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Mathlib.Algebra.GroupWithZero.Divisibility

Divisibility in groups with zero. #

Lemmas about divisibility in groups and monoids with zero.

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

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@[simp]
theorem dvd_zero {α : Type u_1} [SemigroupWithZero α] (a : α) :
a ∣ 0
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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.

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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.

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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.

Equations
Instances For
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    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
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    theorem isRelPrime_zero_left {α : Type u_1} [CommMonoidWithZero α] {x : α} :
    IsRelPrime 0 x ↔ IsUnit x
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    theorem isRelPrime_zero_right {α : Type u_1} [CommMonoidWithZero α] {x : α} :
    IsRelPrime x 0 ↔ IsUnit x
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    theorem not_isRelPrime_zero_zero {α : Type u_1} [CommMonoidWithZero α] [Nontrivial α] :
    ¬IsRelPrime 0 0
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    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
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    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
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    theorem dvd_and_not_dvd_iff {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {x y : α} :
    x ∣ y ∧ ¬y ∣ x ↔ DvdNotUnit x y
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    theorem ne_zero_of_dvd_ne_zero {α : Type u_1} [MonoidWithZero α] {p q : α} (h₁ : q ≠ 0) (h₂ : p ∣ q) :
    p ≠ 0
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    theorem isPrimal_zero {α : Type u_1} [MonoidWithZero α] :
    IsPrimal 0
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    theorem IsPrimal.mul {α : Type u_2} [CommMonoidWithZero α] [IsCancelMulZero α] {m n : α} (hm : IsPrimal m) (hn : IsPrimal n) :
    IsPrimal (m * n)
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    theorem dvd_antisymm {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] :
    a ∣ b → b ∣ a → a = b
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    theorem dvd_antisymm' {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} [Subsingleton αˣ] :
    a ∣ b → b ∣ a → b = a
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    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.

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    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'.

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    instance instIsPartialOrderDvd {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] [Subsingleton αˣ] :
    IsPartialOrder α fun (x1 x2 : α) => x1 ∣ x2
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    instance instIsPartialOrderRightDvd {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] [Subsingleton αˣ] :
    IsPartialOrder α RightDvd
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    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
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    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
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    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
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    theorem mul_dvd_left_iff_isUnit {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} (ha0 : a ≠ 0) :
    a * b ∣ a ↔ IsUnit b
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    theorem mul_dvd_right_iff_isUnit {α : Type u_1} [CommMonoidWithZero α] [IsCancelMulZero α] {a b : α} (ha0 : a ≠ 0) :
    b * a ∣ a ↔ IsUnit b
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    @[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.