Associated elements.

In this file we define an equivalence relation Associated saying that two elements of a monoid differ by a multiplication by a unit. Then we show that the quotient type Associates is a monoid and prove basic properties of this quotient.

def Associated {M : Type u_1} [Monoid M] (x y : M) : Prop

Two elements of a Monoid are Associated if one of them is another one multiplied by a unit on the right.

theorem Associated.refl {M : Type u_1} [Monoid M] (x : M) : Associated x x

@[simp]

theorem Associated.rfl {M : Type u_1} [Monoid M] {x : M} : Associated x x

instance Associated.instRefl {M : Type u_1} [Monoid M] : Std.Refl Associated

theorem Associated.symm {M : Type u_1} [Monoid M] {x y : M} : Associated x y → Associated y x

instance Associated.instSymm {M : Type u_1} [Monoid M] : Std.Symm Associated

theorem Associated.comm {M : Type u_1} [Monoid M] {x y : M} : Associated x y ↔ Associated y x

theorem Associated.trans {M : Type u_1} [Monoid M] {x y z : M} : Associated x y → Associated y z → Associated x z

instance Associated.instIsTrans {M : Type u_1} [Monoid M] : IsTrans M Associated

def Associated.setoid (M : Type u_2) [Monoid M] : Setoid M

The setoid of the relation x ~ᵤ y iff there is a unit u such that x * u = y

theorem Associated.map {M : Type u_2} {N : Type u_3} [Monoid M] [Monoid N] {F : Type u_4} [FunLike F M N] [MonoidHomClass F M N] (f : F) {x y : M} (ha : Associated x y) : Associated (f x) (f y)

theorem Associated.of_eq {M : Type u_1} [Monoid M] {a b : M} (h : a = b) : Associated a b

theorem unit_associated_one {M : Type u_1} [Monoid M] {u : Mˣ} : Associated (↑u) 1

@[simp]

theorem associated_one_iff_isUnit {M : Type u_1} [Monoid M] {a : M} : Associated a 1 ↔ IsUnit a

@[simp]

theorem associated_zero_iff_eq_zero {M : Type u_1} [MonoidWithZero M] (a : M) : Associated a 0 ↔ a = 0

theorem associated_one_of_mul_eq_one {M : Type u_1} [CommMonoid M] {a : M} (b : M) (hab : a * b = 1) : Associated a 1

theorem associated_one_of_associated_mul_one {M : Type u_1} [CommMonoid M] {a b : M} : Associated (a * b) 1 → Associated a 1

theorem associated_mul_unit_left {N : Type u_2} [Monoid N] (a u : N) (hu : IsUnit u) : Associated (a * u) a

theorem associated_unit_mul_left {N : Type u_2} [CommMonoid N] (a u : N) (hu : IsUnit u) : Associated (u * a) a

theorem associated_mul_unit_right {N : Type u_2} [Monoid N] (a u : N) (hu : IsUnit u) : Associated a (a * u)

theorem associated_unit_mul_right {N : Type u_2} [CommMonoid N] (a u : N) (hu : IsUnit u) : Associated a (u * a)

theorem associated_mul_isUnit_left_iff {N : Type u_2} [Monoid N] {a u b : N} (hu : IsUnit u) : Associated (a * u) b ↔ Associated a b

theorem associated_isUnit_mul_left_iff {N : Type u_2} [CommMonoid N] {u a b : N} (hu : IsUnit u) : Associated (u * a) b ↔ Associated a b

theorem associated_mul_isUnit_right_iff {N : Type u_2} [Monoid N] {a b u : N} (hu : IsUnit u) : Associated a (b * u) ↔ Associated a b

theorem associated_isUnit_mul_right_iff {N : Type u_2} [CommMonoid N] {a u b : N} (hu : IsUnit u) : Associated a (u * b) ↔ Associated a b

@[simp]

theorem associated_mul_unit_left_iff {N : Type u_2} [Monoid N] {a b : N} {u : Nˣ} : Associated (a * ↑u) b ↔ Associated a b

@[simp]

theorem associated_unit_mul_left_iff {N : Type u_2} [CommMonoid N] {a b : N} {u : Nˣ} : Associated (↑u * a) b ↔ Associated a b

@[simp]

theorem associated_mul_unit_right_iff {N : Type u_2} [Monoid N] {a b : N} {u : Nˣ} : Associated a (b * ↑u) ↔ Associated a b

@[simp]

theorem associated_unit_mul_right_iff {N : Type u_2} [CommMonoid N] {a b : N} {u : Nˣ} : Associated a (↑u * b) ↔ Associated a b

theorem Associated.mul_left {M : Type u_1} [Monoid M] (a : M) {b c : M} (h : Associated b c) : Associated (a * b) (a * c)

theorem Associated.mul_right {M : Type u_1} [CommMonoid M] {a b : M} (h : Associated a b) (c : M) : Associated (a * c) (b * c)

theorem Associated.mul_mul {M : Type u_1} [CommMonoid M] {a₁ a₂ b₁ b₂ : M} (h₁ : Associated a₁ b₁) (h₂ : Associated a₂ b₂) : Associated (a₁ * a₂) (b₁ * b₂)

theorem Associated.pow_pow {M : Type u_1} [CommMonoid M] {a b : M} {n : ℕ} (h : Associated a b) : Associated (a ^ n) (b ^ n)

theorem Associated.dvd {M : Type u_1} [Monoid M] {a b : M} : Associated a b → a ∣ b

theorem Associated.dvd' {M : Type u_1} [Monoid M] {a b : M} (h : Associated a b) : b ∣ a

theorem Associated.dvd_dvd {M : Type u_1} [Monoid M] {a b : M} (h : Associated a b) : a ∣ b ∧ b ∣ a

theorem associated_of_dvd_dvd {M : Type u_1} [MonoidWithZero M] [IsLeftCancelMulZero M] {a b : M} (hab : a ∣ b) (hba : b ∣ a) : Associated a b

theorem dvd_dvd_iff_associated {M : Type u_1} [MonoidWithZero M] [IsLeftCancelMulZero M] {a b : M} : a ∣ b ∧ b ∣ a ↔ Associated a b

instance instDecidableRelAssociatedOfIsLeftCancelMulZeroOfDvd {M : Type u_1} [MonoidWithZero M] [IsLeftCancelMulZero M] [DecidableRel fun (x1 x2 : M) => x1 ∣ x2] : DecidableRel fun (x1 x2 : M) => Associated x1 x2

theorem Associated.dvd_iff_dvd_left {M : Type u_1} [Monoid M] {a b c : M} (h : Associated a b) : a ∣ c ↔ b ∣ c

theorem Associated.dvd_iff_dvd_right {M : Type u_1} [Monoid M] {a b c : M} (h : Associated b c) : a ∣ b ↔ a ∣ c

theorem Associated.eq_zero_iff {M : Type u_1} [MonoidWithZero M] {a b : M} (h : Associated a b) : a = 0 ↔ b = 0

theorem Associated.ne_zero_iff {M : Type u_1} [MonoidWithZero M] {a b : M} (h : Associated a b) : a ≠ 0 ↔ b ≠ 0

theorem Associated.prime {M : Type u_1} [CommMonoidWithZero M] {p q : M} (h : Associated p q) (hp : Prime p) : Prime q

theorem prime_mul_iff {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {x y : M} : Prime (x * y) ↔ Prime x ∧ IsUnit y ∨ IsUnit x ∧ Prime y

@[simp]

theorem prime_pow_iff {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p : M} {n : ℕ} : Prime (p ^ n) ↔ Prime p ∧ n = 1

theorem Irreducible.dvd_iff {M : Type u_1} [Monoid M] {x y : M} (hx : Irreducible x) : y ∣ x ↔ IsUnit y ∨ Associated x y

theorem Irreducible.associated_of_dvd {M : Type u_1} [Monoid M] {p q : M} (p_irr : Irreducible p) (q_irr : Irreducible q) (dvd : p ∣ q) : Associated p q

theorem Irreducible.dvd_irreducible_iff_associated {M : Type u_1} [Monoid M] {p q : M} (pp : Irreducible p) (qp : Irreducible q) : p ∣ q ↔ Associated p q

theorem Prime.associated_of_dvd {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p q : M} (p_prime : Prime p) (q_prime : Prime q) (dvd : p ∣ q) : Associated p q

theorem Prime.dvd_prime_iff_associated {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p q : M} (pp : Prime p) (qp : Prime q) : p ∣ q ↔ Associated p q

theorem Associated.prime_iff {M : Type u_1} [CommMonoidWithZero M] {p q : M} (h : Associated p q) : Prime p ↔ Prime q

theorem Associated.isUnit {M : Type u_1} [Monoid M] {a b : M} (h : Associated a b) : IsUnit a → IsUnit b

theorem Associated.isUnit_iff {M : Type u_1} [Monoid M] {a b : M} (h : Associated a b) : IsUnit a ↔ IsUnit b

theorem Irreducible.isUnit_iff_not_associated_of_dvd {M : Type u_1} [Monoid M] {x y : M} (hx : Irreducible x) (hy : y ∣ x) : IsUnit y ↔ ¬Associated x y

theorem Associated.irreducible {M : Type u_1} [Monoid M] {p q : M} (h : Associated p q) (hp : Irreducible p) : Irreducible q

theorem Associated.irreducible_iff {M : Type u_1} [Monoid M] {p q : M} (h : Associated p q) : Irreducible p ↔ Irreducible q

theorem Associated.of_mul_left {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {a b c d : M} (h : Associated (a * b) (c * d)) (h₁ : Associated a c) (ha : a ≠ 0) : Associated b d

theorem Associated.of_mul_right {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {a b c d : M} : Associated (a * b) (c * d) → Associated b d → b ≠ 0 → Associated a c

theorem Associated.of_pow_associated_of_prime {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p₁ p₂ : M} {k₁ k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁) (h : Associated (p₁ ^ k₁) (p₂ ^ k₂)) : Associated p₁ p₂

theorem Associated.of_pow_associated_of_prime' {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p₁ p₂ : M} {k₁ k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₂ : 0 < k₂) (h : Associated (p₁ ^ k₁) (p₂ ^ k₂)) : Associated p₁ p₂

theorem Irreducible.isRelPrime_iff_not_dvd {M : Type u_1} [Monoid M] {p n : M} (hp : Irreducible p) : IsRelPrime p n ↔ ¬p ∣ n

See also Irreducible.coprime_iff_not_dvd.

theorem Irreducible.dvd_or_isRelPrime {M : Type u_1} [Monoid M] {p n : M} (hp : Irreducible p) : p ∣ n ∨ IsRelPrime p n

theorem associated_iff_eq {M : Type u_1} [Monoid M] [Subsingleton Mˣ] {x y : M} : Associated x y ↔ x = y

theorem associated_eq_eq {M : Type u_1} [Monoid M] [Subsingleton Mˣ] : Associated = Eq

theorem prime_dvd_prime_iff_eq {M : Type u_2} [CommMonoidWithZero M] [IsCancelMulZero M] [Subsingleton Mˣ] {p q : M} (pp : Prime p) (qp : Prime q) : p ∣ q ↔ p = q

theorem eq_of_prime_pow_eq {R : Type u_2} [CommMonoidWithZero R] [IsCancelMulZero R] [Subsingleton Rˣ] {p₁ p₂ : R} {k₁ k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂

theorem eq_of_prime_pow_eq' {R : Type u_2} [CommMonoidWithZero R] [IsCancelMulZero R] [Subsingleton Rˣ] {p₁ p₂ : R} {k₁ k₂ : ℕ} (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₂) (h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂

@[reducible, inline]

abbrev Associates (M : Type u_2) [Monoid M] : Type u_2

The quotient of a monoid by the Associated relation. Two elements x and y are associated iff there is a unit u such that x * u = y. There is a natural monoid structure on Associates M.

@[reducible, inline]

abbrev Associates.mk {M : Type u_2} [Monoid M] (a : M) : Associates M

The canonical quotient map from a monoid M into the Associates of M

instance Associates.instInhabited {M : Type u_1} [Monoid M] : Inhabited (Associates M)

theorem Associates.mk_eq_mk_iff_associated {M : Type u_1} [Monoid M] {a b : M} : Associates.mk a = Associates.mk b ↔ Associated a b

theorem Associates.quotient_mk_eq_mk {M : Type u_1} [Monoid M] (a : M) : ⟦a⟧ = Associates.mk a

theorem Associates.quot_mk_eq_mk {M : Type u_1} [Monoid M] (a : M) : Quot.mk (⇑(Associated.setoid M)) a = Associates.mk a

@[simp]

theorem Associates.quot_out {M : Type u_1} [Monoid M] (a : Associates M) : Associates.mk (Quot.out a) = a

theorem Associates.mk_quot_out {M : Type u_1} [Monoid M] (a : M) : Associated (Quot.out (Associates.mk a)) a

theorem Associates.forall_associated {M : Type u_1} [Monoid M] {p : Associates M → Prop} : (∀ (a : Associates M), p a) ↔ ∀ (a : M), p (Associates.mk a)

theorem Associates.mk_surjective {M : Type u_1} [Monoid M] : Function.Surjective Associates.mk

instance Associates.instOne {M : Type u_1} [Monoid M] : One (Associates M)

@[simp]

theorem Associates.mk_one {M : Type u_1} [Monoid M] : Associates.mk 1 = 1

theorem Associates.one_eq_mk_one {M : Type u_1} [Monoid M] : 1 = Associates.mk 1

@[simp]

theorem Associates.mk_eq_one {M : Type u_1} [Monoid M] {a : M} : Associates.mk a = 1 ↔ IsUnit a

instance Associates.instBot {M : Type u_1} [Monoid M] : Bot (Associates M)

theorem Associates.bot_eq_one {M : Type u_1} [Monoid M] : ⊥ = 1

theorem Associates.exists_rep {M : Type u_1} [Monoid M] (a : Associates M) : ∃ (a0 : M), Associates.mk a0 = a

instance Associates.instUniqueOfSubsingleton {M : Type u_1} [Monoid M] [Subsingleton M] : Unique (Associates M)

theorem Associates.mk_injective {M : Type u_1} [Monoid M] [Subsingleton Mˣ] : Function.Injective Associates.mk

instance Associates.instMul {M : Type u_1} [CommMonoid M] : Mul (Associates M)

theorem Associates.mk_mul_mk {M : Type u_1} [CommMonoid M] {x y : M} : Associates.mk x * Associates.mk y = Associates.mk (x * y)

instance Associates.instCommMonoid {M : Type u_1} [CommMonoid M] : CommMonoid (Associates M)

instance Associates.instPreorder {M : Type u_1} [CommMonoid M] : Preorder (Associates M)

def Associates.mkMonoidHom {M : Type u_1} [CommMonoid M] : M →* Associates M

Associates.mk as a MonoidHom.

@[simp]

theorem Associates.mkMonoidHom_apply {M : Type u_1} [CommMonoid M] (a : M) : Associates.mkMonoidHom a = Associates.mk a

theorem Associates.associated_map_mk {M : Type u_1} [CommMonoid M] {f : Associates M →* M} (hinv : Function.RightInverse (⇑f) Associates.mk) (a : M) : Associated a (f (Associates.mk a))

theorem Associates.mk_pow {M : Type u_1} [CommMonoid M] (a : M) (n : ℕ) : Associates.mk (a ^ n) = Associates.mk a ^ n

theorem Associates.dvd_eq_le {M : Type u_1} [CommMonoid M] : (fun (x1 x2 : Associates M) => x1 ∣ x2) = fun (x1 x2 : Associates M) => x1 ≤ x2

instance Associates.uniqueUnits {M : Type u_1} [CommMonoid M] : Unique (Associates M)ˣ

@[simp]

theorem Associates.coe_unit_eq_one {M : Type u_1} [CommMonoid M] (u : (Associates M)ˣ) : ↑u = 1

theorem Associates.isUnit_iff_eq_one {M : Type u_1} [CommMonoid M] (a : Associates M) : IsUnit a ↔ a = 1

theorem Associates.isUnit_iff_eq_bot {M : Type u_1} [CommMonoid M] {a : Associates M} : IsUnit a ↔ a = ⊥

theorem Associates.isUnit_mk {M : Type u_1} [CommMonoid M] {a : M} : IsUnit (Associates.mk a) ↔ IsUnit a

theorem Associates.mul_mono {M : Type u_1} [CommMonoid M] {a b c d : Associates M} (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d

theorem Associates.one_le {M : Type u_1} [CommMonoid M] {a : Associates M} : 1 ≤ a

theorem Associates.le_mul_right {M : Type u_1} [CommMonoid M] {a b : Associates M} : a ≤ a * b

theorem Associates.le_mul_left {M : Type u_1} [CommMonoid M] {a b : Associates M} : a ≤ b * a

instance Associates.instOrderBot {M : Type u_1} [CommMonoid M] : OrderBot (Associates M)

@[simp]

theorem Associates.mk_dvd_mk {M : Type u_1} [CommMonoid M] {a b : M} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b

theorem Associates.dvd_of_mk_le_mk {M : Type u_1} [CommMonoid M] {a b : M} : Associates.mk a ≤ Associates.mk b → a ∣ b

theorem Associates.mk_le_mk_of_dvd {M : Type u_1} [CommMonoid M] {a b : M} : a ∣ b → Associates.mk a ≤ Associates.mk b

theorem Associates.mk_le_mk_iff_dvd {M : Type u_1} [CommMonoid M] {a b : M} : Associates.mk a ≤ Associates.mk b ↔ a ∣ b

@[simp]

theorem Associates.isPrimal_mk {M : Type u_1} [CommMonoid M] {a : M} : IsPrimal (Associates.mk a) ↔ IsPrimal a

@[simp]

theorem Associates.decompositionMonoid_iff {M : Type u_1} [CommMonoid M] : DecompositionMonoid (Associates M) ↔ DecompositionMonoid M

instance Associates.instDecompositionMonoid {M : Type u_1} [CommMonoid M] [DecompositionMonoid M] : DecompositionMonoid (Associates M)

@[simp]

theorem Associates.mk_isRelPrime_iff {M : Type u_1} [CommMonoid M] {a b : M} : IsRelPrime (Associates.mk a) (Associates.mk b) ↔ IsRelPrime a b

instance Associates.instZero {M : Type u_1} [Zero M] [Monoid M] : Zero (Associates M)

instance Associates.instTopOfZero {M : Type u_1} [Zero M] [Monoid M] : Top (Associates M)

@[simp]

theorem Associates.mk_zero {M : Type u_1} [Zero M] [Monoid M] : Associates.mk 0 = 0

@[simp]

theorem Associates.mk_eq_zero {M : Type u_1} [MonoidWithZero M] {a : M} : Associates.mk a = 0 ↔ a = 0

@[simp]

theorem Associates.quot_out_zero {M : Type u_1} [MonoidWithZero M] : Quot.out 0 = 0

theorem Associates.mk_ne_zero {M : Type u_1} [MonoidWithZero M] {a : M} : Associates.mk a ≠ 0 ↔ a ≠ 0

instance Associates.instNontrivial {M : Type u_1} [MonoidWithZero M] [Nontrivial M] : Nontrivial (Associates M)

theorem Associates.exists_non_zero_rep {M : Type u_1} [MonoidWithZero M] {a : Associates M} : a ≠ 0 → ∃ (a0 : M), a0 ≠ 0 ∧ Associates.mk a0 = a

instance Associates.instCommMonoidWithZero {M : Type u_1} [CommMonoidWithZero M] : CommMonoidWithZero (Associates M)

instance Associates.instOrderTop {M : Type u_1} [CommMonoidWithZero M] : OrderTop (Associates M)

@[simp]

theorem Associates.le_zero {M : Type u_1} [CommMonoidWithZero M] (a : Associates M) : a ≤ 0

instance Associates.instBoundedOrder {M : Type u_1} [CommMonoidWithZero M] : BoundedOrder (Associates M)

instance Associates.instDecidableRelDvd {M : Type u_1} [CommMonoidWithZero M] [DecidableRel fun (x1 x2 : M) => x1 ∣ x2] : DecidableRel fun (x1 x2 : Associates M) => x1 ∣ x2

theorem Associates.Prime.le_or_le {M : Type u_1} [CommMonoidWithZero M] {p : Associates M} (hp : Prime p) {a b : Associates M} (h : p ≤ a * b) : p ≤ a ∨ p ≤ b

@[simp]

theorem Associates.prime_mk {M : Type u_1} [CommMonoidWithZero M] {p : M} : Prime (Associates.mk p) ↔ Prime p

@[simp]

theorem Associates.irreducible_mk {M : Type u_1} [CommMonoidWithZero M] {a : M} : Irreducible (Associates.mk a) ↔ Irreducible a

@[simp]

theorem Associates.mk_dvdNotUnit_mk_iff {M : Type u_1} [CommMonoidWithZero M] {a b : M} : DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b

theorem Associates.dvdNotUnit_of_lt {M : Type u_1} [CommMonoidWithZero M] {a b : Associates M} (hlt : a < b) : DvdNotUnit a b

theorem Associates.irreducible_iff_prime_iff {M : Type u_1} [CommMonoidWithZero M] : (∀ (a : M), Irreducible a ↔ Prime a) ↔ ∀ (a : Associates M), Irreducible a ↔ Prime a

instance Associates.instPartialOrder {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] : PartialOrder (Associates M)

instance Associates.instIsCancelMulZero {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] : IsCancelMulZero (Associates M)

theorem associates_irreducible_iff_prime {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] [DecompositionMonoid M] {p : Associates M} : Irreducible p ↔ Prime p

instance Associates.instNoZeroDivisors {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] : NoZeroDivisors (Associates M)

theorem Associates.le_of_mul_le_mul_left {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] (a b c : Associates M) (ha : a ≠ 0) : a * b ≤ a * c → b ≤ c

theorem Associates.one_or_eq_of_le_of_prime {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p m : Associates M} (hp : Prime p) (hle : m ≤ p) : m = 1 ∨ m = p

theorem Associates.dvdNotUnit_iff_lt {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {a b : Associates M} : DvdNotUnit a b ↔ a < b

theorem Associates.le_one_iff {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p : Associates M} : p ≤ 1 ↔ p = 1

theorem dvdNotUnit_of_dvdNotUnit_associated {M : Type u_1} [CommMonoidWithZero M] [Nontrivial M] {p q r : M} (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r

theorem isUnit_of_associated_mul {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p b : M} (h : Associated (p * b) p) (hp : p ≠ 0) : IsUnit b

theorem DvdNotUnit.not_associated {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p q : M} (h : DvdNotUnit p q) : ¬Associated p q

theorem dvd_prime_pow {M : Type u_1} [CommMonoidWithZero M] [IsCancelMulZero M] {p q : M} (hp : Prime p) (n : ℕ) : q ∣ p ^ n ↔ ∃ i ≤ n, Associated q (p ^ i)