Documentation

Mathlib.Algebra.Ring.Subsemiring.Basic

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] :
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
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.

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
    @[simp]
    theorem Subsemiring.topEquiv_symm_apply_coe {R : Type u} [NonAssocSemiring R] (r : R) :
    (Subsemiring.topEquiv.symm r) = r
    @[simp]
    theorem Subsemiring.topEquiv_apply {R : Type u} [NonAssocSemiring R] (r : ) :
    Subsemiring.topEquiv r = r
    def Subsemiring.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' := }
    Instances For
      @[simp]
      theorem Subsemiring.coe_comap {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring S) :
      (Subsemiring.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) :
      (Subsemiring.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} :
      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' := }
      Instances For
        @[simp]
        theorem Subsemiring.map_toSubmonoid {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) :
        (Subsemiring.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) :
        (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' := }
        Instances For
          @[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
          Instances For
            @[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.

            Equations
            Equations
            Equations
            • Subsemiring.instInhabited = { default := }
            theorem Subsemiring.mem_bot {R : Type u} [NonAssocSemiring R] {x : R} :
            x ∃ (n : ), n = x
            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
            Instances For
              @[simp]
              theorem Subsemiring.center_toSubmonoid (R : Type u) [NonAssocSemiring R] :
              (Subsemiring.center R).toSubmonoid = { carrier := (NonUnitalSubsemiring.center R).carrier, mul_mem' := , one_mem' := }
              @[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
              Instances For

                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' := }
                Instances For
                  @[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
                  Instances For
                    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.

                    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' := }
                    Instances For
                      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 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
                      Instances For

                        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' := }
                        Instances For
                          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
                          @[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' := }
                          Instances For
                            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.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
                              @[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 : 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
                              Instances For
                                @[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
                                Instances For
                                  @[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.

                                  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

                                  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.

                                  Equations
                                  Instances For
                                    @[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.

                                    Equations
                                    Instances For
                                      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.
                                      Instances For
                                        @[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
                                        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' := }
                                        Instances For
                                          @[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

                                          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
                                          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 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 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.

                                          The center of a semiring acts commutatively on that semiring.

                                          @[reducible, inline]
                                          abbrev Subsemiring.closureCommSemiringOfComm {R' : Type u_1} [Semiring R'] {s : Set R'} (hcomm : xs, ys, x * y = y * x) :

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

                                          Equations
                                          Instances For
                                            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) :