File: Mathlib/Algebra/Order/Ring/Abs.lean

full_name: mabs_zpow
code: @[to_additive] lemma mabs_zpow (n : ℤ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ |n| 
goal: mabs (a ^ n) = mabs a ^ |n|

full_name: abs_zsmul
code: @[to_additive] lemma mabs_zpow (n : ℤ) (a : α) : |a ^ n|ₘ = |a|ₘ ^ |n| 
goal: |n • a| = |n| • |a|

full_name: odd_abs
code: lemma odd_abs [LinearOrder α] [Ring α] {a : α} : Odd (abs a) ↔ Odd a 
goal: Odd |a| ↔ Odd a

full_name: abs_one
code: @[simp] lemma abs_one : |(1 : α)| = 1 
goal: |1| = 1

full_name: abs_two
code: lemma abs_two : |(2 : α)| = 2 
goal: |2| = 2

full_name: abs_mul
code: lemma abs_mul (a b : α) : |a * b| = |a| * |b| 
goal: |a * b| = |a| * |b|

full_name: absHom
code: /-- `abs` as a `MonoidWithZeroHom`. -/
def absHom : α →*₀ α 
goal: α →*₀ α

full_name: abs_pow
code: @[simp]
lemma abs_pow (a : α) (n : ℕ) : |a ^ n| = |a| ^ n 
goal: |a ^ n| = |a| ^ n

full_name: pow_abs
code: lemma pow_abs (a : α) (n : ℕ) : |a| ^ n = |a ^ n| 
goal: |a| ^ n = |a ^ n|

full_name: Even.pow_abs
code: lemma Even.pow_abs (hn : Even n) (a : α) : |a| ^ n = a ^ n 
goal: |a| ^ n = a ^ n

full_name: abs_neg_one_pow
code: lemma abs_neg_one_pow (n : ℕ) : |(-1 : α) ^ n| = 1 
goal: |(-1) ^ n| = 1

full_name: abs_pow_eq_one
code: lemma abs_pow_eq_one (a : α) (h : n ≠ 0) : |a ^ n| = 1 ↔ |a| = 1 
goal: |a ^ n| = 1 ↔ |a| = 1

full_name: abs_mul_abs_self
code: @[simp] lemma abs_mul_abs_self (a : α) : |a| * |a| = a * a 
goal: |a| * |a| = a * a

full_name: abs_mul_self
code: @[simp]
lemma abs_mul_self (a : α) : |a * a| = a * a 
goal: |a * a| = a * a

full_name: abs_eq_iff_mul_self_eq
code: lemma abs_eq_iff_mul_self_eq : |a| = |b| ↔ a * a = b * b 
goal: |a| = |b| ↔ a * a = b * b

full_name: abs_lt_iff_mul_self_lt
code: lemma abs_lt_iff_mul_self_lt : |a| < |b| ↔ a * a < b * b 
goal: |a| < |b| ↔ a * a < b * b

full_name: abs_le_iff_mul_self_le
code: lemma abs_le_iff_mul_self_le : |a| ≤ |b| ↔ a * a ≤ b * b 
goal: |a| ≤ |b| ↔ a * a ≤ b * b

full_name: abs_le_one_iff_mul_self_le_one
code: lemma abs_le_one_iff_mul_self_le_one : |a| ≤ 1 ↔ a * a ≤ 1 
goal: |a| ≤ 1 ↔ a * a ≤ 1

full_name: sq_abs
code: @[simp] lemma sq_abs (a : α) : |a| ^ 2 = a ^ 2 
goal: |a| ^ 2 = a ^ 2

full_name: abs_sq
code: lemma abs_sq (x : α) : |x ^ 2| = x ^ 2 
goal: |x ^ 2| = x ^ 2

full_name: sq_lt_sq
code: lemma sq_lt_sq : a ^ 2 < b ^ 2 ↔ |a| < |b| 
goal: a ^ 2 < b ^ 2 ↔ |a| < |b|

full_name: sq_lt_sq'
code: lemma sq_lt_sq' (h1 : -b < a) (h2 : a < b) : a ^ 2 < b ^ 2 
goal: a ^ 2 < b ^ 2

full_name: sq_le_sq
code: lemma sq_le_sq : a ^ 2 ≤ b ^ 2 ↔ |a| ≤ |b| 
goal: a ^ 2 ≤ b ^ 2 ↔ |a| ≤ |b|

full_name: sq_le_sq'
code: lemma sq_le_sq' (h1 : -b ≤ a) (h2 : a ≤ b) : a ^ 2 ≤ b ^ 2 
goal: a ^ 2 ≤ b ^ 2

full_name: abs_lt_of_sq_lt_sq
code: lemma abs_lt_of_sq_lt_sq (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b) : |a| < b 
goal: |a| < b

full_name: abs_lt_of_sq_lt_sq'
code: lemma abs_lt_of_sq_lt_sq' (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b) : -b < a ∧ a < b 
goal: -b < a ∧ a < b

full_name: abs_le_of_sq_le_sq
code: lemma abs_le_of_sq_le_sq (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b) : |a| ≤ b 
goal: |a| ≤ b

full_name: abs_le_of_sq_le_sq'
code: lemma abs_le_of_sq_le_sq' (h : a ^ 2 ≤ b ^ 2) (hb : 0 ≤ b) : -b ≤ a ∧ a ≤ b 
goal: -b ≤ a ∧ a ≤ b

full_name: sq_eq_sq_iff_abs_eq_abs
code: lemma sq_eq_sq_iff_abs_eq_abs (a b : α) : a ^ 2 = b ^ 2 ↔ |a| = |b| 
goal: a ^ 2 = b ^ 2 ↔ |a| = |b|

full_name: sq_le_one_iff_abs_le_one
code: @[simp] lemma sq_le_one_iff_abs_le_one (a : α) : a ^ 2 ≤ 1 ↔ |a| ≤ 1 
goal: a ^ 2 ≤ 1 ↔ |a| ≤ 1

full_name: sq_lt_one_iff_abs_lt_one
code: @[simp] lemma sq_lt_one_iff_abs_lt_one (a : α) : a ^ 2 < 1 ↔ |a| < 1 
goal: a ^ 2 < 1 ↔ |a| < 1

full_name: one_le_sq_iff_one_le_abs
code: @[simp] lemma one_le_sq_iff_one_le_abs (a : α) : 1 ≤ a ^ 2 ↔ 1 ≤ |a| 
goal: 1 ≤ a ^ 2 ↔ 1 ≤ |a|

full_name: one_lt_sq_iff_one_lt_abs
code: @[simp] lemma one_lt_sq_iff_one_lt_abs (a : α) : 1 < a ^ 2 ↔ 1 < |a| 
goal: 1 < a ^ 2 ↔ 1 < |a|

full_name: exists_abs_lt
code: lemma exists_abs_lt {α : Type*} [LinearOrderedRing α] (a : α) : ∃ b > 0, |a| < b 
goal: ∃ b > 0, |a| < b

full_name: abs_sub_sq
code: theorem abs_sub_sq (a b : α) : |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b 
goal: |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b

full_name: abs_dvd
code: @[simp]
theorem abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b 
goal: |a| ∣ b ↔ a ∣ b

full_name: abs_dvd_self
code: theorem abs_dvd_self (a : α) : |a| ∣ a 
goal: |a| ∣ a

full_name: dvd_abs
code: @[simp]
theorem dvd_abs (a b : α) : a ∣ |b| ↔ a ∣ b 
goal: a ∣ |b| ↔ a ∣ b

full_name: self_dvd_abs
code: theorem self_dvd_abs (a : α) : a ∣ |a| 
goal: a ∣ |a|

full_name: abs_dvd_abs
code: theorem abs_dvd_abs (a b : α) : |a| ∣ |b| ↔ a ∣ b 
goal: |a| ∣ |b| ↔ a ∣ b

full_name: pow_eq_pow_iff_of_ne_zero
code: lemma pow_eq_pow_iff_of_ne_zero (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b ∨ a = -b ∧ Even n 
goal: a ^ n = b ^ n ↔ a = b ∨ a = -b ∧ Even n

full_name: pow_eq_pow_iff_cases
code: lemma pow_eq_pow_iff_cases : a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n 
goal: a ^ n = b ^ n ↔ n = 0 ∨ a = b ∨ a = -b ∧ Even n

full_name: pow_eq_one_iff_of_ne_zero
code: lemma pow_eq_one_iff_of_ne_zero (hn : n ≠ 0) : a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n 
goal: a ^ n = 1 ↔ a = 1 ∨ a = -1 ∧ Even n

full_name: pow_eq_one_iff_cases
code: lemma pow_eq_one_iff_cases : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n 
goal: a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n

full_name: pow_eq_neg_pow_iff
code: lemma pow_eq_neg_pow_iff (hb : b ≠ 0) : a ^ n = -b ^ n ↔ a = -b ∧ Odd n 
goal: a ^ n = -b ^ n ↔ a = -b ∧ Odd n

full_name: pow_eq_neg_one_iff
code: lemma pow_eq_neg_one_iff : a ^ n = -1 ↔ a = -1 ∧ Odd n 
goal: a ^ n = -1 ↔ a = -1 ∧ Odd n

full_name: Odd.mod_even_iff
code: /-- If `a` is even, then `n` is odd iff `n % a` is odd. -/
lemma Odd.mod_even_iff (ha : Even a) : Odd (n % a) ↔ Odd n 
goal: Odd (n % a) ↔ Odd n

full_name: Even.mod_even_iff
code: /-- If `a` is even, then `n` is even iff `n % a` is even. -/
lemma Even.mod_even_iff (ha : Even a) : Even (n % a) ↔ Even n 
goal: Even (n % a) ↔ Even n

full_name: Odd.mod_even
code: /-- If `n` is odd and `a` is even, then `n % a` is odd. -/
lemma Odd.mod_even (hn : Odd n) (ha : Even a) : Odd (n % a) 
goal: Odd (n % a)

full_name: Even.mod_even
code: /-- If `n` is even and `a` is even, then `n % a` is even. -/
lemma Even.mod_even (hn : Even n) (ha : Even a) : Even (n % a) 
goal: Even (n % a)

full_name: Odd.of_dvd_nat
code: lemma Odd.of_dvd_nat (hn : Odd n) (hm : m ∣ n) : Odd m 
goal: Odd m

full_name: Odd.ne_two_of_dvd_nat
code: /-- `2` is not a factor of an odd natural number. -/
lemma Odd.ne_two_of_dvd_nat {m n : ℕ} (hn : Odd n) (hm : m ∣ n) : m ≠ 2 
goal: m ≠ 2