Bundled subsemirings #

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

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

CharZero ↥s

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

StrictMono toSubmonoid

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

Monotone toSubmonoid

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

StrictMono toAddSubmonoid

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

Monotone toAddSubmonoid

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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.

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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.

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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.

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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.

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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.

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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.

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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' := ⋯ }

Instances For

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@[simp]

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

↑(topEquiv.symm r) = r

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@[simp]

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

topEquiv r = ↑r

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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' := ⋯ }

Instances For

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@[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

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@[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' := ⋯ }

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@[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

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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

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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' := ⋯ }

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@[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' := ⋯ }

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@[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

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@[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

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@[simp]

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

map (RingHom.id R) s = s

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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

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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

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

GaloisConnection (map f) (comap f)

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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

Equations

s.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := ⋯, map_add' := ⋯ }

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@[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

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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) ⋯

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@[simp]

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

↑f.rangeS = Set.range ⇑f

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@[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' := ⋯ }

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@[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

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

f.rangeS = Subsemiring.map f ⊤

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

f x ∈ f.rangeS

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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

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theorem RingHom.rangeS_eq_top {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} :

f.rangeS = ⊤ ↔ Function.Surjective ⇑f

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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

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

Bot (Subsemiring R)

Equations

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

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

Inhabited (Subsemiring R)

Equations

Subsemiring.instInhabited = { default := ⊥ }

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theorem Subsemiring.coe_bot {R : Type u} [NonAssocSemiring R] :

↑⊥ = Set.range Nat.cast

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theorem Subsemiring.mem_bot {R : Type u} [NonAssocSemiring R] {x : R} :

x ∈ ⊥ ↔ ∃ (n : ℕ), ↑n = x

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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) ⋯ }

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@[simp]

theorem Subsemiring.coe_sInf {R : Type u} [NonAssocSemiring R] (S : Set (Subsemiring R)) :

↑(sInf S) = ⋂ s ∈ S, ↑s

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@[simp]

theorem Subsemiring.mem_sInf {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} {x : R} :

x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

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@[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)

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@[simp]

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

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@[simp]

theorem Subsemiring.sInf_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) :

(sInf s).toSubmonoid = ⨅ t ∈ s, t.toSubmonoid

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@[simp]

theorem Subsemiring.sInf_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) :

(sInf s).toAddSubmonoid = ⨅ t ∈ s, t.toAddSubmonoid

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

CompleteLattice (Subsemiring R)

Subsemirings of a semiring form a complete lattice.

Equations

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

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theorem Subsemiring.eq_top_iff' {R : Type u} [NonAssocSemiring R] (A : Subsemiring R) :

A = ⊤ ↔ ∀ (x : R), x ∈ A

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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' := ⋯ }

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@[simp]

theorem Subsemiring.center_toSubmonoid (R : Type u) [NonAssocSemiring R] :

(center R).toSubmonoid = { carrier := (NonUnitalSubsemiring.center R).carrier, mul_mem' := ⋯, one_mem' := ⋯ }

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@[simp]

theorem Subsemiring.coe_center (R : Type u) [NonAssocSemiring R] :

↑(center R) = (NonUnitalSubsemiring.center R).carrier

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@[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.

Equations

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

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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.

Equations

Subsemiring.centerCongr e = NonUnitalSubsemiring.centerCongr e

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@[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

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@[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

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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.

Equations

Subsemiring.centerToMulOpposite = NonUnitalSubsemiring.centerToMulOpposite

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@[simp]

theorem Subsemiring.centerToMulOpposite_apply_coe {R : Type u} [NonAssocSemiring R] (r : ↥(Subsemigroup.center R)) :

↑(centerToMulOpposite r) = MulOpposite.op ↑r

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@[simp]

theorem Subsemiring.centerToMulOpposite_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : ↥(Subsemigroup.center Rᵐᵒᵖ)) :

↑(centerToMulOpposite.symm r) = MulOpposite.unop ↑r

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instance Subsemiring.center.commSemiring {R : Type u_1} [Semiring R] :

CommSemiring ↥(center R)

The center is commutative.

Equations

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

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

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

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instance Subsemiring.decidableMemCenter {R : Type u_1} [Semiring R] [DecidableEq R] [Fintype R] :

DecidablePred fun (x : R) => x ∈ center R

Equations

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

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@[simp]

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

center R = ⊤

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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' := ⋯ }

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@[simp]

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

↑(centralizer s) = s.centralizer

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

(centralizer s).toSubmonoid = Submonoid.centralizer s

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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

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

center R ≤ centralizer s

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

centralizer t ≤ centralizer s

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@[simp]

theorem Subsemiring.centralizer_eq_top_iff_subset {R : Type u_1} [Semiring R] {s : Set R} :

centralizer s = ⊤ ↔ s ⊆ ↑(center R)

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@[simp]

theorem Subsemiring.centralizer_univ {R : Type u_1} [Semiring R] :

centralizer Set.univ = center R

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theorem Subsemiring.le_centralizer_centralizer {R : Type u_1} [Semiring R] {s : Subsemiring R} :

s ≤ centralizer ↑(centralizer ↑s)

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@[simp]

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

centralizer s.centralizer.centralizer = centralizer s

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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}

Instances For

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theorem Subsemiring.mem_closure {R : Type u} [NonAssocSemiring R] {x : R} {s : Set R} :

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

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@[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.

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theorem Subsemiring.mem_closure_of_mem {R : Type u} [NonAssocSemiring R] {s : Set R} {x : R} (hx : x ∈ s) :

x ∈ closure s

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

P ∉ s

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@[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.

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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.

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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

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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

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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

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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

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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' := ⋯ }

Instances For

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theorem Submonoid.subsemiringClosure_coe {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

↑M.subsemiringClosure = ↑(AddSubmonoid.closure ↑M)

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theorem Submonoid.subsemiringClosure_mem {R : Type u} [NonAssocSemiring R] (M : Submonoid R) {x : R} :

x ∈ M.subsemiringClosure ↔ x ∈ AddSubmonoid.closure ↑M

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theorem Submonoid.subsemiringClosure_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

M.subsemiringClosure.toAddSubmonoid = AddSubmonoid.closure ↑M

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@[simp]

theorem Submonoid.subsemiringClosure_toNonUnitalSubsemiring {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :

M.subsemiringClosure.toNonUnitalSubsemiring = NonUnitalSubsemiring.closure ↑M

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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 .

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@[simp]

theorem Subsemiring.closure_submonoid_closure {R : Type u} [NonAssocSemiring R] (s : Set R) :

closure ↑(Submonoid.closure s) = closure s

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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.

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theorem Subsemiring.mem_closure_iff {R : Type u} [NonAssocSemiring R] {s : Set R} {x : R} :

x ∈ closure s ↔ x ∈ AddSubmonoid.closure ↑(Submonoid.closure s)

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@[simp]

theorem Subsemiring.closure_addSubmonoid_closure {R : Type u} [NonAssocSemiring R] {s : Set R} :

closure ↑(AddSubmonoid.closure s) = closure s

source

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.

source

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.

source

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

source

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

GaloisInsertion 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 := ⋯ }

Instances For

source

@[simp]

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

closure ↑s = s

Closure of a subsemiring S equals S.

source

@[simp]

theorem Subsemiring.closure_empty {R : Type u} [NonAssocSemiring R] :

closure ∅ = ⊥

source

@[simp]

theorem Subsemiring.closure_univ {R : Type u} [NonAssocSemiring R] :

closure Set.univ = ⊤

source

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

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

source

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)

source

theorem Subsemiring.closure_sUnion {R : Type u} [NonAssocSemiring R] (s : Set (Set R)) :

closure (⋃₀ s) = ⨆ t ∈ s, closure t

source

@[simp]

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

closure {↑n} = ⊥

source

@[simp]

theorem Subsemiring.closure_singleton_zero {R : Type u} [NonAssocSemiring R] :

closure {0} = ⊥

source

@[simp]

theorem Subsemiring.closure_singleton_one {R : Type u} [NonAssocSemiring R] :

closure {1} = ⊥

source

@[simp]

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

closure (insert (↑n) s) = closure s

source

@[simp]

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

closure (insert 0 s) = closure s

source

@[simp]

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

closure (insert 1 s) = closure s

source

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

source

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)

source

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

source

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)

source

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

source

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)

source

@[simp]

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

map f ⊥ = ⊥

source

@[simp]

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

comap f ⊤ = ⊤

source

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' := ⋯ }

Instances For

source

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

source

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

source

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

source

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

source

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

source

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

source

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

source

@[simp]

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

⊤.prod ⊤ = ⊤

source

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' := ⋯ }

Instances For

source

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

source

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)

source

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

source

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

source

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' := ⋯ }

Instances For

source

@[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

source

theorem RingHom.injective_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} :

Function.Injective ⇑(f.codRestrict s h) ↔ Function.Injective ⇑f

source

theorem RingHom.rangeS_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} :

(f.codRestrict s h).rangeS = Subsemiring.comap (SubsemiringClass.subtype s) f.rangeS

source

theorem RingHom.surjective_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} :

Function.Surjective ⇑(f.codRestrict s h) ↔ f.rangeS = Subsemiring.ofClass s

source

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 ⋯

Instances For

source

@[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

source

@[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')

source

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 ⋯

Instances For

source

@[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

source

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

Function.Surjective ⇑f.rangeSRestrict

source

theorem RingHom.rangeS_top_iff_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} :

f.rangeS = ⊤ ↔ Function.Surjective ⇑f

source

@[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.

source

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.

source

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

source

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

source

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)

source

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.

source

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.

Equations

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

Instances For

source

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

Function.Injective ⇑(inclusion h)

source

@[simp]

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

s.subtype.rangeS = s

source

@[simp]

theorem Subsemiring.range_fst {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :

(RingHom.fst R S).rangeS = ⊤

source

@[simp]

theorem Subsemiring.range_snd {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :

(RingHom.snd R S).rangeS = ⊤

source

@[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

source

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.

Equations

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

Instances For

source

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.

Instances For

source

@[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

source

@[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

source

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

Equations

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

Instances For

source

@[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

source

@[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 `Subsemiring`s #

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.

source

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 = inferInstance

source

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

source

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

source

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

source

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.

source

@[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

source

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

source

@[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

Equations

Subsemiring.instSMulWithZeroSubtypeMem s = { toSMul := Submonoid.instSMulSubtypeMem s, smul_zero := ⋯, zero_smul := ⋯ }

source

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.

Equations

S.instSMulWithZeroSubtypeMem_1 = inferInstance

source

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 = inferInstance

source

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 = inferInstance

source

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 = inferInstance

source

@[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

Equations

Subsemiring.instMulActionWithZeroSubtypeMem s = { toMulAction := Submonoid.instMulActionSubtypeMem s, smul_zero := ⋯, zero_smul := ⋯ }

source

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 = inferInstance

source

@[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

Equations

Subsemiring.instModuleSubtypeMem s = { toDistribMulAction := inferInstance, add_smul := ⋯, zero_smul := ⋯ }

source

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 = inferInstance

source

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 = inferInstance

source

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.

source

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.

source

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

closure s ≤ centralizer ↑(centralizer s)

source

@[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.

Equations

Subsemiring.closureCommSemiringOfComm hcomm = { toSemiring := (Subsemiring.closure s).toSemiring, mul_comm := ⋯ }

Instances For

source

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

source

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

source

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

source

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

Documentation

Mathlib.Algebra.Ring.Subsemiring.Basic

Bundled subsemirings #

Actions by `Subsemiring`s #

Bundled subsemirings #

Actions by Subsemirings #

Actions by `Subsemiring`s #