# mathlib3documentation

algebra.group_with_zero.divisibility

# Divisibility in groups with zero. #

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Lemmas about divisibility in groups and monoids with zero.

theorem eq_zero_of_zero_dvd {α : Type u_1} {a : α} (h : 0 a) :
a = 0
@[simp]
theorem zero_dvd_iff {α : Type u_1} {a : α} :
0 a a = 0

Given an element a of a commutative semigroup with zero, there exists another element whose product with zero equals a iff a equals zero.

@[simp]
theorem dvd_zero {α : Type u_1} (a : α) :
a 0
theorem mul_dvd_mul_iff_left {α : Type u_1} {a b c : α} (ha : a 0) :
a * b a * c b c

Given two elements b, c of a cancel_monoid_with_zero and a nonzero element a, a*b divides a*c iff b divides c.

theorem mul_dvd_mul_iff_right {α : Type u_1} {a b c : α} (hc : c 0) :
a * c b * c a b

Given two elements a, b of a commutative cancel_monoid_with_zero and a nonzero element c, a*c divides b*c iff a divides b.

def dvd_not_unit {α : Type u_1} (a b : α) :
Prop

dvd_not_unit a b expresses that a divides b "strictly", i.e. that b divided by a is not a unit.

Equations
theorem dvd_not_unit_of_dvd_of_not_dvd {α : Type u_1} {a b : α} (hd : a b) (hnd : ¬b a) :
b
theorem dvd_and_not_dvd_iff {α : Type u_1} {x y : α} :
x y ¬y x y
theorem ne_zero_of_dvd_ne_zero {α : Type u_1} {p q : α} (h₁ : q 0) (h₂ : p q) :
p 0
theorem dvd_antisymm {α : Type u_1} {a b : α} :
a b b a a = b
theorem dvd_antisymm' {α : Type u_1} {a b : α} :
a b b a b = a
theorem has_dvd.dvd.antisymm {α : Type u_1} {a b : α} :
a b b a a = b

Alias of dvd_antisymm.

theorem has_dvd.dvd.antisymm' {α : Type u_1} {a b : α} :
a b b a b = a

Alias of dvd_antisymm'.

theorem eq_of_forall_dvd {α : Type u_1} {a b : α} (h : (c : α), a c b c) :
a = b
theorem eq_of_forall_dvd' {α : Type u_1} {a b : α} (h : (c : α), c a c b) :
a = b