Divisibility in groups with zero.

Lemmas about divisibility in groups and monoids with zero.

theorem eq_zero_of_zero_dvd {α : Type u_1} [SemigroupWithZero α] {a : α} (h : 0 ∣ a) : a = 0

@[simp]

theorem zero_dvd_iff {α : Type u_1} [SemigroupWithZero α] {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} [SemigroupWithZero α] (a : α) : a ∣ 0

theorem mul_dvd_mul_iff_left {α : Type u_1} [MonoidWithZero α] [IsLeftCancelMulZero α] {a b c : α} (ha : a ≠ 0) : a * b ∣ a * c ↔ b ∣ c

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

theorem mul_dvd_mul_iff_right {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b c : α} (hc : c ≠ 0) : a * c ∣ b * c ↔ a ∣ b

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

def DvdNotUnit {α : Type u_1} [CommMonoidWithZero α] (a b : α) : Prop

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

theorem dvdNotUnit_of_dvd_of_not_dvd {α : Type u_1} [CommMonoidWithZero α] {a b : α} (hd : a ∣ b) (hnd : ¬b ∣ a) : DvdNotUnit a b

theorem isRelPrime_zero_left {α : Type u_1} [CommMonoidWithZero α] {x : α} : IsRelPrime 0 x ↔ IsUnit x

theorem isRelPrime_zero_right {α : Type u_1} [CommMonoidWithZero α] {x : α} : IsRelPrime x 0 ↔ IsUnit x

theorem not_isRelPrime_zero_zero {α : Type u_1} [CommMonoidWithZero α] [Nontrivial α] : ¬IsRelPrime 0 0

theorem IsRelPrime.ne_zero_or_ne_zero {α : Type u_1} [CommMonoidWithZero α] {x y : α} [Nontrivial α] (h : IsRelPrime x y) : x ≠ 0 ∨ y ≠ 0

theorem isRelPrime_of_no_nonunits_factors {α : Type u_1} [MonoidWithZero α] {x y : α} (nonzero : ¬(x = 0 ∧ y = 0)) (H : ∀ (z : α), ¬IsUnit z → z ≠ 0 → z ∣ x → ¬z ∣ y) : IsRelPrime x y

theorem dvd_and_not_dvd_iff {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {x y : α} : x ∣ y ∧ ¬y ∣ x ↔ DvdNotUnit x y

theorem ne_zero_of_dvd_ne_zero {α : Type u_1} [MonoidWithZero α] {p q : α} (h₁ : q ≠ 0) (h₂ : p ∣ q) : p ≠ 0

theorem isPrimal_zero {α : Type u_1} [MonoidWithZero α] : IsPrimal 0

theorem IsPrimal.mul {α : Type u_2} [CommMonoidWithZero α] [IsCancelMulZero α] {m n : α} (hm : IsPrimal m) (hn : IsPrimal n) : IsPrimal (m * n)

theorem dvd_antisymm {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] : a ∣ b → b ∣ a → a = b

theorem dvd_antisymm' {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] : a ∣ b → b ∣ a → b = a

theorem Dvd.dvd.antisymm {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] : a ∣ b → b ∣ a → a = b

Alias of dvd_antisymm.

theorem Dvd.dvd.antisymm' {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] : a ∣ b → b ∣ a → b = a

Alias of dvd_antisymm'.

theorem eq_of_forall_dvd {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] (h : ∀ (c : α), a ∣ c ↔ b ∣ c) : a = b

theorem eq_of_forall_dvd' {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] (h : ∀ (c : α), c ∣ a ↔ c ∣ b) : a = b

theorem pow_dvd_pow_iff {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a : α} {m n : ℕ} (ha₀ : a ≠ 0) (ha : ¬IsUnit a) : a ^ n ∣ a ^ m ↔ n ≤ m

@[simp]

theorem GroupWithZero.dvd_iff {α : Type u_1} [GroupWithZero α] {m n : α} : m ∣ n ↔ m = 0 → n = 0

∣ is not a useful definition if an inverse is available.