Semiring, ring etc structures on R × S× S

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

• fst R S : R × S →+* R× S →+* R→+* R, snd R S : R × S →+* S× S →+* S→+* S: projections Prod.fst and Prod.snd as RingHoms;
• f.prod g : R →+* S × T→+* S × T× T: sends x to (f x, g x);
• f.prod_map g : R × S → R' × S'× S → R' × S'→ R' × S'× S': Prod.map f g as a RingHom, sends (x, y) to (f x, g y).

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instance Prod.instDistribProd {R : Type u_1} {S : Type u_2} [inst : Distrib R] [inst : Distrib S] : Distrib (R × S)

Product of two distributive types is distributive.

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instance Prod.instNonUnitalNonAssocSemiringProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] : NonUnitalNonAssocSemiring (R × S)

Product of two NonUnitalNonAssocSemirings is a NonUnitalNonAssocSemiring.

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instance Prod.instNonUnitalSemiringProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalSemiring R] [inst : NonUnitalSemiring S] : NonUnitalSemiring (R × S)

Product of two NonUnitalSemirings is a NonUnitalSemiring.

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instance Prod.instNonAssocSemiringProd {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : NonAssocSemiring (R × S)

Product of two NonAssocSemirings is a NonAssocSemiring.

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instance Prod.instSemiringProd {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] : Semiring (R × S)

Product of two semirings is a semiring.

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instance Prod.instNonUnitalCommSemiringProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalCommSemiring R] [inst : NonUnitalCommSemiring S] : NonUnitalCommSemiring (R × S)

Product of two NonUnitalCommSemirings is a NonUnitalCommSemiring.

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instance Prod.instCommSemiringProd {R : Type u_1} {S : Type u_2} [inst : CommSemiring R] [inst : CommSemiring S] : CommSemiring (R × S)

Product of two commutative semirings is a commutative semiring.

Equations

• Prod.instCommSemiringProd = let src := inferInstanceAs (Semiring (R × S)); let src_1 := inferInstanceAs (CommMonoid (R × S)); CommSemiring.mk (_ : ∀ (a b : R × S), a * b = b * a)

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instance Prod.instNonUnitalNonAssocRingProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocRing R] [inst : NonUnitalNonAssocRing S] : NonUnitalNonAssocRing (R × S)

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instance Prod.instNonUnitalRingProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalRing R] [inst : NonUnitalRing S] : NonUnitalRing (R × S)

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instance Prod.instNonAssocRingProd {R : Type u_1} {S : Type u_2} [inst : NonAssocRing R] [inst : NonAssocRing S] : NonAssocRing (R × S)

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instance Prod.instRingProd {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst : Ring S] : Ring (R × S)

Product of two rings is a ring.

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instance Prod.instNonUnitalCommRingProd {R : Type u_1} {S : Type u_2} [inst : NonUnitalCommRing R] [inst : NonUnitalCommRing S] : NonUnitalCommRing (R × S)

Product of two NonUnitalCommRings is a NonUnitalCommRing.

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instance Prod.instCommRingProd {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst : CommRing S] : CommRing (R × S)

Product of two commutative rings is a commutative ring.

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• Prod.instCommRingProd = let src := inferInstanceAs (Ring (R × S)); let src_1 := inferInstanceAs (CommMonoid (R × S)); CommRing.mk (_ : ∀ (a b : R × S), a * b = b * a)

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def NonUnitalRingHom.fst (R : Type u_1) (S : Type u_2) [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] : R × S →ₙ+* R

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

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def NonUnitalRingHom.snd (R : Type u_1) (S : Type u_2) [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] : R × S →ₙ+* S

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

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theorem NonUnitalRingHom.coe_fst {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] : ↑(NonUnitalRingHom.fst R S) = Prod.fst

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theorem NonUnitalRingHom.coe_snd {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] : ↑(NonUnitalRingHom.snd R S) = Prod.snd

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def NonUnitalRingHom.prod {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) : R →ₙ+* S × T

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

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theorem NonUnitalRingHom.prod_apply {R : Type u_3} {S : Type u_1} {T : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) (x : R) : ↑(NonUnitalRingHom.prod f g) x = (↑f x, ↑g x)

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theorem NonUnitalRingHom.fst_comp_prod {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) : NonUnitalRingHom.comp (NonUnitalRingHom.fst S T) (NonUnitalRingHom.prod f g) = f

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theorem NonUnitalRingHom.snd_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring T] (f : R →ₙ+* S) (g : R →ₙ+* T) : NonUnitalRingHom.comp (NonUnitalRingHom.snd S T) (NonUnitalRingHom.prod f g) = g

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theorem NonUnitalRingHom.prod_unique {R : Type u_1} {S : Type u_3} {T : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring T] (f : R →ₙ+* S × T) : NonUnitalRingHom.prod (NonUnitalRingHom.comp (NonUnitalRingHom.fst S T) f) (NonUnitalRingHom.comp (NonUnitalRingHom.snd S T) f) = f

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def NonUnitalRingHom.prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring R'] [inst : NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') : R × S →ₙ+* R' × S'

prod.map as a NonUnitalRingHom.

Equations

• NonUnitalRingHom.prodMap f g = NonUnitalRingHom.prod (NonUnitalRingHom.comp f (NonUnitalRingHom.fst R S)) (NonUnitalRingHom.comp g (NonUnitalRingHom.snd R S))

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theorem NonUnitalRingHom.prodMap_def {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring R'] [inst : NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') : NonUnitalRingHom.prodMap f g = NonUnitalRingHom.prod (NonUnitalRingHom.comp f (NonUnitalRingHom.fst R S)) (NonUnitalRingHom.comp g (NonUnitalRingHom.snd R S))

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theorem NonUnitalRingHom.coe_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring R'] [inst : NonUnitalNonAssocSemiring S'] (f : R →ₙ+* R') (g : S →ₙ+* S') : ↑(NonUnitalRingHom.prodMap f g) = Prod.map ↑f ↑g

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theorem NonUnitalRingHom.prod_comp_prodMap {R : Type u_2} {R' : Type u_4} {S : Type u_3} {S' : Type u_5} {T : Type u_1} [inst : NonUnitalNonAssocSemiring R] [inst : NonUnitalNonAssocSemiring S] [inst : NonUnitalNonAssocSemiring R'] [inst : NonUnitalNonAssocSemiring S'] [inst : NonUnitalNonAssocSemiring T] (f : T →ₙ+* R) (g : T →ₙ+* S) (f' : R →ₙ+* R') (g' : S →ₙ+* S') : NonUnitalRingHom.comp (NonUnitalRingHom.prodMap f' g') (NonUnitalRingHom.prod f g) = NonUnitalRingHom.prod (NonUnitalRingHom.comp f' f) (NonUnitalRingHom.comp g' g)

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def RingHom.fst (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : R × S →+* R

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

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def RingHom.snd (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : R × S →+* S

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

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theorem RingHom.coe_fst {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : ↑(RingHom.fst R S) = Prod.fst

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theorem RingHom.coe_snd {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : ↑(RingHom.snd R S) = Prod.snd

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def RingHom.prod {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring T] (f : R →+* S) (g : R →+* T) : R →+* S × T

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

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theorem RingHom.prod_apply {R : Type u_3} {S : Type u_1} {T : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring T] (f : R →+* S) (g : R →+* T) (x : R) : ↑(RingHom.prod f g) x = (↑f x, ↑g x)

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theorem RingHom.fst_comp_prod {R : Type u_1} {S : Type u_2} {T : Type u_3} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring T] (f : R →+* S) (g : R →+* T) : RingHom.comp (RingHom.fst S T) (RingHom.prod f g) = f

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theorem RingHom.snd_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring T] (f : R →+* S) (g : R →+* T) : RingHom.comp (RingHom.snd S T) (RingHom.prod f g) = g

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theorem RingHom.prod_unique {R : Type u_1} {S : Type u_3} {T : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring T] (f : R →+* S × T) : RingHom.prod (RingHom.comp (RingHom.fst S T) f) (RingHom.comp (RingHom.snd S T) f) = f

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def RingHom.prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring R'] [inst : NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') : R × S →+* R' × S'

Prod.map as a RingHom.

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• RingHom.prodMap f g = RingHom.prod (RingHom.comp f (RingHom.fst R S)) (RingHom.comp g (RingHom.snd R S))

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theorem RingHom.prodMap_def {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring R'] [inst : NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') : RingHom.prodMap f g = RingHom.prod (RingHom.comp f (RingHom.fst R S)) (RingHom.comp g (RingHom.snd R S))

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theorem RingHom.coe_prodMap {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring R'] [inst : NonAssocSemiring S'] (f : R →+* R') (g : S →+* S') : ↑(RingHom.prodMap f g) = Prod.map ↑f ↑g

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theorem RingHom.prod_comp_prodMap {R : Type u_2} {R' : Type u_4} {S : Type u_3} {S' : Type u_5} {T : Type u_1} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : NonAssocSemiring R'] [inst : NonAssocSemiring S'] [inst : NonAssocSemiring T] (f : T →+* R) (g : T →+* S) (f' : R →+* R') (g' : S →+* S') : RingHom.comp (RingHom.prodMap f' g') (RingHom.prod f g) = RingHom.prod (RingHom.comp f' f) (RingHom.comp g' g)

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def RingEquiv.prodComm {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : R × S ≃+* S × R

Swapping components as an equivalence of (semi)rings.

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theorem RingEquiv.coe_prod_comm {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : ↑RingEquiv.prodComm = Prod.swap

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theorem RingEquiv.coe_prod_comm_symm {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : ↑(RingEquiv.symm RingEquiv.prodComm) = Prod.swap

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theorem RingEquiv.fst_comp_coe_prod_comm {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : RingHom.comp (RingHom.fst S R) ↑RingEquiv.prodComm = RingHom.snd R S

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theorem RingEquiv.snd_comp_coe_prod_comm {R : Type u_1} {S : Type u_2} [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] : RingHom.comp (RingHom.snd S R) ↑RingEquiv.prodComm = RingHom.fst R S

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theorem RingEquiv.prodZeroRing_apply (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] (x : R) : ↑(RingEquiv.prodZeroRing R S) x = (x, 0)

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theorem RingEquiv.prodZeroRing_symm_apply (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] (self : R × S) : ↑(RingEquiv.symm (RingEquiv.prodZeroRing R S)) self = self.fst

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def RingEquiv.prodZeroRing (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] : R ≃+* R × S

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

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theorem RingEquiv.zeroRingProd_apply (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] (x : R) : ↑(RingEquiv.zeroRingProd R S) x = (0, x)

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theorem RingEquiv.zeroRingProd_symm_apply (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] (self : S × R) : ↑(RingEquiv.symm (RingEquiv.zeroRingProd R S)) self = self.snd

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def RingEquiv.zeroRingProd (R : Type u_1) (S : Type u_2) [inst : NonAssocSemiring R] [inst : NonAssocSemiring S] [inst : Subsingleton S] : R ≃+* S × R

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

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theorem false_of_nontrivial_of_product_domain (R : Type u_1) (S : Type u_2) [inst : Ring R] [inst : Ring S] [inst : IsDomain (R × S)] [inst : Nontrivial R] [inst : Nontrivial S] : False

The product of two nontrivial rings is not a domain

Order

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instance instOrderedSemiringProd {α : Type u_1} {β : Type u_2} [inst : OrderedSemiring α] [inst : OrderedSemiring β] : OrderedSemiring (α × β)

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instance instOrderedCommSemiringProd {α : Type u_1} {β : Type u_2} [inst : OrderedCommSemiring α] [inst : OrderedCommSemiring β] : OrderedCommSemiring (α × β)

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instance instOrderedRingProd {α : Type u_1} {β : Type u_2} [inst : OrderedRing α] [inst : OrderedRing β] : OrderedRing (α × β)

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instance instOrderedCommRingProd {α : Type u_1} {β : Type u_2} [inst : OrderedCommRing α] [inst : OrderedCommRing β] : OrderedCommRing (α × β)

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• instOrderedCommRingProd = let src := inferInstanceAs (OrderedRing (α × β)); let src_1 := inferInstanceAs (CommRing (α × β)); OrderedCommRing.mk (_ : ∀ (a b : α × β), a * b = b * a)