mathlib docs: data.int.modeq

data.int.modeq

int.mod_mul_left_mod

int.mod_mul_right_mod

int.modeq.coe_nat_modeq_iff

int.modeq.decidable

int.modeq.exists_unique_equiv

int.modeq.exists_unique_equiv_nat

int.modeq.gcd_a_modeq

int.modeq.mod_coprime

int.modeq.mod_modeq

int.modeq.modeq_add

int.modeq.modeq_add_cancel_left

int.modeq.modeq_add_cancel_right

int.modeq.modeq_add_fac

int.modeq.modeq_and_modeq_iff_modeq_mul

int.modeq.modeq_iff_dvd

int.modeq.modeq_mul

int.modeq.modeq_mul_left

int.modeq.modeq_mul_left'

int.modeq.modeq_mul_right

int.modeq.modeq_mul_right'

int.modeq.modeq_neg

int.modeq.modeq_of_dvd_of_modeq

int.modeq.modeq_of_modeq_mul_left

int.modeq.modeq_of_modeq_mul_right

int.modeq.modeq_sub

int.modeq.modeq_zero_iff

int.modeq.trans

def int.modeq :

ℤ → ℤ → ℤ → Prop

Equations

a ≡ b [ZMOD n] = (a % n = b % n)

theorem int.modeq.refl {n : ℤ} (a : ℤ) :

a ≡ a [ZMOD n]

theorem int.modeq.symm {n a b : ℤ} :

a ≡ b [ZMOD n] → b ≡ a [ZMOD n]

theorem int.modeq.trans {n a b c : ℤ} :

a ≡ b [ZMOD n] → b ≡ c [ZMOD n] → a ≡ c [ZMOD n]

theorem int.modeq.coe_nat_modeq_iff {a b n : ℕ} :

↑a ≡ ↑b [ZMOD ↑n] ↔ a ≡ b [MOD n]

@[instance]

def int.modeq.decidable {n a b : ℤ} :

decidable a ≡ b [ZMOD n]

Equations

int.modeq.decidable = int.modeq.decidable._proof_1.mpr (int.decidable_eq (a % n) (b % n))

theorem int.modeq.modeq_zero_iff {n a : ℤ} :

a ≡ 0 [ZMOD n] ↔ n ∣ a

theorem int.modeq.modeq_iff_dvd {n a b : ℤ} :

a ≡ b [ZMOD n] ↔ n ∣ b - a

theorem int.modeq.modeq_of_dvd_of_modeq {n m a b : ℤ} :

m ∣ n → a ≡ b [ZMOD n] → a ≡ b [ZMOD m]

theorem int.modeq.modeq_mul_left' {n a b c : ℤ} :

0 ≤ c → a ≡ b [ZMOD n] → c * a ≡ c * b [ZMOD c * n]

theorem int.modeq.modeq_mul_right' {n a b c : ℤ} :

0 ≤ c → a ≡ b [ZMOD n] → a * c ≡ b * c [ZMOD n * c]

theorem int.modeq.modeq_add {n a b c d : ℤ} :

a ≡ b [ZMOD n] → c ≡ d [ZMOD n] → a + c ≡ b + d [ZMOD n]

theorem int.modeq.modeq_add_cancel_left {n a b c d : ℤ} :

a ≡ b [ZMOD n] → a + c ≡ b + d [ZMOD n] → c ≡ d [ZMOD n]

theorem int.modeq.modeq_add_cancel_right {n a b c d : ℤ} :

c ≡ d [ZMOD n] → a + c ≡ b + d [ZMOD n] → a ≡ b [ZMOD n]

theorem int.modeq.mod_modeq (a n : ℤ) :

a % n ≡ a [ZMOD n]

theorem int.modeq.modeq_neg {n a b : ℤ} :

a ≡ b [ZMOD n] → -a ≡ -b [ZMOD n]

theorem int.modeq.modeq_sub {n a b c d : ℤ} :

a ≡ b [ZMOD n] → c ≡ d [ZMOD n] → a - c ≡ b - d [ZMOD n]

theorem int.modeq.modeq_mul_left {n a b : ℤ} (c : ℤ) :

a ≡ b [ZMOD n] → c * a ≡ c * b [ZMOD n]

theorem int.modeq.modeq_mul_right {n a b : ℤ} (c : ℤ) :

a ≡ b [ZMOD n] → a * c ≡ b * c [ZMOD n]

theorem int.modeq.modeq_mul {n a b c d : ℤ} :

a ≡ b [ZMOD n] → c ≡ d [ZMOD n] → a * c ≡ b * d [ZMOD n]

theorem int.modeq.modeq_of_modeq_mul_left {n a b : ℤ} (m : ℤ) :

a ≡ b [ZMOD m * n] → a ≡ b [ZMOD n]

theorem int.modeq.modeq_of_modeq_mul_right {n a b : ℤ} (m : ℤ) :

a ≡ b [ZMOD n * m] → a ≡ b [ZMOD n]

theorem int.modeq.modeq_and_modeq_iff_modeq_mul {a b m n : ℤ} :

m.nat_abs.coprime n.nat_abs → (a ≡ b [ZMOD m] ∧ a ≡ b [ZMOD n] ↔ a ≡ b [ZMOD m * n])

theorem int.modeq.gcd_a_modeq (a b : ℕ) :

↑a * a.gcd_a b ≡ ↑(a.gcd b) [ZMOD ↑b]

theorem int.modeq.modeq_add_fac {a b n : ℤ} (c : ℤ) :

a ≡ b [ZMOD n] → a + n * c ≡ b [ZMOD n]

theorem int.modeq.mod_coprime {a b : ℕ} :

a.coprime b → (∃ (y : ℤ), ↑a * y ≡ 1 [ZMOD ↑b])

theorem int.modeq.exists_unique_equiv (a : ℤ) {b : ℤ} :

0 < b → (∃ (z : ℤ), 0 ≤ z ∧ z < b ∧ z ≡ a [ZMOD b])

theorem int.modeq.exists_unique_equiv_nat (a : ℤ) {b : ℤ} :

0 < b → (∃ (z : ℕ), ↑z < b ∧ ↑z ≡ a [ZMOD b])

@[simp]

theorem int.mod_mul_right_mod (a b c : ℤ) :

a % (b * c) % b = a % b

@[simp]

theorem int.mod_mul_left_mod (a b c : ℤ) :

a % (b * c) % c = a % c

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