Documentation

Mathlib.Algebra.Ring.Subsemiring.Basic

Bundled subsemirings #

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

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

    Instances
      theorem natCast_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] (n : ) :
      n s
      @[deprecated natCast_mem]
      theorem coe_nat_mem {S : Type u_1} {R : Type u_2} [AddMonoidWithOne R] [SetLike S R] (s : S) [AddSubmonoidWithOneClass S R] (n : ) :
      n s

      Alias of natCast_mem.

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

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

        Instances
          @[instance 100]
          Equations
          • =
          @[instance 100]
          Equations
          • =
          @[instance 75]
          instance SubsemiringClass.toNonAssocSemiring {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :

          A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

          Equations
          instance SubsemiringClass.nontrivial {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [Nontrivial R] :
          Equations
          • =
          instance SubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [NoZeroDivisors R] :
          Equations
          • =
          def SubsemiringClass.subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :
          s →+* R

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

          Equations
          @[simp]
          theorem SubsemiringClass.coe_subtype {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) :
          (SubsemiringClass.subtype s) = Subtype.val
          @[instance 75]
          instance SubsemiringClass.toSemiring {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] :

          A subsemiring of a Semiring is a Semiring.

          Equations
          @[simp]
          theorem SubsemiringClass.coe_pow {S : Type v} (s : S) {R : Type u_1} [Semiring R] [SetLike S R] [SubsemiringClass S R] (x : s) (n : ) :
          (x ^ n) = x ^ n
          instance SubsemiringClass.toCommSemiring {S : Type v} (s : S) {R : Type u_1} [CommSemiring R] [SetLike S R] [SubsemiringClass S R] :

          A subsemiring of a CommSemiring is a CommSemiring.

          Equations
          instance SubsemiringClass.instCharZero {R : Type u} {S : Type v} [NonAssocSemiring R] [SetLike S R] [hSR : SubsemiringClass S R] (s : S) [CharZero R] :
          Equations
          • =
          structure Subsemiring (R : Type u) [NonAssocSemiring R] extends Submonoid :

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

          • carrier : Set R
          • mul_mem' : ∀ {a b : R}, a self.carrierb self.carriera * b self.carrier
          • one_mem' : 1 self.carrier
          • add_mem' : ∀ {a b : R}, a self.carrierb self.carriera + b self.carrier

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

          • zero_mem' : 0 self.carrier

            An additive submonoid contains 0.

          @[reducible]

          Reinterpret a Subsemiring as an AddSubmonoid.

          Equations
          • self.toAddSubmonoid = { carrier := self.carrier, add_mem' := , zero_mem' := }
          Equations
          • Subsemiring.instSetLike = { coe := fun (s : Subsemiring R) => s.carrier, coe_injective' := }
          @[simp]
          theorem Subsemiring.mem_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :
          x s.toSubmonoid x s
          theorem Subsemiring.mem_carrier {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :
          x s.carrier x s
          theorem Subsemiring.ext_iff {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} :
          S = T ∀ (x : R), x S x T
          theorem Subsemiring.ext {R : Type u} [NonAssocSemiring R] {S : Subsemiring R} {T : Subsemiring R} (h : ∀ (x : R), x S x T) :
          S = T

          Two subsemirings are equal if they have the same elements.

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

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

          Equations
          • S.copy s hs = { carrier := s, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem Subsemiring.coe_copy {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = S) :
          (S.copy s hs) = s
          theorem Subsemiring.copy_eq {R : Type u} [NonAssocSemiring R] (S : Subsemiring R) (s : Set R) (hs : s = S) :
          S.copy s hs = S
          theorem Subsemiring.toSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] :
          StrictMono Subsemiring.toSubmonoid
          theorem Subsemiring.toSubmonoid_mono {R : Type u} [NonAssocSemiring R] :
          Monotone Subsemiring.toSubmonoid
          theorem Subsemiring.toAddSubmonoid_strictMono {R : Type u} [NonAssocSemiring R] :
          StrictMono Subsemiring.toAddSubmonoid
          theorem Subsemiring.toAddSubmonoid_mono {R : Type u} [NonAssocSemiring R] :
          Monotone Subsemiring.toAddSubmonoid
          def Subsemiring.mk' {R : Type u} [NonAssocSemiring R] (s : Set R) (sm : Submonoid R) (hm : sm = s) (sa : AddSubmonoid R) (ha : sa = s) :

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

          Equations
          • Subsemiring.mk' s sm hm sa ha = { carrier := s, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem Subsemiring.coe_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : sm = s) {sa : AddSubmonoid R} (ha : sa = s) :
          (Subsemiring.mk' s sm hm sa ha) = s
          @[simp]
          theorem Subsemiring.mem_mk' {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : sm = s) {sa : AddSubmonoid R} (ha : sa = s) {x : R} :
          x Subsemiring.mk' s sm hm sa ha x s
          @[simp]
          theorem Subsemiring.mk'_toSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : sm = s) {sa : AddSubmonoid R} (ha : sa = s) :
          (Subsemiring.mk' s sm hm sa ha).toSubmonoid = sm
          @[simp]
          theorem Subsemiring.mk'_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Set R} {sm : Submonoid R} (hm : sm = s) {sa : AddSubmonoid R} (ha : sa = s) :
          (Subsemiring.mk' s sm hm sa ha).toAddSubmonoid = sa
          theorem Subsemiring.one_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          1 s

          A subsemiring contains the semiring's 1.

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

          A subsemiring contains the semiring's 0.

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

          A subsemiring is closed under multiplication.

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

          A subsemiring is closed under addition.

          theorem Subsemiring.list_prod_mem {R : Type u_1} [Semiring R] (s : Subsemiring R) {l : List R} :
          (∀ xl, 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} :
          (∀ xl, 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) :
          (∀ am, 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) :
          (∀ am, a s)m.sum s

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

          theorem Subsemiring.prod_mem {R : Type u_1} [CommSemiring R] (s : Subsemiring R) {ι : Type u_2} {t : Finset ι} {f : ιR} (h : ct, f c s) :
          it, 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 : ιR} (h : ct, f c s) :
          it, f i s

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

          A subsemiring of a NonAssocSemiring inherits a NonAssocSemiring structure

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

          A subsemiring of a Semiring is a Semiring.

          Equations
          • s.toSemiring = Semiring.mk Monoid.npow
          @[simp]
          theorem Subsemiring.coe_pow {R : Type u_1} [Semiring R] (s : Subsemiring R) (x : s) (n : ) :
          (x ^ n) = x ^ n

          A subsemiring of a CommSemiring is a CommSemiring.

          Equations

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

          Equations
          • s.subtype = { toFun := Subtype.val, map_one' := , map_mul' := , map_zero' := , map_add' := }
          @[simp]
          theorem Subsemiring.coe_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          s.subtype = Subtype.val
          theorem Subsemiring.nsmul_mem {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) {x : R} (hx : x s) (n : ) :
          n x s
          @[simp]
          theorem Subsemiring.coe_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          s.toSubmonoid = s
          @[simp]
          theorem Subsemiring.coe_carrier_toSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          s.carrier = s
          theorem Subsemiring.mem_toAddSubmonoid {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {x : R} :
          x s.toAddSubmonoid x s
          theorem Subsemiring.coe_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          s.toAddSubmonoid = s

          The subsemiring R of the semiring R.

          Equations
          • Subsemiring.instTop = { top := let __src := ; let __src_1 := ; { toSubmonoid := __src, add_mem' := , zero_mem' := } }
          @[simp]
          theorem Subsemiring.mem_top {R : Type u} [NonAssocSemiring R] (x : R) :
          @[simp]
          theorem Subsemiring.coe_top {R : Type u} [NonAssocSemiring R] :
          = Set.univ
          @[simp]
          theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : R) :
          (Subsemiring.topEquiv.symm r) = r
          @[simp]
          theorem Subsemiring.topEquiv_apply {R : Type u} [NonAssocSemiring R] (r : ) :
          Subsemiring.topEquiv r = r

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

          Equations
          • Subsemiring.topEquiv = { toFun := fun (r : ) => r, invFun := fun (r : R) => r, , left_inv := , right_inv := , map_mul' := , map_add' := }
          def Subsemiring.comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring S) :

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

          Equations
          • Subsemiring.comap f s = { carrier := f ⁻¹' s, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring S) (f : R →+* S) :
          (Subsemiring.comap f s) = f ⁻¹' s
          @[simp]
          theorem Subsemiring.mem_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring S} {f : R →+* S} {x : R} :
          def Subsemiring.map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) :

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

          Equations
          • Subsemiring.map f s = { carrier := f '' s, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem Subsemiring.coe_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) :
          (Subsemiring.map f s) = f '' s
          @[simp]
          theorem Subsemiring.mem_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {y : S} :
          y Subsemiring.map f s xs, f x = y
          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) :
          noncomputable def Subsemiring.equivMapOfInjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective f) :
          s ≃+* (Subsemiring.map f s)

          A subsemiring is isomorphic to its image under an injective function

          Equations
          • s.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑s) hf, map_mul' := , map_add' := }
          @[simp]
          theorem Subsemiring.coe_equivMapOfInjective_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective f) (x : s) :
          ((s.equivMapOfInjective f hf) x) = f x
          def RingHom.rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :

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

          Equations
          @[simp]
          theorem RingHom.coe_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :
          f.rangeS = Set.range f
          @[simp]
          theorem RingHom.mem_rangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {y : S} :
          y f.rangeS ∃ (x : R), f x = y
          theorem RingHom.rangeS_eq_map {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :
          f.rangeS = Subsemiring.map f
          theorem RingHom.mem_rangeS_self {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) :
          f x f.rangeS
          theorem RingHom.map_rangeS {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) :
          Subsemiring.map g f.rangeS = (g.comp f).rangeS
          instance RingHom.fintypeRangeS {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] [Fintype R] [DecidableEq S] (f : R →+* S) :
          Fintype f.rangeS

          The range of a morphism of semirings is a fintype, if the domain is a fintype. Note: this instance can form a diamond with Subtype.fintype in the presence of Fintype S.

          Equations
          Equations
          Equations
          • Subsemiring.instInhabited = { default := }
          theorem Subsemiring.mem_bot {R : Type u} [NonAssocSemiring R] {x : R} :
          x ∃ (n : ), n = x

          The inf of two subsemirings is their intersection.

          Equations
          • One or more equations did not get rendered due to their size.
          @[simp]
          theorem Subsemiring.coe_inf {R : Type u} [NonAssocSemiring R] (p : Subsemiring R) (p' : Subsemiring R) :
          (p p') = p p'
          @[simp]
          theorem Subsemiring.mem_inf {R : Type u} [NonAssocSemiring R] {p : Subsemiring R} {p' : Subsemiring R} {x : R} :
          x p p' x p x p'
          Equations
          • Subsemiring.instInfSet = { sInf := fun (s : Set (Subsemiring R)) => Subsemiring.mk' (⋂ ts, t) (⨅ ts, t.toSubmonoid) (⨅ ts, t.toAddSubmonoid) }
          @[simp]
          theorem Subsemiring.coe_sInf {R : Type u} [NonAssocSemiring R] (S : Set (Subsemiring R)) :
          (sInf S) = sS, s
          theorem Subsemiring.mem_sInf {R : Type u} [NonAssocSemiring R] {S : Set (Subsemiring R)} {x : R} :
          x sInf S pS, x p
          @[simp]
          theorem Subsemiring.coe_iInf {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} {S : ιSubsemiring R} :
          (⨅ (i : ι), S i) = ⋂ (i : ι), (S i)
          theorem Subsemiring.mem_iInf {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} {S : ιSubsemiring 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 = ts, t.toSubmonoid
          @[simp]
          theorem Subsemiring.sInf_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (s : Set (Subsemiring R)) :
          (sInf s).toAddSubmonoid = ts, t.toAddSubmonoid

          Subsemirings of a semiring form a complete lattice.

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

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

          Equations
          @[reducible, inline]

          The center is commutative and associative.

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

          Equations

          The center is commutative.

          Equations
          theorem Subsemiring.mem_center_iff {R : Type u_1} [Semiring R] {z : R} :
          z Subsemiring.center R ∀ (g : R), g * z = z * g
          Equations
          def Subsemiring.centralizer {R : Type u_1} [Semiring R] (s : Set R) :

          The centralizer of a set as subsemiring.

          Equations
          • Subsemiring.centralizer s = { carrier := s.centralizer, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem Subsemiring.coe_centralizer {R : Type u_1} [Semiring R] (s : Set R) :
          (Subsemiring.centralizer s) = s.centralizer
          theorem Subsemiring.mem_centralizer_iff {R : Type u_1} [Semiring R] {s : Set R} {z : R} :
          z Subsemiring.centralizer s gs, g * z = z * g

          The Subsemiring generated by a set.

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

          The subsemiring generated by a set includes the set.

          theorem Subsemiring.not_mem_of_not_mem_closure {R : Type u} [NonAssocSemiring R] {s : Set R} {P : R} (hP : PSubsemiring.closure s) :
          Ps
          @[simp]
          theorem Subsemiring.closure_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} :

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

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

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

          theorem Subsemiring.closure_eq_of_le {R : Type u} [NonAssocSemiring R] {s : Set R} {t : Subsemiring R} (h₁ : s t) (h₂ : t Subsemiring.closure s) :
          theorem Subsemiring.mem_map_equiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R ≃+* S} {K : Subsemiring R} {x : S} :
          x Subsemiring.map (↑f) K f.symm x K

          The additive closure of a submonoid is a subsemiring.

          Equations
          • M.subsemiringClosure = { carrier := (AddSubmonoid.closure M).carrier, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          theorem Submonoid.subsemiringClosure_coe {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :
          M.subsemiringClosure = (AddSubmonoid.closure M)
          theorem Submonoid.subsemiringClosure_toAddSubmonoid {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :
          M.subsemiringClosure.toAddSubmonoid = AddSubmonoid.closure M
          theorem Submonoid.subsemiringClosure_eq_closure {R : Type u} [NonAssocSemiring R] (M : Submonoid R) :
          M.subsemiringClosure = Subsemiring.closure M

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

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

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

          An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

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

          Alias of Subsemiring.closure_induction.


          An induction principle for closure membership. If p holds for 0, 1, and all elements of s, and is preserved under addition and multiplication, then p holds for all elements of the closure of s.

          theorem Subsemiring.closure_induction₂ {R : Type u} [NonAssocSemiring R] {s : Set R} {p : (x y : R) → x Subsemiring.closure sy Subsemiring.closure sProp} (mem_mem : ∀ (x y : R) (hx : x s) (hy : y s), p x y ) (zero_left : ∀ (x : R) (hx : x Subsemiring.closure s), p 0 x hx) (zero_right : ∀ (x : R) (hx : x Subsemiring.closure s), p x 0 hx ) (one_left : ∀ (x : R) (hx : x Subsemiring.closure s), p 1 x hx) (one_right : ∀ (x : R) (hx : x Subsemiring.closure s), p x 1 hx ) (add_left : ∀ (x y z : R) (hx : x Subsemiring.closure s) (hy : y Subsemiring.closure s) (hz : z Subsemiring.closure s), p x z hx hzp y z hy hzp (x + y) z hz) (add_right : ∀ (x y z : R) (hx : x Subsemiring.closure s) (hy : y Subsemiring.closure s) (hz : z Subsemiring.closure s), p x y hx hyp x z hx hzp x (y + z) hx ) (mul_left : ∀ (x y z : R) (hx : x Subsemiring.closure s) (hy : y Subsemiring.closure s) (hz : z Subsemiring.closure s), p x z hx hzp y z hy hzp (x * y) z hz) (mul_right : ∀ (x y z : R) (hx : x Subsemiring.closure s) (hy : y Subsemiring.closure s) (hz : z Subsemiring.closure s), p x y hx hyp x z hx hzp x (y * z) hx ) {x : R} {y : R} (hx : x Subsemiring.closure s) (hy : y Subsemiring.closure s) :
          p x y hx hy

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

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

          closure forms a Galois insertion with the coercion to set.

          Equations

          Closure of a subsemiring S equals S.

          theorem Subsemiring.closure_iUnion {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} (s : ιSet R) :
          Subsemiring.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Subsemiring.closure (s i)
          theorem Subsemiring.map_iSup {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ιSubsemiring R) :
          Subsemiring.map f (iSup s) = ⨆ (i : ι), Subsemiring.map f (s i)
          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 : ιSubsemiring R) :
          Subsemiring.map f (iInf s) = ⨅ (i : ι), Subsemiring.map f (s i)
          theorem Subsemiring.comap_iInf {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ιSubsemiring S) :
          Subsemiring.comap f (iInf s) = ⨅ (i : ι), Subsemiring.comap f (s i)
          @[simp]

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

          Equations
          • s.prod t = { carrier := s ×ˢ t, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          theorem Subsemiring.coe_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :
          (s.prod t) = s ×ˢ t
          theorem Subsemiring.mem_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {s : Subsemiring R} {t : Subsemiring S} {p : R × S} :
          p s.prod t p.1 s p.2 t
          theorem Subsemiring.prod_mono {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] ⦃s₁ : Subsemiring R ⦃s₂ : Subsemiring R (hs : s₁ s₂) ⦃t₁ : Subsemiring S ⦃t₂ : Subsemiring S (ht : t₁ t₂) :
          s₁.prod t₁ s₂.prod t₂
          theorem Subsemiring.prod_mono_right {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) :
          Monotone fun (t : Subsemiring S) => s.prod t
          theorem Subsemiring.prod_mono_left {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (t : Subsemiring S) :
          Monotone fun (s : Subsemiring R) => s.prod t
          @[simp]
          def Subsemiring.prodEquiv {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :
          (s.prod t) ≃+* s × t

          Product of subsemirings is isomorphic to their product as monoids.

          Equations
          • s.prodEquiv t = { toEquiv := Equiv.Set.prod s t, map_mul' := , map_add' := }
          theorem Subsemiring.mem_iSup_of_directed {R : Type u} [NonAssocSemiring R] {ι : Sort u_1} [hι : Nonempty ι] {S : ιSubsemiring 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 : ιSubsemiring 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 sS, 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) = sS, s
          def RingHom.domRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) (s : σR) :
          s →+* S

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

          Equations
          @[simp]
          theorem RingHom.restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} [SetLike σR R] [SubsemiringClass σR R] (f : R →+* S) {s : σR} (x : s) :
          (f.domRestrict s) x = f x
          def RingHom.codRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σS : Type u_2} [SetLike σS S] [SubsemiringClass σS S] (f : R →+* S) (s : σS) (h : ∀ (x : R), f x s) :
          R →+* s

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

          Equations
          • f.codRestrict s h = { toFun := fun (n : R) => f n, , map_one' := , map_mul' := , map_zero' := , map_add' := }
          def RingHom.restrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : xs', f x s) :
          s' →+* s

          The ring homomorphism from the preimage of s to s.

          Equations
          • f.restrict s' s h = (f.domRestrict s').codRestrict s
          @[simp]
          theorem RingHom.coe_restrict_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {σR : Type u_1} {σS : Type u_2} [SetLike σR R] [SetLike σS S] [SubsemiringClass σR R] [SubsemiringClass σS S] (f : R →+* S) (s' : σR) (s : σS) (h : xs', 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 : xs', f x s) :
          (SubsemiringClass.subtype s).comp (f.restrict s' s h) = f.comp (SubsemiringClass.subtype s')
          def RingHom.rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :
          R →+* f.rangeS

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

          This is the bundled version of Set.rangeFactorization.

          Equations
          • f.rangeSRestrict = f.codRestrict f.rangeS
          @[simp]
          theorem RingHom.coe_rangeSRestrict {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (x : R) :
          (f.rangeSRestrict x) = f x
          theorem RingHom.rangeSRestrict_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :
          Function.Surjective f.rangeSRestrict
          @[simp]
          theorem RingHom.rangeS_top_of_surjective {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (hf : Function.Surjective f) :
          f.rangeS =

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

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

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

          Equations
          • f.eqLocusS g = { carrier := {x : R | f x = g x}, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := }
          @[simp]
          theorem RingHom.eqLocusS_same {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) :
          f.eqLocusS f =
          theorem RingHom.eqOn_sclosure {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {g : R →+* S} {s : Set R} (h : Set.EqOn (⇑f) (⇑g) s) :
          Set.EqOn f g (Subsemiring.closure s)

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

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

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

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

          The ring homomorphism associated to an inclusion of subsemirings.

          Equations
          @[simp]
          theorem Subsemiring.rangeS_subtype {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) :
          s.subtype.rangeS = s
          @[simp]
          theorem Subsemiring.range_fst {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :
          (RingHom.fst R S).rangeS =
          @[simp]
          theorem Subsemiring.range_snd {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] :
          (RingHom.snd R S).rangeS =
          @[simp]
          theorem Subsemiring.prod_bot_sup_bot_prod {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (t : Subsemiring S) :
          s.prod .prod t = s.prod t
          def RingEquiv.subsemiringCongr {R : Type u} [NonAssocSemiring R] {s : Subsemiring R} {t : Subsemiring R} (h : s = t) :
          s ≃+* t

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

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

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

          Equations
          • One or more equations did not get rendered due to their size.
          @[simp]
          theorem RingEquiv.ofLeftInverseS_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : SR} {f : R →+* S} (h : Function.LeftInverse g f) (x : R) :
          @[simp]
          theorem RingEquiv.ofLeftInverseS_symm_apply {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {g : SR} {f : R →+* S} (h : Function.LeftInverse g f) (x : f.rangeS) :
          (RingEquiv.ofLeftInverseS h).symm x = g x
          @[simp]
          theorem RingEquiv.subsemiringMap_symm_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (y : (e.toAddEquiv '' s.toAddSubmonoid)) :
          ((e.subsemiringMap s).symm y) = (↑e).symm y
          @[simp]
          theorem RingEquiv.subsemiringMap_apply_coe {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (x : s.toAddSubmonoid) :
          ((e.subsemiringMap s) x) = e x
          def RingEquiv.subsemiringMap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) :
          s ≃+* (Subsemiring.map e.toRingHom s)

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

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

          Actions by Subsemirings #

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

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

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

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

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

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

          Equations
          • =
          instance Subsemiring.faithfulSMul {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [SMul R' α] [FaithfulSMul R' α] (S : Subsemiring R') :
          FaithfulSMul (↥S) α
          Equations
          • =
          instance Subsemiring.instSMulWithZeroSubtypeMem {R' : Type u_1} {α : Type u_2} [NonAssocSemiring R'] [Zero α] [SMulWithZero R' α] (S : Subsemiring R') :
          SMulWithZero (↥S) α

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

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

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

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

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

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

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

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

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

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

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

          Equations
          • S.instMulSemiringActionSubtypeMem = S.mulSemiringAction

          The center of a semiring acts commutatively on that semiring.

          Equations
          • =

          The center of a semiring acts commutatively on that semiring.

          Equations
          • =
          def Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [Semiring R'] {s : Set R'} (hcomm : as, bs, a * b = b * a) :

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

          Equations
          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) :