Documentation

Mathlib.Algebra.Ring.Prod

Semiring, ring etc structures on R × S #

In this file we define two-binop (Semiring, Ring etc) structures on R × S. We also prove trivial simp lemmas, and define the following operations on RingHoms and similarly for NonUnitalRingHoms:

instance Prod.instDistrib {R : Type u_3} {S : Type u_5} [Distrib R] [Distrib S] :
Distrib (R × S)

Product of two distributive types is distributive.

Equations

Product of two NonUnitalSemirings is a NonUnitalSemiring.

Equations

Product of two NonAssocSemirings is a NonAssocSemiring.

Equations
instance Prod.instSemiring {R : Type u_3} {S : Type u_5} [Semiring R] [Semiring S] :
Semiring (R × S)

Product of two semirings is a semiring.

Equations
  • Prod.instSemiring = Semiring.mk Monoid.npow

Product of two NonUnitalCommSemirings is a NonUnitalCommSemiring.

Equations
instance Prod.instCommSemiring {R : Type u_3} {S : Type u_5} [CommSemiring R] [CommSemiring S] :

Product of two commutative semirings is a commutative semiring.

Equations
Equations
instance Prod.instNonUnitalRing {R : Type u_3} {S : Type u_5} [NonUnitalRing R] [NonUnitalRing S] :
Equations
instance Prod.instNonAssocRing {R : Type u_3} {S : Type u_5} [NonAssocRing R] [NonAssocRing S] :
Equations
instance Prod.instRing {R : Type u_3} {S : Type u_5} [Ring R] [Ring S] :
Ring (R × S)

Product of two rings is a ring.

Equations
  • Prod.instRing = Ring.mk SubNegMonoid.zsmul

Product of two NonUnitalCommRings is a NonUnitalCommRing.

Equations
instance Prod.instCommRing {R : Type u_3} {S : Type u_5} [CommRing R] [CommRing S] :
CommRing (R × S)

Product of two commutative rings is a commutative ring.

Equations

Given non-unital semirings R, S, the natural projection homomorphism from R × S to R.

Equations
Instances For

    Given non-unital semirings R, S, the natural projection homomorphism from R × S to S.

    Equations
    Instances For

      Combine two non-unital ring homomorphisms f : R →ₙ+* S, g : R →ₙ+* T into f.prod g : R →ₙ+* S × T given by (f.prod g) x = (f x, g x)

      Equations
      • f.prod g = { toFun := fun (x : R) => (f x, g x), map_mul' := , map_zero' := , map_add' := }
      Instances For
        @[simp]
        theorem NonUnitalRingHom.prod_apply {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) (x : R) :
        (f.prod g) x = (f x, g x)
        @[simp]
        theorem NonUnitalRingHom.fst_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) :
        (NonUnitalRingHom.fst S T).comp (f.prod g) = f
        @[simp]
        theorem NonUnitalRingHom.snd_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) :
        (NonUnitalRingHom.snd S T).comp (f.prod g) = g

        Prod.map as a NonUnitalRingHom.

        Equations
        Instances For
          theorem NonUnitalRingHom.prodMap_def {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') :
          f.prodMap g = (f.comp (NonUnitalRingHom.fst R S)).prod (g.comp (NonUnitalRingHom.snd R S))
          @[simp]
          theorem NonUnitalRingHom.coe_prodMap {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') :
          (f.prodMap g) = Prod.map f g
          theorem NonUnitalRingHom.prod_comp_prodMap {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} {T : Type u_7} [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [NonUnitalNonAssocSemiring R'] [NonUnitalNonAssocSemiring S'] [NonUnitalNonAssocSemiring T] (f : T →ₙ+* R) (g : T →ₙ+* S) (f' : R →ₙ+* R') (g' : S →ₙ+* S') :
          (f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
          def RingHom.fst (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] :
          R × S →+* R

          Given semirings R, S, the natural projection homomorphism from R × S to R.

          Equations
          • RingHom.fst R S = { toFun := Prod.fst, map_one' := , map_mul' := , map_zero' := , map_add' := }
          Instances For
            def RingHom.snd (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] :
            R × S →+* S

            Given semirings R, S, the natural projection homomorphism from R × S to S.

            Equations
            • RingHom.snd R S = { toFun := Prod.snd, map_one' := , map_mul' := , map_zero' := , map_add' := }
            Instances For
              Equations
              • =
              Equations
              • =
              @[simp]
              theorem RingHom.coe_fst {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
              (RingHom.fst R S) = Prod.fst
              @[simp]
              theorem RingHom.coe_snd {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
              (RingHom.snd R S) = Prod.snd
              def RingHom.prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) :
              R →+* S × T

              Combine two ring homomorphisms f : R →+* S, g : R →+* T into f.prod g : R →+* S × T given by (f.prod g) x = (f x, g x)

              Equations
              • f.prod g = { toFun := fun (x : R) => (f x, g x), map_one' := , map_mul' := , map_zero' := , map_add' := }
              Instances For
                @[simp]
                theorem RingHom.prod_apply {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) (x : R) :
                (f.prod g) x = (f x, g x)
                @[simp]
                theorem RingHom.fst_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) :
                (RingHom.fst S T).comp (f.prod g) = f
                @[simp]
                theorem RingHom.snd_comp_prod {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S) (g : R →+* T) :
                (RingHom.snd S T).comp (f.prod g) = g
                theorem RingHom.prod_unique {R : Type u_3} {S : Type u_5} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (f : R →+* S × T) :
                ((RingHom.fst S T).comp f).prod ((RingHom.snd S T).comp f) = f
                def RingHom.prodMap {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
                R × S →+* R' × S'

                Prod.map as a RingHom.

                Equations
                Instances For
                  theorem RingHom.prodMap_def {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
                  f.prodMap g = (f.comp (RingHom.fst R S)).prod (g.comp (RingHom.snd R S))
                  @[simp]
                  theorem RingHom.coe_prodMap {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') :
                  (f.prodMap g) = Prod.map f g
                  theorem RingHom.prod_comp_prodMap {R : Type u_3} {R' : Type u_4} {S : Type u_5} {S' : Type u_6} {T : Type u_7} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] [NonAssocSemiring T] (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') :
                  (f'.prodMap g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)
                  def RingEquiv.prodComm {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
                  R × S ≃+* S × R

                  Swapping components as an equivalence of (semi)rings.

                  Equations
                  • RingEquiv.prodComm = { toEquiv := AddEquiv.prodComm.toEquiv, map_mul' := , map_add' := }
                  Instances For
                    @[simp]
                    theorem RingEquiv.coe_prodComm {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
                    RingEquiv.prodComm = Prod.swap
                    @[simp]
                    theorem RingEquiv.coe_prodComm_symm {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
                    RingEquiv.prodComm.symm = Prod.swap
                    @[simp]
                    theorem RingEquiv.fst_comp_coe_prodComm {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
                    (RingHom.fst S R).comp RingEquiv.prodComm = RingHom.snd R S
                    @[simp]
                    theorem RingEquiv.snd_comp_coe_prodComm {R : Type u_3} {S : Type u_5} [NonAssocSemiring R] [NonAssocSemiring S] :
                    (RingHom.snd S R).comp RingEquiv.prodComm = RingHom.fst R S
                    @[simp]
                    theorem RingEquiv.prodProdProdComm_apply (R : Type u_3) (R' : Type u_4) (S : Type u_5) (S' : Type u_6) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] (rrss : (R × R') × S × S') :
                    (RingEquiv.prodProdProdComm R R' S S') rrss = ((rrss.1.1, rrss.2.1), rrss.1.2, rrss.2.2)
                    def RingEquiv.prodProdProdComm (R : Type u_3) (R' : Type u_4) (S : Type u_5) (S' : Type u_6) [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring R'] [NonAssocSemiring S'] :
                    (R × R') × S × S' ≃+* (R × S) × R' × S'

                    Four-way commutativity of Prod. The name matches mul_mul_mul_comm.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For
                      @[simp]
                      @[simp]
                      @[simp]
                      theorem RingEquiv.prodZeroRing_apply (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (x : R) :
                      (RingEquiv.prodZeroRing R S) x = (x, 0)
                      @[simp]
                      theorem RingEquiv.prodZeroRing_symm_apply (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (self : R × S) :
                      (RingEquiv.prodZeroRing R S).symm self = self.1

                      A ring R is isomorphic to R × S when S is the zero ring

                      Equations
                      • RingEquiv.prodZeroRing R S = { toFun := fun (x : R) => (x, 0), invFun := Prod.fst, left_inv := , right_inv := , map_mul' := , map_add' := }
                      Instances For
                        @[simp]
                        theorem RingEquiv.zeroRingProd_symm_apply (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (self : S × R) :
                        (RingEquiv.zeroRingProd R S).symm self = self.2
                        @[simp]
                        theorem RingEquiv.zeroRingProd_apply (R : Type u_3) (S : Type u_5) [NonAssocSemiring R] [NonAssocSemiring S] [Subsingleton S] (x : R) :
                        (RingEquiv.zeroRingProd R S) x = (0, x)

                        A ring R is isomorphic to S × R when S is the zero ring

                        Equations
                        • RingEquiv.zeroRingProd R S = { toFun := fun (x : R) => (0, x), invFun := Prod.snd, left_inv := , right_inv := , map_mul' := , map_add' := }
                        Instances For
                          theorem false_of_nontrivial_of_product_domain (R : Type u_9) (S : Type u_10) [Ring R] [Ring S] [IsDomain (R × S)] [Nontrivial R] [Nontrivial S] :

                          The product of two nontrivial rings is not a domain