set.EqOn f g ↑()

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

theorem RingHom.eq_of_eqOn_stop {R : Type u} {S : Type v} [inst : ] [inst : ] {f : R →+* S} {g : R →+* S} (h : Set.EqOn f g ) :
f = g
theorem RingHom.eq_of_eqOn_sdense {R : Type u} {S : Type v} [inst : ] [inst : ] {s : Set R} (hs : ) {f : R →+* S} {g : R →+* S} (h : Set.EqOn (f) (g) s) :
f = g
theorem RingHom.sclosure_preimage_le {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (s : Set S) :
theorem RingHom.map_closureS {R : Type u} {S : Type v} [inst : ] [inst : ] (f : R →+* S) (s : Set R) :

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

def Subsemiring.inclusion {R : Type u} [inst : ] {S : } {T : } (h : S T) :
{ x // x S } →+* { x // x T }

The ring homomorphism associated to an inclusion of subsemirings.

Equations
@[simp]
theorem Subsemiring.rangeS_subtype {R : Type u} [inst : ] (s : ) :
@[simp]
theorem Subsemiring.range_fst {R : Type u} {S : Type v} [inst : ] [inst : ] :
@[simp]
theorem Subsemiring.range_snd {R : Type u} {S : Type v} [inst : ] [inst : ] :
@[simp]
theorem Subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [inst : ] [inst : ] (s : ) (t : ) :
def RingEquiv.subsemiringCongr {R : Type u} [inst : ] {s : } {t : } (h : s = t) :
{ x // x s } ≃+* { x // x t }

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

Equations
• One or more equations did not get rendered due to their size.
def RingEquiv.ofLeftInverseS {R : Type u} {S : Type v} [inst : ] [inst : ] {g : SR} {f : R →+* S} (h : ) :
R ≃+* { x // }

Restrict a ring homomorphism with a left inverse to a ring isomorphism to its RingHom.rangeS.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem RingEquiv.ofLeftInverseS_apply {R : Type u} {S : Type v} [inst : ] [inst : ] {g : SR} {f : R →+* S} (h : ) (x : R) :
↑(↑() x) = f x
@[simp]
theorem RingEquiv.ofLeftInverseS_symm_apply {R : Type u} {S : Type v} [inst : ] [inst : ] {g : SR} {f : R →+* S} (h : ) (x : { x // }) :
= g x
@[simp]
theorem RingEquiv.subsemiringMap_apply_coe {R : Type u} {S : Type v} [inst : ] [inst : ] (e : R ≃+* S) (s : ) :
∀ (a : ), ↑(↑() a) = e a
@[simp]
theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [inst : ] [inst : ] (e : R ≃+* S) (s : ) :
∀ (a : ↑()), ↑(↑() a) = ↑() a
def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [inst : ] [inst : ] (e : R ≃+* S) (s : ) :
{ x // x s } ≃+* { x // }

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

Equations
• One or more equations did not get rendered due to their size.

### Actions by Subsemirings #

These are just copies of the definitions about Submonoid starting from submonoid.mul_action. The only new result is subsemiring.module.

When R is commutative, algebra.of_subsemiring provides a stronger result than those found in this file, which uses the same scalar action.

instance Subsemiring.smul {R' : Type u_1} {α : Type u_2} [inst : ] [inst : SMul R' α] (S : ) :
SMul { x // x S } α

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

Equations
theorem Subsemiring.smul_def {R' : Type u_1} {α : Type u_2} [inst : ] [inst : SMul R' α] {S : } (g : { x // x S }) (m : α) :
g m = g m
instance Subsemiring.smulCommClass_left {R' : Type u_1} {α : Type u_2} {β : Type u_3} [inst : ] [inst : SMul R' β] [inst : SMul α β] [inst : SMulCommClass R' α β] (S : ) :
SMulCommClass { x // x S } α β
Equations
instance Subsemiring.smulCommClass_right {R' : Type u_1} {α : Type u_2} {β : Type u_3} [inst : ] [inst : SMul α β] [inst : SMul R' β] [inst : SMulCommClass α R' β] (S : ) :
SMulCommClass α { x // x S } β
Equations
instance Subsemiring.isScalarTower {R' : Type u_1} {α : Type u_2} {β : Type u_3} [inst : ] [inst : SMul α β] [inst : SMul R' α] [inst : SMul R' β] [inst : IsScalarTower R' α β] (S : ) :
IsScalarTower { x // x S } α β

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

Equations
instance Subsemiring.faithfulSMul {R' : Type u_1} {α : Type u_2} [inst : ] [inst : SMul R' α] [inst : FaithfulSMul R' α] (S : ) :
FaithfulSMul { x // x S } α
Equations

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

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.mulAction {R' : Type u_1} {α : Type u_2} [inst : Semiring R'] [inst : MulAction R' α] (S : ) :
MulAction { x // x S } α

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

Equations
instance Subsemiring.distribMulAction {R' : Type u_1} {α : Type u_2} [inst : Semiring R'] [inst : ] [inst : ] (S : ) :
DistribMulAction { x // x S } α

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

Equations
instance Subsemiring.mulDistribMulAction {R' : Type u_1} {α : Type u_2} [inst : Semiring R'] [inst : ] [inst : ] (S : ) :
MulDistribMulAction { x // x S } α

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

Equations
instance Subsemiring.mulActionWithZero {R' : Type u_1} {α : Type u_2} [inst : Semiring R'] [inst : Zero α] [inst : ] (S : ) :
MulActionWithZero { x // x S } α

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

Equations
instance Subsemiring.module {R' : Type u_1} {α : Type u_2} [inst : Semiring R'] [inst : ] [inst : Module R' α] (S : ) :
Module { x // x S } α

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

Equations

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

Equations
• One or more equations did not get rendered due to their size.
instance Subsemiring.center.smulCommClass_left {R' : Type u_1} [inst : Semiring R'] :
SMulCommClass { x // } R' R'

The center of a semiring acts commutatively on that semiring.

Equations
instance Subsemiring.center.smulCommClass_right {R' : Type u_1} [inst : Semiring R'] :
SMulCommClass R' { x // } R'

The center of a semiring acts commutatively on that semiring.

Equations
def Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [inst : Semiring R'] {s : Set R'} (hcomm : ∀ (a : R'), a s∀ (b : R'), b sa * b = b * a) :
CommSemiring { x // }

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

Equations
def posSubmonoid (R : Type u_1) [inst : ] :

Submonoid of positive elements of an ordered semiring.

Equations
• = { toSubsemigroup := { carrier := { x | 0 < x }, mul_mem' := (_ : ∀ {x y : R}, 0 < x0 < y0 < x * y) }, one_mem' := (_ : 0 < 1) }
@[simp]
theorem mem_posSubmonoid {R : Type u_1} [inst : ] (u : Rˣ) :
u 0 < u