Bundled non-unital subsemirings

We define bundled non-unital subsemirings and some standard constructions: CompleteLattice structure, subtype and inclusion ring homomorphisms, non-unital subsemiring map, comap and range (srange) of a NonUnitalRingHom etc.

class NonUnitalSubsemiringClass (S : Type u_1) (R : outParam (Type u)) [NonUnitalNonAssocSemiring R] [SetLike S R] extends AddSubmonoidClass : Prop

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

• add_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a + b ∈ s
• zero_mem : ∀ (s : S), 0 ∈ s
• mul_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s

theorem NonUnitalSubsemiringClass.mul_mem {S : Type u_1} {R : outParam (Type u)} : ∀ {inst : NonUnitalNonAssocSemiring R} {inst_1 : SetLike S R} [self : NonUnitalSubsemiringClass S R] {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s

@[instance 100]

instance NonUnitalSubsemiringClass.mulMemClass (S : Type u_1) (R : Type u) [NonUnitalNonAssocSemiring R] [SetLike S R] [h : NonUnitalSubsemiringClass S R] : MulMemClass S R

Equations

• ⋯ = ⋯

@[instance 75]

instance NonUnitalSubsemiringClass.toNonUnitalNonAssocSemiring {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : NonUnitalNonAssocSemiring ↥s

A non-unital subsemiring of a NonUnitalNonAssocSemiring inherits a NonUnitalNonAssocSemiring structure

Equations

• NonUnitalSubsemiringClass.toNonUnitalNonAssocSemiring s = Function.Injective.nonUnitalNonAssocSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

instance NonUnitalSubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) [NoZeroDivisors R] : NoZeroDivisors ↥s

Equations

• ⋯ = ⋯

def NonUnitalSubsemiringClass.subtype {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : ↥s →ₙ+* R

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

Equations

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

@[simp]

theorem NonUnitalSubsemiringClass.coeSubtype {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : ⇑(NonUnitalSubsemiringClass.subtype s) = Subtype.val

instance NonUnitalSubsemiringClass.toNonUnitalSemiring {S : Type v} (s : S) {R : Type u_1} [NonUnitalSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] : NonUnitalSemiring ↥s

A non-unital subsemiring of a NonUnitalSemiring is a NonUnitalSemiring.

Equations

• NonUnitalSubsemiringClass.toNonUnitalSemiring s = Function.Injective.nonUnitalSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

instance NonUnitalSubsemiringClass.toNonUnitalCommSemiring {S : Type v} (s : S) {R : Type u_1} [NonUnitalCommSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] : NonUnitalCommSemiring ↥s

A non-unital subsemiring of a NonUnitalCommSemiring is a NonUnitalCommSemiring.

Equations

• NonUnitalSubsemiringClass.toNonUnitalCommSemiring s = Function.Injective.nonUnitalCommSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

Note: currently, there are no ordered versions of non-unital rings.

structure NonUnitalSubsemiring (R : Type u) [NonUnitalNonAssocSemiring R] extends AddSubmonoid : Type u

A non-unital subsemiring of a non-unital semiring R is a subset s that is both an additive submonoid and a semigroup.

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

  The product of two elements of a subsemigroup belongs to the subsemigroup.

@[reducible]

abbrev NonUnitalSubsemiring.toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (self : NonUnitalSubsemiring R) : Subsemigroup R

Reinterpret a NonUnitalSubsemiring as a Subsemigroup.

Equations

• self.toSubsemigroup = { carrier := self.carrier, mul_mem' := ⋯ }

instance NonUnitalSubsemiring.instSetLike {R : Type u} [NonUnitalNonAssocSemiring R] : SetLike (NonUnitalSubsemiring R) R

Equations

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

instance NonUnitalSubsemiring.instNonUnitalSubsemiringClass {R : Type u} [NonUnitalNonAssocSemiring R] : NonUnitalSubsemiringClass (NonUnitalSubsemiring R) R

Equations

• ⋯ = ⋯

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

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

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

Two non-unital subsemirings are equal if they have the same elements.

def NonUnitalSubsemiring.copy {R : Type u} [NonUnitalNonAssocSemiring R] (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) : NonUnitalSubsemiring R

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

Equations

• S.copy s hs = { carrier := s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

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

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

theorem NonUnitalSubsemiring.toSubsemigroup_injective {R : Type u} [NonUnitalNonAssocSemiring R] : Function.Injective NonUnitalSubsemiring.toSubsemigroup

theorem NonUnitalSubsemiring.toSubsemigroup_strictMono {R : Type u} [NonUnitalNonAssocSemiring R] : StrictMono NonUnitalSubsemiring.toSubsemigroup

theorem NonUnitalSubsemiring.toSubsemigroup_mono {R : Type u} [NonUnitalNonAssocSemiring R] : Monotone NonUnitalSubsemiring.toSubsemigroup

theorem NonUnitalSubsemiring.toAddSubmonoid_injective {R : Type u} [NonUnitalNonAssocSemiring R] : Function.Injective NonUnitalSubsemiring.toAddSubmonoid

theorem NonUnitalSubsemiring.toAddSubmonoid_strictMono {R : Type u} [NonUnitalNonAssocSemiring R] : StrictMono NonUnitalSubsemiring.toAddSubmonoid

theorem NonUnitalSubsemiring.toAddSubmonoid_mono {R : Type u} [NonUnitalNonAssocSemiring R] : Monotone NonUnitalSubsemiring.toAddSubmonoid

def NonUnitalSubsemiring.mk' {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) (sg : Subsemigroup R) (hg : ↑sg = s) (sa : AddSubmonoid R) (ha : ↑sa = s) : NonUnitalSubsemiring R

Construct a NonUnitalSubsemiring R from a set s, a subsemigroup sg, and an additive submonoid sa such that x ∈ s ↔ x ∈ sg ↔ x ∈ sa.

Equations

• NonUnitalSubsemiring.mk' s sg hg sa ha = { carrier := s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.coe_mk' {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : ↑(NonUnitalSubsemiring.mk' s sg hg sa ha) = s

@[simp]

theorem NonUnitalSubsemiring.mem_mk' {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) {x : R} : x ∈ NonUnitalSubsemiring.mk' s sg hg sa ha ↔ x ∈ s

@[simp]

theorem NonUnitalSubsemiring.mk'_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (NonUnitalSubsemiring.mk' s sg hg sa ha).toSubsemigroup = sg

@[simp]

theorem NonUnitalSubsemiring.mk'_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (NonUnitalSubsemiring.mk' s sg hg sa ha).toAddSubmonoid = sa

@[simp]

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

@[simp]

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

@[simp]

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

Note: currently, there are no ordered versions of non-unital rings.

@[simp]

theorem NonUnitalSubsemiring.mem_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {x : R} : x ∈ s.toSubsemigroup ↔ x ∈ s

@[simp]

theorem NonUnitalSubsemiring.coe_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : ↑s.toSubsemigroup = ↑s

@[simp]

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

@[simp]

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

instance NonUnitalSubsemiring.instTop {R : Type u} [NonUnitalNonAssocSemiring R] : Top (NonUnitalSubsemiring R)

The non-unital subsemiring R of the non-unital semiring R.

Equations

• NonUnitalSubsemiring.instTop = { top := let __src := ⊤; let __src_1 := ⊤; { carrier := __src.carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ } }

@[simp]

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

@[simp]

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

@[simp]

theorem NonUnitalSubsemiring.topEquiv_symm_apply_coe {R : Type u} [NonUnitalNonAssocSemiring R] (x : R) : ↑(NonUnitalSubsemiring.topEquiv.symm x) = x

@[simp]

theorem NonUnitalSubsemiring.topEquiv_apply {R : Type u} [NonUnitalNonAssocSemiring R] (x : ↥⊤) : NonUnitalSubsemiring.topEquiv x = ↑x

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

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

Equations

• NonUnitalSubsemiring.topEquiv = { toEquiv := Subsemigroup.topEquiv.toEquiv, map_mul' := ⋯, map_add' := ⋯ }

def NonUnitalSubsemiring.comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring S) : NonUnitalSubsemiring R

The preimage of a non-unital subsemiring along a non-unital ring homomorphism is a non-unital subsemiring.

Equations

• NonUnitalSubsemiring.comap f s = { carrier := ⇑f ⁻¹' ↑s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.coe_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring S) (f : F) : ↑(NonUnitalSubsemiring.comap f s) = ⇑f ⁻¹' ↑s

@[simp]

theorem NonUnitalSubsemiring.mem_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {s : NonUnitalSubsemiring S} {f : F} {x : R} : x ∈ NonUnitalSubsemiring.comap f s ↔ f x ∈ s

theorem NonUnitalSubsemiring.comap_comap {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] {F : Type u_1} {G : Type u_2} [FunLike F R S] [NonUnitalRingHomClass F R S] [FunLike G S T] [NonUnitalRingHomClass G S T] (s : NonUnitalSubsemiring T) (g : G) (f : F) : NonUnitalSubsemiring.comap f (NonUnitalSubsemiring.comap g s) = NonUnitalSubsemiring.comap ((↑g).comp ↑f) s

def NonUnitalSubsemiring.map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring R) : NonUnitalSubsemiring S

The image of a non-unital subsemiring along a ring homomorphism is a non-unital subsemiring.

Equations

• NonUnitalSubsemiring.map f s = { carrier := ⇑f '' ↑s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.coe_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (s : NonUnitalSubsemiring R) : ↑(NonUnitalSubsemiring.map f s) = ⇑f '' ↑s

@[simp]

theorem NonUnitalSubsemiring.mem_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubsemiring R} {y : S} : y ∈ NonUnitalSubsemiring.map f s ↔ ∃ x ∈ s, f x = y

@[simp]

theorem NonUnitalSubsemiring.map_id {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : NonUnitalSubsemiring.map (NonUnitalRingHom.id R) s = s

theorem NonUnitalSubsemiring.map_map {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] {F : Type u_1} {G : Type u_2} [FunLike F R S] [NonUnitalRingHomClass F R S] [FunLike G S T] [NonUnitalRingHomClass G S T] (s : NonUnitalSubsemiring R) (g : G) (f : F) : NonUnitalSubsemiring.map (↑g) (NonUnitalSubsemiring.map (↑f) s) = NonUnitalSubsemiring.map ((↑g).comp ↑f) s

theorem NonUnitalSubsemiring.map_le_iff_le_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring S} : NonUnitalSubsemiring.map f s ≤ t ↔ s ≤ NonUnitalSubsemiring.comap f t

theorem NonUnitalSubsemiring.gc_map_comap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : GaloisConnection (NonUnitalSubsemiring.map f) (NonUnitalSubsemiring.comap f)

noncomputable def NonUnitalSubsemiring.equivMapOfInjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) : ↥s ≃+* ↥(NonUnitalSubsemiring.map f s)

A non-unital 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 NonUnitalSubsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) (x : ↥s) : ↑((s.equivMapOfInjective f hf) x) = f ↑x

def NonUnitalRingHom.srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : NonUnitalSubsemiring S

The range of a non-unital ring homomorphism is a non-unital subsemiring. See note [range copy pattern].

Equations

• NonUnitalRingHom.srange f = (NonUnitalSubsemiring.map ↑f ⊤).copy (Set.range ⇑f) ⋯

@[simp]

theorem NonUnitalRingHom.coe_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : ↑(NonUnitalRingHom.srange f) = Set.range ⇑f

@[simp]

theorem NonUnitalRingHom.mem_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {f : F} {y : S} : y ∈ NonUnitalRingHom.srange f ↔ ∃ (x : R), f x = y

theorem NonUnitalRingHom.srange_eq_map {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : NonUnitalRingHom.srange f = NonUnitalSubsemiring.map f ⊤

theorem NonUnitalRingHom.mem_srange_self {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) (x : R) : f x ∈ NonUnitalRingHom.srange f

theorem NonUnitalRingHom.map_srange {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (g : S →ₙ+* T) (f : R →ₙ+* S) : NonUnitalSubsemiring.map g (NonUnitalRingHom.srange f) = NonUnitalRingHom.srange (g.comp f)

instance NonUnitalRingHom.finite_srange {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] [Finite R] (f : F) : Finite ↥(NonUnitalRingHom.srange f)

The range of a morphism of non-unital semirings is finite if the domain is a finite.

Equations

• ⋯ = ⋯

instance NonUnitalSubsemiring.instBot {R : Type u} [NonUnitalNonAssocSemiring R] : Bot (NonUnitalSubsemiring R)

Equations

• NonUnitalSubsemiring.instBot = { bot := { carrier := {0}, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ } }

instance NonUnitalSubsemiring.instInhabited {R : Type u} [NonUnitalNonAssocSemiring R] : Inhabited (NonUnitalSubsemiring R)

Equations

• NonUnitalSubsemiring.instInhabited = { default := ⊥ }

theorem NonUnitalSubsemiring.coe_bot {R : Type u} [NonUnitalNonAssocSemiring R] : ↑⊥ = {0}

theorem NonUnitalSubsemiring.mem_bot {R : Type u} [NonUnitalNonAssocSemiring R] {x : R} : x ∈ ⊥ ↔ x = 0

instance NonUnitalSubsemiring.instInf {R : Type u} [NonUnitalNonAssocSemiring R] : Inf (NonUnitalSubsemiring R)

The inf of two non-unital subsemirings is their intersection.

Equations

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

@[simp]

theorem NonUnitalSubsemiring.coe_inf {R : Type u} [NonUnitalNonAssocSemiring R] (p : NonUnitalSubsemiring R) (p' : NonUnitalSubsemiring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

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

instance NonUnitalSubsemiring.instInfSet {R : Type u} [NonUnitalNonAssocSemiring R] : InfSet (NonUnitalSubsemiring R)

Equations

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

@[simp]

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

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

@[simp]

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

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

@[simp]

theorem NonUnitalSubsemiring.sInf_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set (NonUnitalSubsemiring R)) : (sInf s).toSubsemigroup = ⨅ t ∈ s, t.toSubsemigroup

@[simp]

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

instance NonUnitalSubsemiring.instCompleteLattice {R : Type u} [NonUnitalNonAssocSemiring R] : CompleteLattice (NonUnitalSubsemiring R)

Non-unital subsemirings of a non-unital semiring form a complete lattice.

Equations

• NonUnitalSubsemiring.instCompleteLattice = CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

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

def NonUnitalSubsemiring.center (R : Type u) [NonUnitalNonAssocSemiring R] : NonUnitalSubsemiring R

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

Equations

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

theorem NonUnitalSubsemiring.coe_center (R : Type u) [NonUnitalNonAssocSemiring R] : ↑(NonUnitalSubsemiring.center R) = Set.center R

@[simp]

theorem NonUnitalSubsemiring.center_toSubsemigroup (R : Type u) [NonUnitalNonAssocSemiring R] : (NonUnitalSubsemiring.center R).toSubsemigroup = Subsemigroup.center R

instance NonUnitalSubsemiring.center.instNonUnitalCommSemiring (R : Type u) [NonUnitalNonAssocSemiring R] : NonUnitalCommSemiring ↥(NonUnitalSubsemiring.center R)

The center is commutative and associative.

Equations

• NonUnitalSubsemiring.center.instNonUnitalCommSemiring R = NonUnitalCommSemiring.mk ⋯

theorem Set.mem_center_iff_addMonoidHom (R : Type u) [NonUnitalNonAssocSemiring R] (a : R) : a ∈ Set.center R ↔ AddMonoidHom.mulLeft a = AddMonoidHom.mulRight a ∧ AddMonoidHom.mul.compr₂ (AddMonoidHom.mulLeft a) = AddMonoidHom.mul.comp (AddMonoidHom.mulLeft a) ∧ AddMonoidHom.mul.comp (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulLeft a) ∧ AddMonoidHom.mul.compr₂ (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulRight a)

A point-free means of proving membership in the center, for a non-associative ring.

This can be helpful when working with types that have ext lemmas for R →+ R.

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

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

Equations

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

@[simp]

theorem NonUnitalSubsemiring.center_eq_top (R : Type u_1) [NonUnitalCommSemiring R] : NonUnitalSubsemiring.center R = ⊤

def NonUnitalSubsemiring.centralizer {R : Type u_1} [NonUnitalSemiring R] (s : Set R) : NonUnitalSubsemiring R

The centralizer of a set as non-unital subsemiring.

Equations

• NonUnitalSubsemiring.centralizer s = { carrier := s.centralizer, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

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

theorem NonUnitalSubsemiring.centralizer_toSubsemigroup {R : Type u_1} [NonUnitalSemiring R] (s : Set R) : (NonUnitalSubsemiring.centralizer s).toSubsemigroup = Subsemigroup.centralizer s

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

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

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

@[simp]

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

@[simp]

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

def NonUnitalSubsemiring.closure {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) : NonUnitalSubsemiring R

The NonUnitalSubsemiring generated by a set.

Equations

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

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

@[simp]

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

The non-unital subsemiring generated by a set includes the set.

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

@[simp]

theorem NonUnitalSubsemiring.closure_le {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {t : NonUnitalSubsemiring R} : NonUnitalSubsemiring.closure s ≤ t ↔ s ⊆ ↑t

A non-unital subsemiring S includes closure s if and only if it includes s.

theorem NonUnitalSubsemiring.closure_mono {R : Type u} [NonUnitalNonAssocSemiring R] ⦃s : Set R⦄ ⦃t : Set R⦄ (h : s ⊆ t) : NonUnitalSubsemiring.closure s ≤ NonUnitalSubsemiring.closure t

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

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

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

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

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

def Subsemigroup.nonUnitalSubsemiringClosure {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : NonUnitalSubsemiring R

The additive closure of a non-unital subsemigroup is a non-unital subsemiring.

Equations

• M.nonUnitalSubsemiringClosure = { toAddSubmonoid := AddSubmonoid.closure ↑M, mul_mem' := ⋯ }

theorem Subsemigroup.nonUnitalSubsemiringClosure_coe {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : ↑M.nonUnitalSubsemiringClosure = ↑(AddSubmonoid.closure ↑M)

theorem Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : M.nonUnitalSubsemiringClosure.toAddSubmonoid = AddSubmonoid.closure ↑M

theorem Subsemigroup.nonUnitalSubsemiringClosure_eq_closure {R : Type u} [NonUnitalNonAssocSemiring R] (M : Subsemigroup R) : M.nonUnitalSubsemiringClosure = NonUnitalSubsemiring.closure ↑M

The NonUnitalSubsemiring generated by a multiplicative subsemigroup coincides with the NonUnitalSubsemiring.closure of the subsemigroup itself .

@[simp]

theorem NonUnitalSubsemiring.closure_subsemigroup_closure {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) : NonUnitalSubsemiring.closure ↑(Subsemigroup.closure s) = NonUnitalSubsemiring.closure s

theorem NonUnitalSubsemiring.coe_closure_eq {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) : ↑(NonUnitalSubsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Subsemigroup.closure s))

The elements of the non-unital subsemiring closure of M are exactly the elements of the additive closure of a multiplicative subsemigroup M.

theorem NonUnitalSubsemiring.mem_closure_iff {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {x : R} : x ∈ NonUnitalSubsemiring.closure s ↔ x ∈ AddSubmonoid.closure ↑(Subsemigroup.closure s)

@[simp]

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

theorem NonUnitalSubsemiring.closure_induction {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {p : R → Prop} {x : R} (h : x ∈ NonUnitalSubsemiring.closure s) (mem : ∀ x ∈ s, p x) (zero : p 0) (add : ∀ (x y : R), p x → p y → p (x + y)) (mul : ∀ (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.

theorem NonUnitalSubsemiring.closure_induction₂ {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {p : R → R → Prop} {x : R} {y : R} (hx : x ∈ NonUnitalSubsemiring.closure s) (hy : y ∈ NonUnitalSubsemiring.closure s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (H0_left : ∀ (x : R), p 0 x) (H0_right : ∀ (x : R), p x 0) (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.

def NonUnitalSubsemiring.gi (R : Type u) [NonUnitalNonAssocSemiring R] : GaloisInsertion NonUnitalSubsemiring.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

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

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

Closure of a non-unital subsemiring S equals S.

@[simp]

theorem NonUnitalSubsemiring.closure_empty {R : Type u} [NonUnitalNonAssocSemiring R] : NonUnitalSubsemiring.closure ∅ = ⊥

@[simp]

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

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

theorem NonUnitalSubsemiring.closure_iUnion {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} (s : ι → Set R) : NonUnitalSubsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), NonUnitalSubsemiring.closure (s i)

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

theorem NonUnitalSubsemiring.map_sup {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring R) (f : F) : NonUnitalSubsemiring.map f (s ⊔ t) = NonUnitalSubsemiring.map f s ⊔ NonUnitalSubsemiring.map f t

theorem NonUnitalSubsemiring.map_iSup {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubsemiring R) : NonUnitalSubsemiring.map f (iSup s) = ⨆ (i : ι), NonUnitalSubsemiring.map f (s i)

theorem NonUnitalSubsemiring.map_inf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective ⇑f) : NonUnitalSubsemiring.map f (s ⊓ t) = NonUnitalSubsemiring.map f s ⊓ NonUnitalSubsemiring.map f t

theorem NonUnitalSubsemiring.map_iInf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} [Nonempty ι] (f : F) (hf : Function.Injective ⇑f) (s : ι → NonUnitalSubsemiring R) : NonUnitalSubsemiring.map f (iInf s) = ⨅ (i : ι), NonUnitalSubsemiring.map f (s i)

theorem NonUnitalSubsemiring.comap_inf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubsemiring S) (t : NonUnitalSubsemiring S) (f : F) : NonUnitalSubsemiring.comap f (s ⊓ t) = NonUnitalSubsemiring.comap f s ⊓ NonUnitalSubsemiring.comap f t

theorem NonUnitalSubsemiring.comap_iInf {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {ι : Sort u_2} (f : F) (s : ι → NonUnitalSubsemiring S) : NonUnitalSubsemiring.comap f (iInf s) = ⨅ (i : ι), NonUnitalSubsemiring.comap f (s i)

@[simp]

theorem NonUnitalSubsemiring.map_bot {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : NonUnitalSubsemiring.map f ⊥ = ⊥

@[simp]

theorem NonUnitalSubsemiring.comap_top {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] (f : F) : NonUnitalSubsemiring.comap f ⊤ = ⊤

def NonUnitalSubsemiring.prod {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (s : NonUnitalSubsemiring R) (t : NonUnitalSubsemiring S) : NonUnitalSubsemiring (R × S)

Given NonUnitalSubsemirings s, t of semirings R, S respectively, s.prod t is s × t as a non-unital subsemiring of R × S.

Equations

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

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

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

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

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

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

theorem NonUnitalSubsemiring.prod_top {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (s : NonUnitalSubsemiring R) : s.prod ⊤ = NonUnitalSubsemiring.comap (NonUnitalRingHom.fst R S) s

theorem NonUnitalSubsemiring.top_prod {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (s : NonUnitalSubsemiring S) : ⊤.prod s = NonUnitalSubsemiring.comap (NonUnitalRingHom.snd R S) s

@[simp]

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

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

Product of non-unital subsemirings is isomorphic to their product as semigroups.

Equations

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

theorem NonUnitalSubsemiring.mem_iSup_of_directed {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring R} (hS : Directed (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) {x : R} : x ∈ ⨆ (i : ι), S i ↔ ∃ (i : ι), x ∈ S i

theorem NonUnitalSubsemiring.coe_iSup_of_directed {R : Type u} [NonUnitalNonAssocSemiring R] {ι : Sort u_2} [hι : Nonempty ι] {S : ι → NonUnitalSubsemiring R} (hS : Directed (fun (x1 x2 : NonUnitalSubsemiring R) => x1 ≤ x2) S) : ↑(⨆ (i : ι), S i) = ⋃ (i : ι), ↑(S i)

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

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

theorem NonUnitalRingHom.eq_of_eqOn_stop {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] {f : F} {g : F} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g

def NonUnitalRingHom.codRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {S' : Type u_2} [SetLike S' S] [NonUnitalSubsemiringClass S' S] (f : F) (s : S') (h : ∀ (x : R), f x ∈ s) : R →ₙ+* ↥s

Restriction of a non-unital ring homomorphism to a non-unital subsemiring of the codomain.

Equations

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

def NonUnitalRingHom.srangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) : R →ₙ+* ↥(NonUnitalRingHom.srange f)

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

This is the bundled version of Set.rangeFactorization.

Equations

• NonUnitalRingHom.srangeRestrict f = NonUnitalRingHom.codRestrict f (NonUnitalRingHom.srange f) ⋯

@[simp]

theorem NonUnitalRingHom.coe_srangeRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (x : R) : ↑((NonUnitalRingHom.srangeRestrict f) x) = f x

theorem NonUnitalRingHom.srangeRestrict_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) : Function.Surjective ⇑(NonUnitalRingHom.srangeRestrict f)

theorem NonUnitalRingHom.srange_top_iff_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {f : F} : NonUnitalRingHom.srange f = ⊤ ↔ Function.Surjective ⇑f

@[simp]

theorem NonUnitalRingHom.srange_top_of_surjective {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (hf : Function.Surjective ⇑f) : NonUnitalRingHom.srange f = ⊤

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

def NonUnitalRingHom.eqSlocus {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (g : F) : NonUnitalSubsemiring R

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

Equations

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

theorem NonUnitalRingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {f : F} {g : F} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(NonUnitalSubsemiring.closure s)

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

theorem NonUnitalRingHom.eq_of_eqOn_sdense {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {s : Set R} (hs : NonUnitalSubsemiring.closure s = ⊤) {f : F} {g : F} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem NonUnitalRingHom.sclosure_preimage_le {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (s : Set S) : NonUnitalSubsemiring.closure (⇑f ⁻¹' s) ≤ NonUnitalSubsemiring.comap f (NonUnitalSubsemiring.closure s)

theorem NonUnitalRingHom.map_sclosure {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f : F) (s : Set R) : NonUnitalSubsemiring.map f (NonUnitalSubsemiring.closure s) = NonUnitalSubsemiring.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 NonUnitalSubsemiring.inclusion {R : Type u} [NonUnitalNonAssocSemiring R] {S : NonUnitalSubsemiring R} {T : NonUnitalSubsemiring R} (h : S ≤ T) : ↥S →ₙ+* ↥T

The non-unital ring homomorphism associated to an inclusion of non-unital subsemirings.

Equations

• NonUnitalSubsemiring.inclusion h = NonUnitalRingHom.codRestrict (NonUnitalSubsemiringClass.subtype S) T ⋯

@[simp]

theorem NonUnitalSubsemiring.srange_subtype {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : NonUnitalRingHom.srange (NonUnitalSubsemiringClass.subtype s) = s

@[simp]

theorem NonUnitalSubsemiring.range_fst {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] : NonUnitalRingHom.srange (NonUnitalRingHom.fst R S) = ⊤

@[simp]

theorem NonUnitalSubsemiring.range_snd {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] : NonUnitalRingHom.srange (NonUnitalRingHom.snd R S) = ⊤

def RingEquiv.nonUnitalSubsemiringCongr {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {t : NonUnitalSubsemiring R} (h : s = t) : ↥s ≃+* ↥t

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

Equations

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

def RingEquiv.sofLeftInverse' {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) : R ≃+* ↥(NonUnitalRingHom.srange f)

Restrict a non-unital ring homomorphism with a left inverse to a ring isomorphism to its NonUnitalRingHom.srange.

Equations

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

@[simp]

theorem RingEquiv.sofLeftInverse'_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) (x : R) : ↑((RingEquiv.sofLeftInverse' h) x) = f x

@[simp]

theorem RingEquiv.sofLeftInverse'_symm_apply {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] {F : Type u_1} [FunLike F R S] [NonUnitalRingHomClass F R S] {g : S → R} {f : F} (h : Function.LeftInverse g ⇑f) (x : ↥(NonUnitalRingHom.srange f)) : (RingEquiv.sofLeftInverse' h).symm x = g ↑x

@[simp]

theorem RingEquiv.nonUnitalSubsemiringMap_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) (x : ↑↑s.toAddSubmonoid) : ↑((e.nonUnitalSubsemiringMap s) x) = e ↑x

@[simp]

theorem RingEquiv.nonUnitalSubsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) (y : ↑(⇑↑e.toAddEquiv '' ↑s.toAddSubmonoid)) : ↑((e.nonUnitalSubsemiringMap s).symm y) = (↑e).symm ↑y

def RingEquiv.nonUnitalSubsemiringMap {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (e : R ≃+* S) (s : NonUnitalSubsemiring R) : ↥s ≃+* ↥(NonUnitalSubsemiring.map e.toNonUnitalRingHom s)

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

Equations

• e.nonUnitalSubsemiringMap s = { toEquiv := (e.toAddEquiv.addSubmonoidMap s.toAddSubmonoid).toEquiv, map_mul' := ⋯, map_add' := ⋯ }