mathlib docs: algebra.group.prod

source

@[instance]

def prod.has_mul {M : Type u_5} {N : Type u_6} [has_mul M] [has_mul N] :

has_mul (M × N)

Equations

prod.has_mul = {mul := λ (p q : M × N), (p.fst * q.fst, p.snd * q.snd)}

source

@[instance]

def prod.has_add {M : Type u_5} {N : Type u_6} [has_add M] [has_add N] :

has_add (M × N)

source

@[simp]

theorem prod.fst_mul {M : Type u_5} {N : Type u_6} [has_mul M] [has_mul N] (p q : M × N) :

(p * q).fst = p.fst * q.fst

source

@[simp]

theorem prod.fst_add {M : Type u_5} {N : Type u_6} [has_add M] [has_add N] (p q : M × N) :

(p + q).fst = p.fst + q.fst

source

@[simp]

theorem prod.snd_mul {M : Type u_5} {N : Type u_6} [has_mul M] [has_mul N] (p q : M × N) :

(p * q).snd = p.snd * q.snd

source

@[simp]

theorem prod.snd_add {M : Type u_5} {N : Type u_6} [has_add M] [has_add N] (p q : M × N) :

(p + q).snd = p.snd + q.snd

source

@[simp]

theorem prod.mk_mul_mk {M : Type u_5} {N : Type u_6} [has_mul M] [has_mul N] (a₁ a₂ : M) (b₁ b₂ : N) :

(a₁, b₁) * (a₂, b₂) = (a₁ * a₂, b₁ * b₂)

source

@[simp]

theorem prod.mk_add_mk {M : Type u_5} {N : Type u_6} [has_add M] [has_add N] (a₁ a₂ : M) (b₁ b₂ : N) :

(a₁, b₁) + (a₂, b₂) = (a₁ + a₂, b₁ + b₂)

source

@[instance]

def prod.has_one {M : Type u_5} {N : Type u_6} [has_one M] [has_one N] :

has_one (M × N)

Equations

prod.has_one = {one := (1, 1)}

source

@[instance]

def prod.has_zero {M : Type u_5} {N : Type u_6} [has_zero M] [has_zero N] :

has_zero (M × N)

source

@[simp]

theorem prod.fst_one {M : Type u_5} {N : Type u_6} [has_one M] [has_one N] :

1.fst = 1

source

@[simp]

theorem prod.fst_zero {M : Type u_5} {N : Type u_6} [has_zero M] [has_zero N] :

0.fst = 0

source

@[simp]

theorem prod.snd_one {M : Type u_5} {N : Type u_6} [has_one M] [has_one N] :

1.snd = 1

source

@[simp]

theorem prod.snd_zero {M : Type u_5} {N : Type u_6} [has_zero M] [has_zero N] :

0.snd = 0

source

theorem prod.one_eq_mk {M : Type u_5} {N : Type u_6} [has_one M] [has_one N] :

1 = (1, 1)

source

theorem prod.zero_eq_mk {M : Type u_5} {N : Type u_6} [has_zero M] [has_zero N] :

0 = (0, 0)

source

@[simp]

theorem prod.mk_eq_zero {M : Type u_5} {N : Type u_6} [has_zero M] [has_zero N] {x : M} {y : N} :

(x, y) = 0 ↔ x = 0 ∧ y = 0

source

@[simp]

theorem prod.mk_eq_one {M : Type u_5} {N : Type u_6} [has_one M] [has_one N] {x : M} {y : N} :

(x, y) = 1 ↔ x = 1 ∧ y = 1

source

theorem prod.fst_add_snd {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] (p : M × N) :

(p.fst, 0) + (0, p.snd) = p

source

theorem prod.fst_mul_snd {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] (p : M × N) :

(p.fst, 1) * (1, p.snd) = p

source

@[instance]

def prod.has_inv {M : Type u_5} {N : Type u_6} [has_inv M] [has_inv N] :

has_inv (M × N)

Equations

prod.has_inv = {inv := λ (p : M × N), ((p.fst)⁻¹, (p.snd)⁻¹)}

source

@[instance]

def prod.has_neg {M : Type u_5} {N : Type u_6} [has_neg M] [has_neg N] :

has_neg (M × N)

source

@[simp]

theorem prod.fst_neg {G : Type u_3} {H : Type u_4} [has_neg G] [has_neg H] (p : G × H) :

(-p).fst = -p.fst

source

@[simp]

theorem prod.fst_inv {G : Type u_3} {H : Type u_4} [has_inv G] [has_inv H] (p : G × H) :

p⁻¹.fst = (p.fst)⁻¹

source

@[simp]

theorem prod.snd_inv {G : Type u_3} {H : Type u_4} [has_inv G] [has_inv H] (p : G × H) :

p⁻¹.snd = (p.snd)⁻¹

source

@[simp]

theorem prod.snd_neg {G : Type u_3} {H : Type u_4} [has_neg G] [has_neg H] (p : G × H) :

(-p).snd = -p.snd

source

@[simp]

theorem prod.inv_mk {G : Type u_3} {H : Type u_4} [has_inv G] [has_inv H] (a : G) (b : H) :

(a, b)⁻¹ = (a⁻¹, b⁻¹)

source

@[simp]

theorem prod.neg_mk {G : Type u_3} {H : Type u_4} [has_neg G] [has_neg H] (a : G) (b : H) :

-(a, b) = (-a, -b)

source

@[instance]

def prod.semigroup {M : Type u_5} {N : Type u_6} [semigroup M] [semigroup N] :

semigroup (M × N)

Equations

prod.semigroup = {mul := has_mul.mul prod.has_mul, mul_assoc := _}

source

@[instance]

def prod.add_semigroup {M : Type u_5} {N : Type u_6} [add_semigroup M] [add_semigroup N] :

add_semigroup (M × N)

source

@[instance]

def prod.monoid {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

monoid (M × N)

Equations

prod.monoid = {mul := semigroup.mul prod.semigroup, mul_assoc := _, one := 1, one_mul := _, mul_one := _}

source

@[instance]

def prod.add_monoid {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

add_monoid (M × N)

source

@[instance]

def prod.add_group {G : Type u_3} {H : Type u_4} [add_group G] [add_group H] :

add_group (G × H)

source

@[instance]

def prod.group {G : Type u_3} {H : Type u_4} [group G] [group H] :

group (G × H)

Equations

prod.group = {mul := monoid.mul prod.monoid, mul_assoc := _, one := monoid.one prod.monoid, one_mul := _, mul_one := _, inv := has_inv.inv prod.has_inv, mul_left_inv := _}

source

@[simp]

theorem prod.fst_sub {A : Type u_1} {B : Type u_2} [add_group A] [add_group B] (a b : A × B) :

(a - b).fst = a.fst - b.fst

source

@[simp]

theorem prod.snd_sub {A : Type u_1} {B : Type u_2} [add_group A] [add_group B] (a b : A × B) :

(a - b).snd = a.snd - b.snd

source

@[simp]

theorem prod.mk_sub_mk {A : Type u_1} {B : Type u_2} [add_group A] [add_group B] (x₁ x₂ : A) (y₁ y₂ : B) :

(x₁, y₁) - (x₂, y₂) = (x₁ - x₂, y₁ - y₂)

source

@[instance]

def prod.comm_semigroup {G : Type u_3} {H : Type u_4} [comm_semigroup G] [comm_semigroup H] :

comm_semigroup (G × H)

Equations

prod.comm_semigroup = {mul := semigroup.mul prod.semigroup, mul_assoc := _, mul_comm := _}

source

@[instance]

def prod.add_comm_semigroup {G : Type u_3} {H : Type u_4} [add_comm_semigroup G] [add_comm_semigroup H] :

add_comm_semigroup (G × H)

source

@[instance]

def prod.comm_monoid {M : Type u_5} {N : Type u_6} [comm_monoid M] [comm_monoid N] :

comm_monoid (M × N)

Equations

prod.comm_monoid = {mul := comm_semigroup.mul prod.comm_semigroup, mul_assoc := _, one := monoid.one prod.monoid, one_mul := _, mul_one := _, mul_comm := _}

source

@[instance]

def prod.add_comm_monoid {M : Type u_5} {N : Type u_6} [add_comm_monoid M] [add_comm_monoid N] :

add_comm_monoid (M × N)

source

@[instance]

def prod.add_comm_group {G : Type u_3} {H : Type u_4} [add_comm_group G] [add_comm_group H] :

add_comm_group (G × H)

source

@[instance]

def prod.comm_group {G : Type u_3} {H : Type u_4} [comm_group G] [comm_group H] :

comm_group (G × H)

Equations

prod.comm_group = {mul := comm_semigroup.mul prod.comm_semigroup, mul_assoc := _, one := group.one prod.group, one_mul := _, mul_one := _, inv := group.inv prod.group, mul_left_inv := _, mul_comm := _}

source

def monoid_hom.fst (M : Type u_5) (N : Type u_6) [monoid M] [monoid N] :

M × N →* M

Given monoids M, N, the natural projection homomorphism from M × N to M.

Equations

monoid_hom.fst M N = {to_fun := prod.fst N, map_one' := _, map_mul' := _}

source

def add_monoid_hom.fst (M : Type u_5) (N : Type u_6) [add_monoid M] [add_monoid N] :

M × N →+ M

Given additive monoids A, B, the natural projection homomorphism from A × B to A

source

def add_monoid_hom.snd (M : Type u_5) (N : Type u_6) [add_monoid M] [add_monoid N] :

M × N →+ N

Given additive monoids A, B, the natural projection homomorphism from A × B to B

source

def monoid_hom.snd (M : Type u_5) (N : Type u_6) [monoid M] [monoid N] :

M × N →* N

Given monoids M, N, the natural projection homomorphism from M × N to N.

Equations

monoid_hom.snd M N = {to_fun := prod.snd N, map_one' := _, map_mul' := _}

source

def monoid_hom.inl (M : Type u_5) (N : Type u_6) [monoid M] [monoid N] :

M →* M × N

Given monoids M, N, the natural inclusion homomorphism from M to M × N.

Equations

monoid_hom.inl M N = {to_fun := λ (x : M), (x, 1), map_one' := _, map_mul' := _}

source

def add_monoid_hom.inl (M : Type u_5) (N : Type u_6) [add_monoid M] [add_monoid N] :

M →+ M × N

Given additive monoids A, B, the natural inclusion homomorphism from A to A × B.

source

def monoid_hom.inr (M : Type u_5) (N : Type u_6) [monoid M] [monoid N] :

N →* M × N

Given monoids M, N, the natural inclusion homomorphism from N to M × N.

Equations

monoid_hom.inr M N = {to_fun := λ (y : N), (1, y), map_one' := _, map_mul' := _}

source

def add_monoid_hom.inr (M : Type u_5) (N : Type u_6) [add_monoid M] [add_monoid N] :

N →+ M × N

Given additive monoids A, B, the natural inclusion homomorphism from B to A × B.

source

@[simp]

theorem monoid_hom.coe_fst {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

⇑(monoid_hom.fst M N) = prod.fst

source

@[simp]

theorem add_monoid_hom.coe_fst {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

⇑(add_monoid_hom.fst M N) = prod.fst

source

@[simp]

theorem add_monoid_hom.coe_snd {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

⇑(add_monoid_hom.snd M N) = prod.snd

source

@[simp]

theorem monoid_hom.coe_snd {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

⇑(monoid_hom.snd M N) = prod.snd

source

@[simp]

theorem add_monoid_hom.inl_apply {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] (x : M) :

⇑(add_monoid_hom.inl M N) x = (x, 0)

source

@[simp]

theorem monoid_hom.inl_apply {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] (x : M) :

⇑(monoid_hom.inl M N) x = (x, 1)

source

@[simp]

theorem add_monoid_hom.inr_apply {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] (y : N) :

⇑(add_monoid_hom.inr M N) y = (0, y)

source

@[simp]

theorem monoid_hom.inr_apply {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] (y : N) :

⇑(monoid_hom.inr M N) y = (1, y)

source

@[simp]

theorem add_monoid_hom.fst_comp_inl {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

(add_monoid_hom.fst M N).comp (add_monoid_hom.inl M N) = add_monoid_hom.id M

source

@[simp]

theorem monoid_hom.fst_comp_inl {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

(monoid_hom.fst M N).comp (monoid_hom.inl M N) = monoid_hom.id M

source

@[simp]

theorem add_monoid_hom.snd_comp_inl {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

(add_monoid_hom.snd M N).comp (add_monoid_hom.inl M N) = 0

source

@[simp]

theorem monoid_hom.snd_comp_inl {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

(monoid_hom.snd M N).comp (monoid_hom.inl M N) = 1

source

@[simp]

theorem add_monoid_hom.fst_comp_inr {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

(add_monoid_hom.fst M N).comp (add_monoid_hom.inr M N) = 0

source

@[simp]

theorem monoid_hom.fst_comp_inr {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

(monoid_hom.fst M N).comp (monoid_hom.inr M N) = 1

source

@[simp]

theorem add_monoid_hom.snd_comp_inr {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] :

(add_monoid_hom.snd M N).comp (add_monoid_hom.inr M N) = add_monoid_hom.id N

source

@[simp]

theorem monoid_hom.snd_comp_inr {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] :

(monoid_hom.snd M N).comp (monoid_hom.inr M N) = monoid_hom.id N

source

def monoid_hom.prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [monoid P] :

(M →* N) → (M →* P) → M →* N × P

Combine two monoid_homs f : M →* N, g : M →* P into f.prod g : M →* N × P given by (f.prod g) x = (f x, g x)

Equations

f.prod g = {to_fun := λ (x : M), (⇑f x, ⇑g x), map_one' := _, map_mul' := _}

source

def add_monoid_hom.prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_monoid P] :

(M →+ N) → (M →+ P) → M →+ N × P

Combine two add_monoid_homs f : M →+ N, g : M →+ P into f.prod g : M →+ N × P given by (f.prod g) x = (f x, g x)

source

@[simp]

theorem monoid_hom.prod_apply {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [monoid P] (f : M →* N) (g : M →* P) (x : M) :

⇑(f.prod g) x = (⇑f x, ⇑g x)

source

@[simp]

theorem add_monoid_hom.prod_apply {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_monoid P] (f : M →+ N) (g : M →+ P) (x : M) :

⇑(f.prod g) x = (⇑f x, ⇑g x)

source

@[simp]

theorem monoid_hom.fst_comp_prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [monoid P] (f : M →* N) (g : M →* P) :

(monoid_hom.fst N P).comp (f.prod g) = f

source

@[simp]

theorem add_monoid_hom.fst_comp_prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_monoid P] (f : M →+ N) (g : M →+ P) :

(add_monoid_hom.fst N P).comp (f.prod g) = f

source

@[simp]

theorem monoid_hom.snd_comp_prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [monoid P] (f : M →* N) (g : M →* P) :

(monoid_hom.snd N P).comp (f.prod g) = g

source

@[simp]

theorem add_monoid_hom.snd_comp_prod {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_monoid P] (f : M →+ N) (g : M →+ P) :

(add_monoid_hom.snd N P).comp (f.prod g) = g

source

@[simp]

theorem monoid_hom.prod_unique {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [monoid P] (f : M →* N × P) :

((monoid_hom.fst N P).comp f).prod ((monoid_hom.snd N P).comp f) = f

source

@[simp]

theorem add_monoid_hom.prod_unique {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_monoid P] (f : M →+ N × P) :

((add_monoid_hom.fst N P).comp f).prod ((add_monoid_hom.snd N P).comp f) = f

source

def monoid_hom.prod_map {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] {M' : Type u_8} {N' : Type u_9} [monoid M'] [monoid N'] :

(M →* M') → (N →* N') → M × N →* M' × N'

prod.map as a monoid_hom.

Equations

f.prod_map g = (f.comp (monoid_hom.fst M N)).prod (g.comp (monoid_hom.snd M N))

source

def add_monoid_hom.prod_map {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] {M' : Type u_8} {N' : Type u_9} [add_monoid M'] [add_monoid N'] :

(M →+ M') → (N →+ N') → M × N →+ M' × N'

prod.map as an add_monoid_hom

source

theorem monoid_hom.prod_map_def {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] {M' : Type u_8} {N' : Type u_9} [monoid M'] [monoid N'] (f : M →* M') (g : N →* N') :

f.prod_map g = (f.comp (monoid_hom.fst M N)).prod (g.comp (monoid_hom.snd M N))

source

theorem add_monoid_hom.prod_map_def {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] {M' : Type u_8} {N' : Type u_9} [add_monoid M'] [add_monoid N'] (f : M →+ M') (g : N →+ N') :

f.prod_map g = (f.comp (add_monoid_hom.fst M N)).prod (g.comp (add_monoid_hom.snd M N))

source

@[simp]

theorem add_monoid_hom.coe_prod_map {M : Type u_5} {N : Type u_6} [add_monoid M] [add_monoid N] {M' : Type u_8} {N' : Type u_9} [add_monoid M'] [add_monoid N'] (f : M →+ M') (g : N →+ N') :

⇑(f.prod_map g) = prod.map ⇑f ⇑g

source

@[simp]

theorem monoid_hom.coe_prod_map {M : Type u_5} {N : Type u_6} [monoid M] [monoid N] {M' : Type u_8} {N' : Type u_9} [monoid M'] [monoid N'] (f : M →* M') (g : N →* N') :

⇑(f.prod_map g) = prod.map ⇑f ⇑g

source

theorem monoid_hom.prod_comp_prod_map {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] {M' : Type u_8} {N' : Type u_9} [monoid M'] [monoid N'] [monoid P] (f : P →* M) (g : P →* N) (f' : M →* M') (g' : N →* N') :

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

source

theorem add_monoid_hom.prod_comp_prod_map {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] {M' : Type u_8} {N' : Type u_9} [add_monoid M'] [add_monoid N'] [add_monoid P] (f : P →+ M) (g : P →+ N) (f' : M →+ M') (g' : N →+ N') :

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

source

def add_monoid_hom.coprod {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_comm_monoid P] :

(M →+ P) → (N →+ P) → M × N →+ P

Coproduct of two add_monoid_homs with the same codomain: f.coprod g (p : M × N) = f p.1 + g p.2.

source

def monoid_hom.coprod {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] :

(M →* P) → (N →* P) → M × N →* P

Coproduct of two monoid_homs with the same codomain: f.coprod g (p : M × N) = f p.1 * g p.2.

Equations

f.coprod g = f.comp (monoid_hom.fst M N) * g.comp (monoid_hom.snd M N)

source

@[simp]

theorem monoid_hom.coprod_apply {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] (f : M →* P) (g : N →* P) (p : M × N) :

⇑(f.coprod g) p = ⇑f p.fst * ⇑g p.snd

source

@[simp]

theorem add_monoid_hom.coprod_apply {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_comm_monoid P] (f : M →+ P) (g : N →+ P) (p : M × N) :

⇑(f.coprod g) p = ⇑f p.fst + ⇑g p.snd

source

@[simp]

theorem add_monoid_hom.coprod_comp_inl {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_comm_monoid P] (f : M →+ P) (g : N →+ P) :

(f.coprod g).comp (add_monoid_hom.inl M N) = f

source

@[simp]

theorem monoid_hom.coprod_comp_inl {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] (f : M →* P) (g : N →* P) :

(f.coprod g).comp (monoid_hom.inl M N) = f

source

@[simp]

theorem add_monoid_hom.coprod_comp_inr {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_comm_monoid P] (f : M →+ P) (g : N →+ P) :

(f.coprod g).comp (add_monoid_hom.inr M N) = g

source

@[simp]

theorem monoid_hom.coprod_comp_inr {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] (f : M →* P) (g : N →* P) :

(f.coprod g).comp (monoid_hom.inr M N) = g

source

@[simp]

theorem monoid_hom.coprod_unique {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] (f : M × N →* P) :

(f.comp (monoid_hom.inl M N)).coprod (f.comp (monoid_hom.inr M N)) = f

source

@[simp]

theorem add_monoid_hom.coprod_unique {M : Type u_5} {N : Type u_6} {P : Type u_7} [add_monoid M] [add_monoid N] [add_comm_monoid P] (f : M × N →+ P) :

(f.comp (add_monoid_hom.inl M N)).coprod (f.comp (add_monoid_hom.inr M N)) = f

source

@[simp]

theorem add_monoid_hom.coprod_inl_inr {M : Type u_1} {N : Type u_2} [add_comm_monoid M] [add_comm_monoid N] :

(add_monoid_hom.inl M N).coprod (add_monoid_hom.inr M N) = add_monoid_hom.id (M × N)

source

@[simp]

theorem monoid_hom.coprod_inl_inr {M : Type u_1} {N : Type u_2} [comm_monoid M] [comm_monoid N] :

(monoid_hom.inl M N).coprod (monoid_hom.inr M N) = monoid_hom.id (M × N)

source

theorem monoid_hom.comp_coprod {M : Type u_5} {N : Type u_6} {P : Type u_7} [monoid M] [monoid N] [comm_monoid P] {Q : Type u_1} [comm_monoid Q] (h : P →* Q) (f : M →* P) (g : N →* P) :

h.comp (f.coprod g) = (h.comp f).coprod (h.comp g)

mathlib documentation

algebra.group.prod

Monoid, group etc structures on `M × N`

General documentation

Additional documentation

Library

mathlib documentation

algebra.​group.​prod

Monoid, group etc structures on M × N

General documentation

Additional documentation

Library

algebra.group.prod

Monoid, group etc structures on `M × N`