Bundled subsemirings

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

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

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

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

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

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

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 NonAssocSemiring is in the 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 NonAssocSemiring indexed by a Finset is in the Subsemiring.

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

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

@[simp]

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

@[simp]

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

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.

@[simp]

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

@[simp]

theorem Subsemiring.comap_toSubmonoid {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring S) : (comap f s).toSubmonoid = { carrier := ⇑f ⁻¹' ↑s, mul_mem' := ⋯, one_mem' := ⋯ }

@[simp]

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

@[simp]

theorem Subsemiring.map_toSubmonoid {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : (map f s).toSubmonoid = { carrier := ⇑f '' ↑s, mul_mem' := ⋯, one_mem' := ⋯ }

@[simp]

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

@[simp]

theorem Subsemiring.map_id {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : 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) : map g (map f s) = 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} : map f s ≤ t ↔ s ≤ comap f t

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

A subsemiring is isomorphic to its image under an injective function

@[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].

@[simp]

theorem RingHom.rangeS_toSubmonoid {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : f.rangeS.toSubmonoid = { carrier := Set.range ⇑f, mul_mem' := ⋯, one_mem' := ⋯ }

@[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.

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

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

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

@[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.

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

@[simp]

theorem Subsemiring.center_toSubmonoid (R : Type u) [NonAssocSemiring R] : (center R).toSubmonoid = { carrier := (NonUnitalSubsemiring.center R).carrier, mul_mem' := ⋯, one_mem' := ⋯ }

@[simp]

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

@[reducible, inline]

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

The center is commutative and associative.

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

def Subsemiring.centerCongr {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) : ↥(center R) ≃+* ↥(center S)

The center of isomorphic (not necessarily associative) semirings are isomorphic.

@[simp]

theorem Subsemiring.centerCongr_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (r : ↥(Subsemigroup.center R)) : ↑((centerCongr e) r) = e ↑r

@[simp]

theorem Subsemiring.centerCongr_symm_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : ↥(Subsemigroup.center S)) : ↑((centerCongr e).symm s) = (↑e).symm ↑s

def Subsemiring.centerToMulOpposite {R : Type u} [NonAssocSemiring R] : ↥(center R) ≃+* ↥(center Rᵐᵒᵖ)

The center of a (not necessarily associative) semiring is isomorphic to the center of its opposite.

@[simp]

theorem Subsemiring.centerToMulOpposite_apply_coe {R : Type u} [NonAssocSemiring R] (r : ↥(Subsemigroup.center R)) : ↑(centerToMulOpposite r) = MulOpposite.op ↑r

@[simp]

theorem Subsemiring.centerToMulOpposite_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : ↥(Subsemigroup.center Rᵐᵒᵖ)) : ↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

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

The center is commutative.

theorem Subsemiring.mem_center_iff {R : Type u_1} [Semiring R] {z : R} : z ∈ 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 ∈ center R

@[simp]

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

def Subsemiring.centralizer {R : Type u_1} [Semiring R] (s : Set R) : Subsemiring R

The centralizer of a set as subsemiring.

@[simp]

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

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

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

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

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

@[simp]

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

@[simp]

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

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

@[simp]

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

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

The Subsemiring generated by a set.

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

@[simp]

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

The subsemiring generated by a set includes the set.

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

@[deprecated Subsemiring.notMem_of_notMem_closure (since := "2025-05-23")]

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

Alias of Subsemiring.notMem_of_notMem_closure.

@[simp]

theorem Subsemiring.closure_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} : 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 t : Set R⦄ (h : s ⊆ t) : closure s ≤ 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 ≤ closure s) : 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 ∈ 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) : map (↑f) K = 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) : comap (↑f) K = 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.

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

theorem Submonoid.subsemiringClosure_mem {R : Type u} [NonAssocSemiring R] (M : Submonoid R) {x : R} : x ∈ M.subsemiringClosure ↔ x ∈ AddSubmonoid.closure ↑M

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

@[simp]

theorem Submonoid.subsemiringClosure_toNonUnitalSubsemiring {R : Type u} [NonAssocSemiring R] (M : Submonoid R) : M.subsemiringClosure.toNonUnitalSubsemiring = NonUnitalSubsemiring.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) : closure ↑(Submonoid.closure s) = closure s

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

@[simp]

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

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

theorem Subsemiring.closure_induction₂ {R : Type u} [NonAssocSemiring R] {s : Set R} {p : (x y : R) → x ∈ closure s → y ∈ closure s → Prop} (mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯) (zero_left : ∀ (x : R) (hx : x ∈ closure s), p 0 x ⋯ hx) (zero_right : ∀ (x : R) (hx : x ∈ closure s), p x 0 hx ⋯) (one_left : ∀ (x : R) (hx : x ∈ closure s), p 1 x ⋯ hx) (one_right : ∀ (x : R) (hx : x ∈ closure s), p x 1 hx ⋯) (add_left : ∀ (x y z : R) (hx : x ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ 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 ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ 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 ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ 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 ∈ closure s) (hy : y ∈ closure s) (hz : z ∈ closure s), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯) {x y : R} (hx : x ∈ closure s) (hy : y ∈ 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 ∈ 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 closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

@[simp]

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

Closure of a subsemiring S equals S.

@[simp]

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

@[simp]

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

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

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

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

@[simp]

theorem Subsemiring.closure_singleton_natCast {R : Type u} [NonAssocSemiring R] (n : ℕ) : closure {↑n} = ⊥

@[simp]

theorem Subsemiring.closure_singleton_zero {R : Type u} [NonAssocSemiring R] : closure {0} = ⊥

@[simp]

theorem Subsemiring.closure_singleton_one {R : Type u} [NonAssocSemiring R] : closure {1} = ⊥

@[simp]

theorem Subsemiring.closure_insert_natCast {R : Type u} [NonAssocSemiring R] (n : ℕ) (s : Set R) : closure (insert (↑n) s) = closure s

@[simp]

theorem Subsemiring.closure_insert_zero {R : Type u} [NonAssocSemiring R] (s : Set R) : closure (insert 0 s) = closure s

@[simp]

theorem Subsemiring.closure_insert_one {R : Type u} [NonAssocSemiring R] (s : Set R) : closure (insert 1 s) = closure s

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

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

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

@[simp]

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

@[simp]

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

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 s₂ : Subsemiring R⦄ (hs : s₁ ≤ s₂) ⦃t₁ 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 ⊤ = 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 = 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.

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.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.

@[simp]

theorem RingHom.codRestrict_apply {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) (x : R) : ↑((f.codRestrict s h) x) = f x

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.

@[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.

@[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.

theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f 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 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 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 T : Subsemiring R} (h : S ≤ T) : ↥S →+* ↥T

The ring homomorphism associated to an inclusion of subsemirings.

theorem Subsemiring.inclusion_injective {R : Type u} [NonAssocSemiring R] {S T : Subsemiring R} (h : S ≤ T) : Function.Injective ⇑(inclusion h)

@[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 t : Subsemiring R} (h : s = t) : ↥s ≃+* ↥t

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

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.

@[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) : ↑((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) : (ofLeftInverseS h).symm x = g ↑x

def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) : ↥s ≃+* ↥(Subsemiring.map (↑e) s)

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

@[simp]

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

@[simp]

theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (x : ↥(Subsemiring.map e.toRingHom s)) : ↑((e.subsemiringMap s).symm x) = e.symm ↑x

Actions by Subsemirings

These are just copies of the definitions about Submonoid starting from Submonoid.mulAction. 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.

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

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

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.

@[instance 100]

instance Subsemiring.instFaithfulSMulSubtypeMem {M' : Type u_5} {α : Type u_6} [SMul M' α] {S' : Type u_7} [SetLike S' M'] (s : S') [FaithfulSMul M' α] : FaithfulSMul (↥s) α

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

@[instance 100]

instance Subsemiring.instSMulWithZeroSubtypeMem {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] {S' : Type u_5} [SetLike S' R'] [SubsemiringClass S' R'] (s : S') [Zero α] [SMulWithZero R' α] : SMulWithZero (↥s) α

instance Subsemiring.instSMulWithZeroSubtypeMem_1 {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.

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.

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.

@[instance 100]

instance Subsemiring.instModuleSubtypeMem {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddCommMonoid α] [Module R' α] {S' : Type u_5} [SetLike S' R'] [SubsemiringClass S' R'] (s : S') : Module (↥s) α

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.

@[instance 100]

instance Subsemiring.instMulActionWithZeroSubtypeMem {R' : Type u_1} {α : Type u_2} [Semiring R'] {S' : Type u_5} [SetLike S' R'] [SubsemiringClass S' R'] (s : S') [Zero α] [MulActionWithZero R' α] : MulActionWithZero (↥s) α

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.

@[instance 100]

instance Subsemiring.instModuleSubtypeMem_1 {R' : Type u_1} {α : Type u_2} [Semiring R'] [AddCommMonoid α] [Module R' α] {S' : Type u_5} [SetLike S' R'] [SubsemiringClass S' R'] (s : S') : Module (↥s) α

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.

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.

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

The center of a semiring acts commutatively on that semiring.

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

The center of a semiring acts commutatively on that semiring.

theorem Subsemiring.closure_le_centralizer_centralizer {R' : Type u_1} [Semiring R'] (s : Set R') : closure s ≤ centralizer ↑(centralizer s)

@[reducible, inline]

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

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

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