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 : outParam (Type u_2)) [AddMonoidWithOne R] [SetLike S R] extends AddSubmonoidClass , OneMemClass : Prop

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

theorem natCast_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] (n : ℕ) : ↑n ∈ s

@[deprecated natCast_mem]

theorem coe_nat_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] (n : ℕ) : ↑n ∈ s

Alias of natCast_mem.

theorem ofNat_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] [AddSubmonoidWithOneClass S R] (s : S) (n : ℕ) [n.AtLeastTwo] : OfNat.ofNat n ∈ s

@[instance 74]

instance AddSubmonoidWithOneClass.toAddMonoidWithOne {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] : AddMonoidWithOne ↥s

Equations

• AddSubmonoidWithOneClass.toAddMonoidWithOne s = AddMonoidWithOne.mk ⋯ ⋯

class SubsemiringClass (S : Type u_1) (R : outParam (Type u)) [NonAssocSemiring R] [SetLike S R] extends SubmonoidClass , AddSubmonoidClass : Prop

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

@[instance 100]

instance SubsemiringClass.addSubmonoidWithOneClass (S : Type u_1) (R : Type u) : ∀ {x : NonAssocSemiring R} [inst : SetLike S R] [h : SubsemiringClass S R], AddSubmonoidWithOneClass S R

Equations

• ⋯ = ⋯

@[instance 100]

instance SubsemiringClass.nonUnitalSubsemiringClass (S : Type u_1) (R : Type u) [NonAssocSemiring R] [SetLike S R] [SubsemiringClass S R] : NonUnitalSubsemiringClass S R

Equations

• ⋯ = ⋯

@[instance 75]

instance SubsemiringClass.toNonAssocSemiring {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) : NonAssocSemiring ↥s

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

Equations

• SubsemiringClass.toNonAssocSemiring s = Function.Injective.nonAssocSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance SubsemiringClass.nontrivial {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [Nontrivial R] : Nontrivial ↥s

Equations

• ⋯ = ⋯

instance SubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [NoZeroDivisors R] : NoZeroDivisors ↥s

Equations

• ⋯ = ⋯

def SubsemiringClass.subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) : ↥s →+* R

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

Equations

• SubsemiringClass.subtype s = { toFun := Subtype.val, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem SubsemiringClass.coe_subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) : ⇑(SubsemiringClass.subtype s) = Subtype.val

@[instance 75]

instance SubsemiringClass.toSemiring {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] : Semiring ↥s

A subsemiring of a Semiring is a Semiring.

Equations

• SubsemiringClass.toSemiring s = Function.Injective.semiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[simp]

theorem SubsemiringClass.coe_pow {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] (x : ↥s) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

instance SubsemiringClass.toCommSemiring {S : Type v} (s : S) {R : Type u_1} [CommSemiring R] [SetLike S R] [SubsemiringClass S R] : CommSemiring ↥s

A subsemiring of a CommSemiring is a CommSemiring.

Equations

• SubsemiringClass.toCommSemiring s = Function.Injective.commSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance SubsemiringClass.instCharZero {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [CharZero R] : CharZero ↥s

Equations

• ⋯ = ⋯

structure Subsemiring (R : Type u) [NonAssocSemiring R] extends Submonoid : Type u

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

• 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

  The sum of two elements of an additive subsemigroup belongs to the subsemigroup.

• zero_mem' : 0 ∈ self.carrier

  An additive submonoid contains 0.

@[reducible]

abbrev Subsemiring.toAddSubmonoid {R : Type u} [NonAssocSemiring R] (self : Subsemiring R) : AddSubmonoid R

Reinterpret a Subsemiring as an AddSubmonoid.

Equations

• self.toAddSubmonoid = { carrier := self.carrier, add_mem' := ⋯, zero_mem' := ⋯ }

instance Subsemiring.instSetLike {R : Type u} [NonAssocSemiring R] : SetLike (Subsemiring R) R

Equations

• Subsemiring.instSetLike = { coe := fun (s : Subsemiring R) => s.carrier, coe_injective' := ⋯ }

instance Subsemiring.instSubsemiringClass {R : Type u} [NonAssocSemiring R] : SubsemiringClass (Subsemiring R) R

Equations

• ⋯ = ⋯

@[simp]

theorem Subsemiring.mem_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ s.toSubmonoid ↔ x ∈ s

theorem Subsemiring.mem_carrier {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ s.carrier ↔ x ∈ s

theorem Subsemiring.ext_iff {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} : S = T ↔ ∀ (x : R), x ∈ S ↔ x ∈ T

theorem Subsemiring.ext {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} (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} [NonAssocSemiring 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' := ⋯ }

@[simp]

theorem Subsemiring.coe_copy {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = ↑S) : ↑(S.copy s hs) = s

theorem Subsemiring.copy_eq {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = ↑S) : S.copy s hs = S

theorem Subsemiring.toSubmonoid_injective {R : Type u} [NonAssocSemiring R] : Function.Injective Subsemiring.toSubmonoid

theorem Subsemiring.toSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] : StrictMono Subsemiring.toSubmonoid

theorem Subsemiring.toSubmonoid_mono {R : Type u} [NonAssocSemiring R] : Monotone Subsemiring.toSubmonoid

theorem Subsemiring.toAddSubmonoid_injective {R : Type u} [NonAssocSemiring R] : Function.Injective Subsemiring.toAddSubmonoid

theorem Subsemiring.toAddSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] : StrictMono Subsemiring.toAddSubmonoid

theorem Subsemiring.toAddSubmonoid_mono {R : Type u} [NonAssocSemiring R] : Monotone Subsemiring.toAddSubmonoid

def Subsemiring.mk' {R : Type u} [NonAssocSemiring R] (s : Set R) (sm : Submonoid R) (hm : ↑sm = s) (sa : AddSubmonoid 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' := ⋯ }

@[simp]

theorem Subsemiring.coe_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : ↑(Subsemiring.mk' s sm hm sa ha) = s

@[simp]

theorem Subsemiring.mem_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) {x : R} : x ∈ Subsemiring.mk' s sm hm sa ha ↔ x ∈ s

@[simp]

theorem Subsemiring.mk'_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (Subsemiring.mk' s sm hm sa ha).toSubmonoid = sm

@[simp]

theorem Subsemiring.mk'_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : ↑sm = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (Subsemiring.mk' s sm hm sa ha).toAddSubmonoid = sa

theorem Subsemiring.one_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : 1 ∈ s

A subsemiring contains the semiring's 1.

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

A subsemiring contains the semiring's 0.

theorem Subsemiring.mul_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} {y : R} : x ∈ s → y ∈ s → x * y ∈ s

A subsemiring is closed under multiplication.

theorem Subsemiring.add_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} {y : R} : x ∈ s → y ∈ s → x + y ∈ s

A subsemiring is closed under addition.

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

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

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

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

theorem Subsemiring.multiset_prod_mem {R : Type u_1} [CommSemiring R] (s : Subsemiring R) (m : Multiset R) : (∀ a ∈ m, a ∈ s) → m.prod ∈ 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} [NonAssocSemiring R] (s : Subsemiring R) (m : Multiset 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.

theorem Subsemiring.prod_mem {R : Type u_1} [CommSemiring R] (s : Subsemiring R) {ι : Type u_2} {t : Finset ι} {f : ι → R} (h : ∀ c ∈ t, f c ∈ s) : ∏ i ∈ t, 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} [NonAssocSemiring R] (s : Subsemiring R) {ι : Type u_1} {t : Finset ι} {f : ι → R} (h : ∀ c ∈ t, f c ∈ s) : ∑ i ∈ t, f i ∈ s

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

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

A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

Equations

• s.toNonAssocSemiring = SubsemiringClass.toNonAssocSemiring s

@[simp]

theorem Subsemiring.coe_one {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ↑1 = 1

@[simp]

theorem Subsemiring.coe_zero {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ↑0 = 0

@[simp]

theorem Subsemiring.coe_add {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) (x : ↥s) (y : ↥s) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem Subsemiring.coe_mul {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) (x : ↥s) (y : ↥s) : ↑(x * y) = ↑x * ↑y

instance Subsemiring.nontrivial {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) [Nontrivial R] : Nontrivial ↥s

Equations

• ⋯ = ⋯

theorem Subsemiring.pow_mem {R : Type u_1} [Semiring R] (s : Subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) : x ^ n ∈ s

instance Subsemiring.noZeroDivisors {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) [NoZeroDivisors R] : NoZeroDivisors ↥s

Equations

• ⋯ = ⋯

instance Subsemiring.toSemiring {R : Type u_1} [Semiring R] (s : Subsemiring R) : Semiring ↥s

A subsemiring of a Semiring is a Semiring.

Equations

• s.toSemiring = Semiring.mk ⋯ ⋯ ⋯ ⋯ Monoid.npow ⋯ ⋯

@[simp]

theorem Subsemiring.coe_pow {R : Type u_1} [Semiring R] (s : Subsemiring R) (x : ↥s) (n : ℕ) : ↑(x ^ n) = ↑x ^ n

instance Subsemiring.toCommSemiring {R : Type u_1} [CommSemiring R] (s : Subsemiring R) : CommSemiring ↥s

A subsemiring of a CommSemiring is a CommSemiring.

Equations

• s.toCommSemiring = CommSemiring.mk ⋯

def Subsemiring.subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ↥s →+* R

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

Equations

• s.subtype = { toFun := Subtype.val, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Subsemiring.coe_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ⇑s.subtype = Subtype.val

theorem Subsemiring.nsmul_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} (hx : x ∈ s) (n : ℕ) : n • x ∈ s

@[simp]

theorem Subsemiring.coe_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ↑s.toSubmonoid = ↑s

@[simp]

theorem Subsemiring.coe_carrier_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : s.carrier = ↑s

theorem Subsemiring.mem_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} : x ∈ s.toAddSubmonoid ↔ x ∈ s

theorem Subsemiring.coe_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : ↑s.toAddSubmonoid = ↑s

instance Subsemiring.instTop {R : Type u} [NonAssocSemiring R] : Top (Subsemiring R)

The subsemiring R of the semiring R.

Equations

• Subsemiring.instTop = { top := let __src := ⊤; let __src_1 := ⊤; { toSubmonoid := __src, add_mem' := ⋯, zero_mem' := ⋯ } }

@[simp]

theorem Subsemiring.mem_top {R : Type u} [NonAssocSemiring R] (x : R) : x ∈ ⊤

@[simp]

theorem Subsemiring.coe_top {R : Type u} [NonAssocSemiring R] : ↑⊤ = Set.univ

@[simp]

theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : R) : ↑(Subsemiring.topEquiv.symm r) = r

@[simp]

theorem Subsemiring.topEquiv_apply {R : Type u} [NonAssocSemiring R] (r : ↥⊤) : Subsemiring.topEquiv r = ↑r

def Subsemiring.topEquiv {R : Type u} [NonAssocSemiring R] : ↥⊤ ≃+* R

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

Equations

• Subsemiring.topEquiv = { toFun := fun (r : ↥⊤) => ↑r, invFun := fun (r : R) => ⟨r, ⋯⟩, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯ }

def Subsemiring.comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring 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' := ⋯ }

@[simp]

theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) (f : R →+* S) : ↑(Subsemiring.comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem Subsemiring.mem_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring S} {f : R →+* S} {x : R} : x ∈ Subsemiring.comap f s ↔ f x ∈ s

theorem Subsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring T) (g : S →+* T) (f : R →+* S) : Subsemiring.comap f (Subsemiring.comap g s) = Subsemiring.comap (g.comp f) s

def Subsemiring.map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring 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' := ⋯ }

@[simp]

theorem Subsemiring.coe_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : ↑(Subsemiring.map f s) = ⇑f '' ↑s

@[simp]

theorem Subsemiring.mem_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {y : S} : y ∈ Subsemiring.map f s ↔ ∃ x ∈ s, f x = y

@[simp]

theorem Subsemiring.map_id {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : Subsemiring.map (RingHom.id R) s = s

theorem Subsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring R) (g : S →+* T) (f : R →+* S) : Subsemiring.map g (Subsemiring.map f s) = Subsemiring.map (g.comp f) s

theorem Subsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {t : Subsemiring S} : Subsemiring.map f s ≤ t ↔ s ≤ Subsemiring.comap f t

theorem Subsemiring.gc_map_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : GaloisConnection (Subsemiring.map f) (Subsemiring.comap f)

noncomputable def Subsemiring.equivMapOfInjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring 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.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := ⋯, map_add' := ⋯ }

@[simp]

theorem Subsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective ⇑f) (x : ↥s) : ↑((s.equivMapOfInjective f hf) x) = f ↑x

def RingHom.rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : Subsemiring S

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

Equations

• f.rangeS = (Subsemiring.map f ⊤).copy (Set.range ⇑f) ⋯

@[simp]

theorem RingHom.coe_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : ↑f.rangeS = Set.range ⇑f

@[simp]

theorem RingHom.mem_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {y : S} : y ∈ f.rangeS ↔ ∃ (x : R), f x = y

theorem RingHom.rangeS_eq_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : f.rangeS = Subsemiring.map f ⊤

theorem RingHom.mem_rangeS_self {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) : f x ∈ f.rangeS

theorem RingHom.map_rangeS {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) : Subsemiring.map g f.rangeS = (g.comp f).rangeS

instance RingHom.fintypeRangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] [Fintype R] [DecidableEq S] (f : R →+* S) : Fintype ↥f.rangeS

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.fintypeRangeS = Set.fintypeRange ⇑f

instance Subsemiring.instBot {R : Type u} [NonAssocSemiring R] : Bot (Subsemiring R)

Equations

• Subsemiring.instBot = { bot := (Nat.castRingHom R).rangeS }

instance Subsemiring.instInhabited {R : Type u} [NonAssocSemiring R] : Inhabited (Subsemiring R)

Equations

• Subsemiring.instInhabited = { default := ⊥ }

theorem Subsemiring.coe_bot {R : Type u} [NonAssocSemiring R] : ↑⊥ = Set.range Nat.cast

theorem Subsemiring.mem_bot {R : Type u} [NonAssocSemiring R] {x : R} : x ∈ ⊥ ↔ ∃ (n : ℕ), ↑n = x

instance Subsemiring.instInf {R : Type u} [NonAssocSemiring R] : Inf (Subsemiring R)

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} [NonAssocSemiring R] (p : Subsemiring R) (p' : Subsemiring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem Subsemiring.mem_inf {R : Type u} [NonAssocSemiring R] {p : Subsemiring R} {p' : Subsemiring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

instance Subsemiring.instInfSet {R : Type u} [NonAssocSemiring R] : InfSet (Subsemiring R)

Equations

• Subsemiring.instInfSet = { sInf := fun (s : Set (Subsemiring R)) => Subsemiring.mk' (⋂ t ∈ s, ↑t) (⨅ t ∈ s, t.toSubmonoid) ⋯ (⨅ t ∈ s, t.toAddSubmonoid) ⋯ }

@[simp]

theorem Subsemiring.coe_sInf {R : Type u} [NonAssocSemiring R] (S : Set (Subsemiring R)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

theorem Subsemiring.mem_sInf {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

@[simp]

theorem Subsemiring.coe_iInf {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} {S : ι → Subsemiring R} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem Subsemiring.mem_iInf {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} {S : ι → Subsemiring R} {x : R} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem Subsemiring.sInf_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) : (sInf s).toSubmonoid = ⨅ t ∈ s, t.toSubmonoid

@[simp]

theorem Subsemiring.sInf_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) : (sInf s).toAddSubmonoid = ⨅ t ∈ s, t.toAddSubmonoid

instance Subsemiring.instCompleteLattice {R : Type u} [NonAssocSemiring R] : CompleteLattice (Subsemiring R)

Subsemirings of a semiring form a complete lattice.

Equations

• Subsemiring.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem Subsemiring.eq_top_iff' {R : Type u} [NonAssocSemiring R] (A : Subsemiring R) : A = ⊤ ↔ ∀ (x : R), x ∈ A

def Subsemiring.center (R : Type u) [NonAssocSemiring R] : Subsemiring R

The center of a non-associative semiring R is the set of elements that commute and associate with everything in R

Equations

• Subsemiring.center R = { carrier := (NonUnitalSubsemiring.center R).carrier, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }

theorem Subsemiring.coe_center (R : Type u) [NonAssocSemiring R] : ↑(Subsemiring.center R) = Set.center R

@[simp]

theorem Subsemiring.center_toSubmonoid (R : Type u) [NonAssocSemiring R] : (Subsemiring.center R).toSubmonoid = Submonoid.center R

@[reducible, inline]

abbrev Subsemiring.center.commSemiring' (R : Type u) [NonAssocSemiring R] : CommSemiring ↥(Subsemiring.center R)

The center is commutative and associative.

This is not an instance as it forms a non-defeq diamond with NonUnitalSubringClass.tNonUnitalring in the npow field.

Equations

• Subsemiring.center.commSemiring' R = CommSemiring.mk ⋯

instance Subsemiring.center.commSemiring {R : Type u_1} [Semiring R] : CommSemiring ↥(Subsemiring.center R)

The center is commutative.

Equations

• Subsemiring.center.commSemiring = CommSemiring.mk ⋯

theorem Subsemiring.mem_center_iff {R : Type u_1} [Semiring R] {z : R} : z ∈ Subsemiring.center R ↔ ∀ (g : R), g * z = z * g

instance Subsemiring.decidableMemCenter {R : Type u_1} [Semiring R] [DecidableEq R] [Fintype R] : DecidablePred fun (x : R) => x ∈ Subsemiring.center R

Equations

• Subsemiring.decidableMemCenter x = decidable_of_iff' (∀ (g : R), g * x = x * g) ⋯

@[simp]

theorem Subsemiring.center_eq_top (R : Type u_1) [CommSemiring R] : Subsemiring.center R = ⊤

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, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }

@[simp]

theorem Subsemiring.coe_centralizer {R : Type u_1} [Semiring R] (s : Set R) : ↑(Subsemiring.centralizer s) = s.centralizer

theorem Subsemiring.centralizer_toSubmonoid {R : Type u_1} [Semiring R] (s : Set R) : (Subsemiring.centralizer s).toSubmonoid = Submonoid.centralizer s

theorem Subsemiring.mem_centralizer_iff {R : Type u_1} [Semiring R] {s : Set R} {z : R} : z ∈ Subsemiring.centralizer s ↔ ∀ g ∈ s, g * z = z * g

theorem Subsemiring.center_le_centralizer {R : Type u_1} [Semiring R] (s : Set R) : Subsemiring.center R ≤ Subsemiring.centralizer s

theorem Subsemiring.centralizer_le {R : Type u_1} [Semiring R] (s : Set R) (t : Set R) (h : s ⊆ t) : Subsemiring.centralizer t ≤ Subsemiring.centralizer s

@[simp]

theorem Subsemiring.centralizer_eq_top_iff_subset {R : Type u_1} [Semiring R] {s : Set R} : Subsemiring.centralizer s = ⊤ ↔ s ⊆ ↑(Subsemiring.center R)

@[simp]

theorem Subsemiring.centralizer_univ {R : Type u_1} [Semiring R] : Subsemiring.centralizer Set.univ = Subsemiring.center R

theorem Subsemiring.le_centralizer_centralizer {R : Type u_1} [Semiring R] {s : Subsemiring R} : s ≤ Subsemiring.centralizer ↑(Subsemiring.centralizer ↑s)

@[simp]

theorem Subsemiring.centralizer_centralizer_centralizer {R : Type u_1} [Semiring R] {s : Set R} : Subsemiring.centralizer s.centralizer.centralizer = Subsemiring.centralizer s

def Subsemiring.closure {R : Type u} [NonAssocSemiring R] (s : Set R) : Subsemiring R

The Subsemiring generated by a set.

Equations

• Subsemiring.closure s = sInf {S : Subsemiring R | s ⊆ ↑S}

theorem Subsemiring.mem_closure {R : Type u} [NonAssocSemiring R] {x : R} {s : Set R} : x ∈ Subsemiring.closure s ↔ ∀ (S : Subsemiring R), s ⊆ ↑S → x ∈ S

@[simp]

theorem Subsemiring.subset_closure {R : Type u} [NonAssocSemiring R] {s : Set R} : s ⊆ ↑(Subsemiring.closure s)

The subsemiring generated by a set includes the set.

theorem Subsemiring.not_mem_of_not_mem_closure {R : Type u} [NonAssocSemiring R] {s : Set R} {P : R} (hP : P ∉ Subsemiring.closure s) : P ∉ s

@[simp]

theorem Subsemiring.closure_le {R : Type u} [NonAssocSemiring 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.

theorem Subsemiring.closure_mono {R : Type u} [NonAssocSemiring R] ⦃s : Set R⦄ ⦃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.

theorem Subsemiring.closure_eq_of_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} (h₁ : s ⊆ ↑t) (h₂ : t ≤ Subsemiring.closure s) : Subsemiring.closure s = t

theorem Subsemiring.mem_map_equiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R ≃+* S} {K : Subsemiring R} {x : S} : x ∈ Subsemiring.map (↑f) K ↔ f.symm x ∈ K

theorem Subsemiring.map_equiv_eq_comap_symm {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring R) : Subsemiring.map (↑f) K = Subsemiring.comap (↑f.symm) K

theorem Subsemiring.comap_equiv_eq_map_symm {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring S) : Subsemiring.comap (↑f) K = Subsemiring.map (↑f.symm) K

def Submonoid.subsemiringClosure {R : Type u} [NonAssocSemiring R] (M : Submonoid R) : Subsemiring R

The additive closure of a submonoid is a subsemiring.

Equations

• M.subsemiringClosure = { carrier := (AddSubmonoid.closure ↑M).carrier, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }

theorem Submonoid.subsemiringClosure_coe {R : Type u} [NonAssocSemiring R] (M : Submonoid R) : ↑M.subsemiringClosure = ↑(AddSubmonoid.closure ↑M)

theorem Submonoid.subsemiringClosure_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (M : Submonoid R) : M.subsemiringClosure.toAddSubmonoid = AddSubmonoid.closure ↑M

theorem Submonoid.subsemiringClosure_eq_closure {R : Type u} [NonAssocSemiring R] (M : Submonoid R) : M.subsemiringClosure = Subsemiring.closure ↑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} [NonAssocSemiring R] (s : Set R) : Subsemiring.closure ↑(Submonoid.closure s) = Subsemiring.closure s

theorem Subsemiring.coe_closure_eq {R : Type u} [NonAssocSemiring R] (s : Set R) : ↑(Subsemiring.closure s) = ↑(AddSubmonoid.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.

theorem Subsemiring.mem_closure_iff {R : Type u} [NonAssocSemiring R] {s : Set R} {x : R} : x ∈ Subsemiring.closure s ↔ x ∈ AddSubmonoid.closure ↑(Submonoid.closure s)

@[simp]

theorem Subsemiring.closure_addSubmonoid_closure {R : Type u} [NonAssocSemiring R] {s : Set R} : Subsemiring.closure ↑(AddSubmonoid.closure s) = Subsemiring.closure s

theorem Subsemiring.closure_induction {R : Type u} [NonAssocSemiring R] {s : Set R} {p : (x : R) → x ∈ Subsemiring.closure s → Prop} (mem : ∀ (x : R) (hx : x ∈ s), p x ⋯) (zero : p 0 ⋯) (one : p 1 ⋯) (add : ∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s), p x hx → p y hy → p (x + y) ⋯) (mul : ∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s), p x hx → p y hy → p (x * y) ⋯) {x : R} (hx : x ∈ Subsemiring.closure s) : p x hx

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.

@[deprecated Subsemiring.closure_induction]

theorem Subsemiring.closure_induction' {R : Type u} [NonAssocSemiring R] {s : Set R} {p : (x : R) → x ∈ Subsemiring.closure s → Prop} (mem : ∀ (x : R) (hx : x ∈ s), p x ⋯) (zero : p 0 ⋯) (one : p 1 ⋯) (add : ∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s), p x hx → p y hy → p (x + y) ⋯) (mul : ∀ (x y : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s), p x hx → p y hy → p (x * y) ⋯) {x : R} (hx : x ∈ Subsemiring.closure s) : p x hx

Alias of Subsemiring.closure_induction.

---

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} [NonAssocSemiring R] {s : Set R} {p : (x y : R) → x ∈ Subsemiring.closure s → y ∈ Subsemiring.closure s → Prop} (mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯) (zero_left : ∀ (x : R) (hx : x ∈ Subsemiring.closure s), p 0 x ⋯ hx) (zero_right : ∀ (x : R) (hx : x ∈ Subsemiring.closure s), p x 0 hx ⋯) (one_left : ∀ (x : R) (hx : x ∈ Subsemiring.closure s), p 1 x ⋯ hx) (one_right : ∀ (x : R) (hx : x ∈ Subsemiring.closure s), p x 1 hx ⋯) (add_left : ∀ (x y z : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) (hz : z ∈ Subsemiring.closure s), p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz) (add_right : ∀ (x y z : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) (hz : z ∈ Subsemiring.closure s), p x y hx hy → p x z hx hz → p x (y + z) hx ⋯) (mul_left : ∀ (x y z : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) (hz : z ∈ Subsemiring.closure s), p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz) (mul_right : ∀ (x y z : R) (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) (hz : z ∈ Subsemiring.closure s), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯) {x : R} {y : R} (hx : x ∈ Subsemiring.closure s) (hy : y ∈ Subsemiring.closure s) : p x y hx hy

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

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 ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x

def Subsemiring.gi (R : Type u) [NonAssocSemiring R] : GaloisInsertion Subsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• Subsemiring.gi R = { choice := fun (s : Set R) (x : ↑(Subsemiring.closure s) ≤ s) => Subsemiring.closure s, gc := ⋯, le_l_u := ⋯, choice_eq := ⋯ }

theorem Subsemiring.closure_eq {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : Subsemiring.closure ↑s = s

Closure of a subsemiring S equals S.

@[simp]

theorem Subsemiring.closure_empty {R : Type u} [NonAssocSemiring R] : Subsemiring.closure ∅ = ⊥

@[simp]

theorem Subsemiring.closure_univ {R : Type u} [NonAssocSemiring R] : Subsemiring.closure Set.univ = ⊤

theorem Subsemiring.closure_union {R : Type u} [NonAssocSemiring R] (s : Set R) (t : Set R) : Subsemiring.closure (s ∪ t) = Subsemiring.closure s ⊔ Subsemiring.closure t

theorem Subsemiring.closure_iUnion {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} (s : ι → Set R) : Subsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Subsemiring.closure (s i)

theorem Subsemiring.closure_sUnion {R : Type u} [NonAssocSemiring R] (s : Set (Set R)) : Subsemiring.closure (⋃₀ s) = ⨆ t ∈ s, Subsemiring.closure t

theorem Subsemiring.map_sup {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring R) (f : R →+* S) : Subsemiring.map f (s ⊔ t) = Subsemiring.map f s ⊔ Subsemiring.map f t

theorem Subsemiring.map_iSup {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring R) : Subsemiring.map f (iSup s) = ⨆ (i : ι), Subsemiring.map f (s i)

theorem Subsemiring.map_inf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring R) (f : R →+* S) (hf : Function.Injective ⇑f) : Subsemiring.map f (s ⊓ t) = Subsemiring.map f s ⊓ Subsemiring.map f t

theorem Subsemiring.map_iInf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} [Nonempty ι] (f : R →+* S) (hf : Function.Injective ⇑f) (s : ι → Subsemiring R) : Subsemiring.map f (iInf s) = ⨅ (i : ι), Subsemiring.map f (s i)

theorem Subsemiring.comap_inf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) (t : Subsemiring S) (f : R →+* S) : Subsemiring.comap f (s ⊓ t) = Subsemiring.comap f s ⊓ Subsemiring.comap f t

theorem Subsemiring.comap_iInf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring S) : Subsemiring.comap f (iInf s) = ⨅ (i : ι), Subsemiring.comap f (s i)

@[simp]

theorem Subsemiring.map_bot {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : Subsemiring.map f ⊥ = ⊥

@[simp]

theorem Subsemiring.comap_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : Subsemiring.comap f ⊤ = ⊤

def Subsemiring.prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring 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' := ⋯ }

theorem Subsemiring.coe_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) : ↑(s.prod t) = ↑s ×ˢ ↑t

theorem Subsemiring.mem_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring R} {t : Subsemiring S} {p : R × S} : p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t

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

theorem Subsemiring.prod_mono_right {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) : Monotone fun (t : Subsemiring S) => s.prod t

theorem Subsemiring.prod_mono_left {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (t : Subsemiring S) : Monotone fun (s : Subsemiring R) => s.prod t

theorem Subsemiring.prod_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) : s.prod ⊤ = Subsemiring.comap (RingHom.fst R S) s

theorem Subsemiring.top_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) : ⊤.prod s = Subsemiring.comap (RingHom.snd R S) s

@[simp]

theorem Subsemiring.top_prod_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] : ⊤.prod ⊤ = ⊤

def Subsemiring.prodEquiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) : ↥(s.prod t) ≃+* ↥s × ↥t

Product of subsemirings is isomorphic to their product as monoids.

Equations

• s.prodEquiv t = { toEquiv := Equiv.Set.prod ↑s ↑t, map_mul' := ⋯, map_add' := ⋯ }

theorem Subsemiring.mem_iSup_of_directed {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subsemiring R} (hS : Directed (fun (x1 x2 : Subsemiring R) => x1 ≤ x2) S) {x : R} : x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

theorem Subsemiring.coe_iSup_of_directed {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ι → Subsemiring R} (hS : Directed (fun (x1 x2 : Subsemiring R) => x1 ≤ x2) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

theorem Subsemiring.mem_sSup_of_directedOn {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subsemiring R) => x1 ≤ x2) S) {x : R} : x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s

theorem Subsemiring.coe_sSup_of_directedOn {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} (Sne : S.Nonempty) (hS : DirectedOn (fun (x1 x2 : Subsemiring R) => x1 ≤ x2) S) : ↑(sSup S) = ⋃ s ∈ S, ↑s

def RingHom.domRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) (s : σR) : ↥s →+* S

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

Equations

• f.domRestrict s = f.comp (SubsemiringClass.subtype s)

@[simp]

theorem RingHom.restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) {s : σR} (x : ↥s) : (f.domRestrict s) x = f ↑x

def RingHom.codRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σS : Type u_2} [SetLike σS S] [SubsemiringClass σS S] (f : R →+* S) (s : σS) (h : ∀ (x : R), f x ∈ s) : R →+* ↥s

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

Equations

• f.codRestrict s h = { toFun := fun (n : R) => ⟨f n, ⋯⟩, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

def RingHom.restrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) : ↥s' →+* ↥s

The ring homomorphism from the preimage of s to s.

Equations

• f.restrict s' s h = (f.domRestrict s').codRestrict s ⋯

@[simp]

theorem RingHom.coe_restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) (x : ↥s') : ↑((f.restrict s' s h) x) = f ↑x

@[simp]

theorem RingHom.comp_restrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : ∀ x ∈ s', f x ∈ s) : (SubsemiringClass.subtype s).comp (f.restrict s' s h) = f.comp (SubsemiringClass.subtype s')

def RingHom.rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : R →+* ↥f.rangeS

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

This is the bundled version of Set.rangeFactorization.

Equations

• f.rangeSRestrict = f.codRestrict f.rangeS ⋯

@[simp]

theorem RingHom.coe_rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) : ↑(f.rangeSRestrict x) = f x

theorem RingHom.rangeSRestrict_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : Function.Surjective ⇑f.rangeSRestrict

theorem RingHom.rangeS_top_iff_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} : f.rangeS = ⊤ ↔ Function.Surjective ⇑f

@[simp]

theorem RingHom.rangeS_top_of_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (hf : Function.Surjective ⇑f) : f.rangeS = ⊤

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

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

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

Equations

• f.eqLocusS g = { carrier := {x : R | f x = g x}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯ }

@[simp]

theorem RingHom.eqLocusS_same {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : f.eqLocusS f = ⊤

theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {g : R →+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subsemiring.closure s)

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} [NonAssocSemiring R] [NonAssocSemiring S] {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} [NonAssocSemiring R] [NonAssocSemiring S] {s : Set R} (hs : Subsemiring.closure s = ⊤) {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} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Set S) : Subsemiring.closure (⇑f ⁻¹' s) ≤ Subsemiring.comap f (Subsemiring.closure s)

theorem RingHom.map_closureS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring 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.

def Subsemiring.inclusion {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} (h : S ≤ T) : ↥S →+* ↥T

The ring homomorphism associated to an inclusion of subsemirings.

Equations

• Subsemiring.inclusion h = S.subtype.codRestrict T ⋯

@[simp]

theorem Subsemiring.rangeS_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : s.subtype.rangeS = s

@[simp]

theorem Subsemiring.range_fst {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] : (RingHom.fst R S).rangeS = ⊤

@[simp]

theorem Subsemiring.range_snd {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] : (RingHom.snd R S).rangeS = ⊤

@[simp]

theorem Subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) : s.prod ⊥ ⊔ ⊥.prod t = s.prod t

def RingEquiv.subsemiringCongr {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {t : Subsemiring R} (h : s = t) : ↥s ≃+* ↥t

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

Equations

• RingEquiv.subsemiringCongr h = { toEquiv := Equiv.setCongr ⋯, map_mul' := ⋯, map_add' := ⋯ }

def RingEquiv.ofLeftInverseS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) : R ≃+* ↥f.rangeS

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} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : R) : ↑((RingEquiv.ofLeftInverseS h) x) = f x

@[simp]

theorem RingEquiv.ofLeftInverseS_symm_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : ↥f.rangeS) : (RingEquiv.ofLeftInverseS h).symm x = g ↑x

@[simp]

theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (y : ↑(⇑↑e.toAddEquiv '' ↑s.toAddSubmonoid)) : ↑((e.subsemiringMap s).symm y) = (↑e).symm ↑y

@[simp]

theorem RingEquiv.subsemiringMap_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (x : ↑↑s.toAddSubmonoid) : ↑((e.subsemiringMap s) x) = e ↑x

def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) : ↥s ≃+* ↥(Subsemiring.map e.toRingHom 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.subsemiringMap s = { toEquiv := (e.toAddEquiv.addSubmonoidMap s.toAddSubmonoid).toEquiv, map_mul' := ⋯, map_add' := ⋯ }

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.ofSubsemiring 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} [NonAssocSemiring R'] [SMul R' α] (S : Subsemiring R') : SMul (↥S) α

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

Equations

• S.smul = S.smul

theorem Subsemiring.smul_def {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [SMul R' α] {S : Subsemiring R'} (g : ↥S) (m : α) : g • m = ↑g • m

instance Subsemiring.smulCommClass_left {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul R' β] [SMul α β] [SMulCommClass R' α β] (S : Subsemiring R') : SMulCommClass (↥S) α β

Equations

• ⋯ = ⋯

instance Subsemiring.smulCommClass_right {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul α β] [SMul R' β] [SMulCommClass α R' β] (S : Subsemiring R') : SMulCommClass α (↥S) β

Equations

• ⋯ = ⋯

instance Subsemiring.isScalarTower {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul α β] [SMul R' α] [SMul R' β] [IsScalarTower R' α β] (S : Subsemiring R') : IsScalarTower (↥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} [NonAssocSemiring R'] [SMul R' α] [FaithfulSMul R' α] (S : Subsemiring R') : FaithfulSMul (↥S) α

Equations

• ⋯ = ⋯

instance Subsemiring.instSMulWithZeroSubtypeMem {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [Zero α] [SMulWithZero R' α] (S : Subsemiring R') : SMulWithZero (↥S) α

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

Equations

• S.instSMulWithZeroSubtypeMem = SMulWithZero.compHom α ↑S.subtype.toMonoidWithZeroHom

instance Subsemiring.mulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [MulAction R' α] (S : Subsemiring R') : MulAction (↥S) α

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

Equations

• S.mulAction = S.mulAction

instance Subsemiring.distribMulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddMonoid α] [DistribMulAction R' α] (S : Subsemiring R') : DistribMulAction (↥S) α

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

Equations

• S.distribMulAction = S.distribMulAction

instance Subsemiring.mulDistribMulAction {R' : Type u_1} {α : Type u_2} [Semiring R'] [Monoid α] [MulDistribMulAction R' α] (S : Subsemiring R') : MulDistribMulAction (↥S) α

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

Equations

• S.mulDistribMulAction = S.mulDistribMulAction

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

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

Equations

• S.mulActionWithZero = MulActionWithZero.compHom α S.subtype.toMonoidWithZeroHom

instance Subsemiring.module {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddCommMonoid 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 = Module.compHom α S.subtype

instance Subsemiring.instMulSemiringActionSubtypeMem {R' : Type u_1} {α : Type u_2} [Semiring R'] [Semiring α] [MulSemiringAction R' α] (S : Subsemiring R') : MulSemiringAction (↥S) α

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

Equations

• S.instMulSemiringActionSubtypeMem = S.mulSemiringAction

instance Subsemiring.center.smulCommClass_left {R' : Type u_1} [Semiring R'] : SMulCommClass (↥(Subsemiring.center R')) R' R'

The center of a semiring acts commutatively on that semiring.

Equations

• ⋯ = ⋯

instance Subsemiring.center.smulCommClass_right {R' : Type u_1} [Semiring R'] : SMulCommClass R' (↥(Subsemiring.center R')) R'

The center of a semiring acts commutatively on that semiring.

Equations

• ⋯ = ⋯

def Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [Semiring R'] {s : Set R'} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : CommSemiring ↥(Subsemiring.closure s)

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

Equations

• Subsemiring.closureCommSemiringOfComm hcomm = CommSemiring.mk ⋯

theorem Subsemiring.map_comap_eq {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (t : Subsemiring S) : Subsemiring.map f (Subsemiring.comap f t) = t ⊓ f.rangeS

theorem Subsemiring.map_comap_eq_self {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {t : Subsemiring S} (h : t ≤ f.rangeS) : Subsemiring.map f (Subsemiring.comap f t) = t

theorem Subsemiring.map_comap_eq_self_of_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (t : Subsemiring S) : Subsemiring.map f (Subsemiring.comap f t) = t

theorem Subsemiring.comap_map_eq_self_of_injective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} (hf : Function.Injective ⇑f) (s : Subsemiring R) : Subsemiring.comap f (Subsemiring.map f s) = s