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} :

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

source

@[simp]

theorem dvd_zero {α : Type u_1} [SemigroupWithZero α] (a : α) :

a ∣ 0

source

theorem mul_dvd_mul_iff_left {α : Type u_1} [CancelMonoidWithZero α] {a : α} {b : α} {c : α} (ha : a ≠ 0) :

a * b ∣ a * c ↔ b ∣ c

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

source

theorem mul_dvd_mul_iff_right {α : Type u_1} [CancelCommMonoidWithZero α] {a : α} {b : α} {c : α} (hc : c ≠ 0) :

a * c ∣ b * c ↔ a ∣ b

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

source

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

Prop

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

Equations

DvdNotUnit a b = (a ≠ 0 ∧ ∃ (x : α), ¬IsUnit x ∧ b = a * x)

Instances For

source

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

DvdNotUnit a b

source

theorem dvd_and_not_dvd_iff {α : Type u_1} [CancelCommMonoidWithZero α] {x : α} {y : α} :

x ∣ y ∧ ¬y ∣ x ↔ DvdNotUnit x y

source

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

p ≠ 0

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theorem dvd_antisymm {α : Type u_1} [CancelCommMonoidWithZero α] [Subsingleton αˣ] {a : α} {b : α} :

a ∣ b → b ∣ a → a = b

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theorem dvd_antisymm' {α : Type u_1} [CancelCommMonoidWithZero α] [Subsingleton αˣ] {a : α} {b : α} :

a ∣ b → b ∣ a → b = a

source

theorem Dvd.dvd.antisymm {α : Type u_1} [CancelCommMonoidWithZero α] [Subsingleton αˣ] {a : α} {b : α} :

a ∣ b → b ∣ a → a = b

Alias of dvd_antisymm.

source

theorem Dvd.dvd.antisymm' {α : Type u_1} [CancelCommMonoidWithZero α] [Subsingleton αˣ] {a : α} {b : α} :

a ∣ b → b ∣ a → b = a

Alias of dvd_antisymm'.

source

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

a = b

source

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

a = b