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 ring_homs:

fst R S : R × S →+* R, snd R S : R × S →+* R: projections prod.fst and prod.snd as ring_homs;
f.prod g :R →+* S × T: sendsxto(f x, g x)`;
f.prod_map g :R × S → R' × S':prod.map f gas aring_hom, sends(x, y)to(f x, g y)`.

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@[instance]

def prod.distrib {R : Type u_1} {S : Type u_3} [distrib R] [distrib S] :

distrib (R × S)

Product of two distributive types is distributive.

Equations

prod.distrib = {mul := has_mul.mul prod.has_mul, add := has_add.add prod.has_add, left_distrib := _, right_distrib := _}

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@[instance]

def prod.semiring {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

semiring (R × S)

Product of two semirings is a semiring.

Equations

prod.semiring = {add := add_comm_monoid.add prod.add_comm_monoid, add_assoc := _, zero := add_comm_monoid.zero prod.add_comm_monoid, zero_add := _, add_zero := _, nsmul := add_comm_monoid.nsmul prod.add_comm_monoid, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := monoid_with_zero.mul prod.monoid_with_zero, mul_assoc := _, one := monoid_with_zero.one prod.monoid_with_zero, one_mul := _, mul_one := _, npow := monoid_with_zero.npow prod.monoid_with_zero, npow_zero' := _, npow_succ' := _, zero_mul := _, mul_zero := _, left_distrib := _, right_distrib := _}

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@[instance]

def prod.comm_semiring {R : Type u_1} {S : Type u_3} [comm_semiring R] [comm_semiring S] :

comm_semiring (R × S)

Product of two commutative semirings is a commutative semiring.

Equations

prod.comm_semiring = {add := semiring.add prod.semiring, add_assoc := _, zero := semiring.zero prod.semiring, zero_add := _, add_zero := _, nsmul := semiring.nsmul prod.semiring, nsmul_zero' := _, nsmul_succ' := _, add_comm := _, mul := semiring.mul prod.semiring, mul_assoc := _, one := semiring.one prod.semiring, one_mul := _, mul_one := _, npow := semiring.npow prod.semiring, npow_zero' := _, npow_succ' := _, zero_mul := _, mul_zero := _, left_distrib := _, right_distrib := _, mul_comm := _}

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@[instance]

def prod.ring {R : Type u_1} {S : Type u_3} [ring R] [ring S] :

ring (R × S)

Product of two rings is a ring.

Equations

prod.ring = {add := add_comm_group.add prod.add_comm_group, add_assoc := _, zero := add_comm_group.zero prod.add_comm_group, zero_add := _, add_zero := _, nsmul := add_comm_group.nsmul prod.add_comm_group, nsmul_zero' := _, nsmul_succ' := _, neg := add_comm_group.neg prod.add_comm_group, sub := add_comm_group.sub prod.add_comm_group, sub_eq_add_neg := _, add_left_neg := _, add_comm := _, mul := semiring.mul prod.semiring, mul_assoc := _, one := semiring.one prod.semiring, one_mul := _, mul_one := _, npow := semiring.npow prod.semiring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _}

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@[instance]

def prod.comm_ring {R : Type u_1} {S : Type u_3} [comm_ring R] [comm_ring S] :

comm_ring (R × S)

Product of two commutative rings is a commutative ring.

Equations

prod.comm_ring = {add := ring.add prod.ring, add_assoc := _, zero := ring.zero prod.ring, zero_add := _, add_zero := _, nsmul := ring.nsmul prod.ring, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg prod.ring, sub := ring.sub prod.ring, sub_eq_add_neg := _, add_left_neg := _, add_comm := _, mul := ring.mul prod.ring, mul_assoc := _, one := ring.one prod.ring, one_mul := _, mul_one := _, npow := ring.npow prod.ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _}

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def ring_hom.fst (R : Type u_1) (S : Type u_3) [semiring R] [semiring S] :

R × S →+* R

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

Equations

ring_hom.fst R S = {to_fun := prod.fst S, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

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def ring_hom.snd (R : Type u_1) (S : Type u_3) [semiring R] [semiring S] :

R × S →+* S

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

Equations

ring_hom.snd R S = {to_fun := prod.snd S, map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

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@[simp]

theorem ring_hom.coe_fst {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

⇑(ring_hom.fst R S) = prod.fst

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@[simp]

theorem ring_hom.coe_snd {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

⇑(ring_hom.snd R S) = prod.snd

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def ring_hom.prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [semiring R] [semiring S] [semiring 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 = {to_fun := λ (x : R), (⇑f x, ⇑g x), map_one' := _, map_mul' := _, map_zero' := _, map_add' := _}

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@[simp]

theorem ring_hom.prod_apply {R : Type u_1} {S : Type u_3} {T : Type u_5} [semiring R] [semiring S] [semiring T] (f : R →+* S) (g : R →+* T) (x : R) :

⇑(f.prod g) x = (⇑f x, ⇑g x)

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@[simp]

theorem ring_hom.fst_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [semiring R] [semiring S] [semiring T] (f : R →+* S) (g : R →+* T) :

(ring_hom.fst S T).comp (f.prod g) = f

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@[simp]

theorem ring_hom.snd_comp_prod {R : Type u_1} {S : Type u_3} {T : Type u_5} [semiring R] [semiring S] [semiring T] (f : R →+* S) (g : R →+* T) :

(ring_hom.snd S T).comp (f.prod g) = g

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theorem ring_hom.prod_unique {R : Type u_1} {S : Type u_3} {T : Type u_5} [semiring R] [semiring S] [semiring T] (f : R →+* S × T) :

((ring_hom.fst S T).comp f).prod ((ring_hom.snd S T).comp f) = f

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def ring_hom.prod_map {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [semiring R] [semiring S] [semiring R'] [semiring S'] (f : R →+* R') (g : S →+* S') :

R × S →* R' × S'

prod.map as a ring_hom.

Equations

f.prod_map g = ↑((f.comp (ring_hom.fst R S)).prod (g.comp (ring_hom.snd R S)))

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theorem ring_hom.prod_map_def {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [semiring R] [semiring S] [semiring R'] [semiring S'] (f : R →+* R') (g : S →+* S') :

f.prod_map g = ↑((f.comp (ring_hom.fst R S)).prod (g.comp (ring_hom.snd R S)))

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@[simp]

theorem ring_hom.coe_prod_map {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} [semiring R] [semiring S] [semiring R'] [semiring S'] (f : R →+* R') (g : S →+* S') :

⇑(f.prod_map g) = prod.map ⇑f ⇑g

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theorem ring_hom.prod_comp_prod_map {R : Type u_1} {R' : Type u_2} {S : Type u_3} {S' : Type u_4} {T : Type u_5} [semiring R] [semiring S] [semiring R'] [semiring S'] [semiring T] (f : T →* R) (g : T →* S) (f' : R →* R') (g' : S →* S') :

(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g)

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def ring_equiv.prod_comm {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

R × S ≃+* S × R

Swapping components as an equivalence of (semi)rings.

Equations

ring_equiv.prod_comm = {to_fun := add_equiv.prod_comm.to_fun, inv_fun := add_equiv.prod_comm.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _}

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@[simp]

theorem ring_equiv.coe_prod_comm {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

⇑ring_equiv.prod_comm = prod.swap

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@[simp]

theorem ring_equiv.coe_prod_comm_symm {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

⇑(ring_equiv.prod_comm.symm) = prod.swap

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@[simp]

theorem ring_equiv.fst_comp_coe_prod_comm {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

(ring_hom.fst S R).comp ↑ring_equiv.prod_comm = ring_hom.snd R S

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@[simp]

theorem ring_equiv.snd_comp_coe_prod_comm {R : Type u_1} {S : Type u_3} [semiring R] [semiring S] :

(ring_hom.snd S R).comp ↑ring_equiv.prod_comm = ring_hom.fst R S

Semiring, ring etc structures on R × S #

Semiring, ring etc structures on `R × S` #