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Mathlib.Algebra.GroupPower.Ring

Power operations on monoids with zero, semirings, and rings #

This file provides additional lemmas about the natural power operator on rings and semirings. Further lemmas about ordered semirings and rings can be found in Algebra.GroupPower.Lemmas.

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theorem zero_pow {M : Type u_1} [inst : MonoidWithZero M] {n : ℕ} :
0 < n → 0 ^ n = 0
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@[simp]
theorem zero_pow' {M : Type u_1} [inst : MonoidWithZero M] (n : ℕ) :
n ≠ 0 → 0 ^ n = 0
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theorem zero_pow_eq {M : Type u_1} [inst : MonoidWithZero M] (n : ℕ) :
0 ^ n = if n = 0 then 1 else 0
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theorem pow_eq_zero_of_le {M : Type u_1} [inst : MonoidWithZero M] {x : M} {n : ℕ} {m : ℕ} (hn : n ≤ m) (hx : x ^ n = 0) :
x ^ m = 0
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theorem pow_eq_zero {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {x : M} {n : ℕ} (H : x ^ n = 0) :
x = 0
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@[simp]
theorem pow_eq_zero_iff {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {a : M} {n : ℕ} (hn : 0 < n) :
a ^ n = 0 ↔ a = 0
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theorem pow_eq_zero_iff' {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] [inst : Nontrivial M] {a : M} {n : ℕ} :
a ^ n = 0 ↔ a = 0 ∧ n ≠ 0
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theorem pow_ne_zero_iff {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {a : M} {n : ℕ} (hn : 0 < n) :
a ^ n ≠ 0 ↔ a ≠ 0
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theorem ne_zero_pow {M : Type u_1} [inst : MonoidWithZero M] {a : M} {n : ℕ} (hn : n ≠ 0) :
a ^ n ≠ 0 → a ≠ 0
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theorem pow_ne_zero {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {a : M} (n : ℕ) (h : a ≠ 0) :
a ^ n ≠ 0
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instance NeZero.pow {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {x : M} [inst : NeZero x] {n : ℕ} :
NeZero (x ^ n)
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theorem sq_eq_zero_iff {M : Type u_1} [inst : MonoidWithZero M] [inst : NoZeroDivisors M] {a : M} :
a ^ 2 = 0 ↔ a = 0
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@[simp]
theorem zero_pow_eq_zero {M : Type u_1} [inst : MonoidWithZero M] [inst : Nontrivial M] {n : ℕ} :
0 ^ n = 0 ↔ 0 < n
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theorem Ring.inverse_pow {M : Type u_1} [inst : MonoidWithZero M] (r : M) (n : ℕ) :
Ring.inverse r ^ n = Ring.inverse (r ^ n)
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def powMonoidWithZeroHom {M : Type u_1} [inst : CommMonoidWithZero M] {n : ℕ} (hn : 0 < n) :
M →*₀ M

We define x ↦ x^n (for positive n : ℕ) as a MonoidWithZeroHom

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  • One or more equations did not get rendered due to their size.
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@[simp]
theorem coe_powMonoidWithZeroHom {M : Type u_1} [inst : CommMonoidWithZero M] {n : ℕ} (hn : 0 < n) :
↑(powMonoidWithZeroHom hn) = fun x => x ^ n
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@[simp]
theorem powMonoidWithZeroHom_apply {M : Type u_1} [inst : CommMonoidWithZero M] {n : ℕ} (hn : 0 < n) (a : M) :
↑(powMonoidWithZeroHom hn) a = a ^ n
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theorem pow_dvd_pow_iff {R : Type u_1} [inst : CancelCommMonoidWithZero R] {x : R} {n : ℕ} {m : ℕ} (h0 : x ≠ 0) (h1 : ¬IsUnit x) :
x ^ n ∣ x ^ m ↔ n ≤ m
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theorem RingHom.map_pow {R : Type u_1} {S : Type u_2} [inst : Semiring R] [inst : Semiring S] (f : R →+* S) (a : R) (n : ℕ) :
↑f (a ^ n) = ↑f a ^ n
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theorem min_pow_dvd_add {R : Type u_1} [inst : Semiring R] {n : ℕ} {m : ℕ} {a : R} {b : R} {c : R} (ha : c ^ n ∣ a) (hb : c ^ m ∣ b) :
c ^ min n m ∣ a + b
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theorem add_sq {R : Type u_1} [inst : CommSemiring R] (a : R) (b : R) :
(a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2
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theorem add_sq' {R : Type u_1} [inst : CommSemiring R] (a : R) (b : R) :
(a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b
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theorem add_pow_two {R : Type u_1} [inst : CommSemiring R] (a : R) (b : R) :
(a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2

Alias of add_sq.

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theorem neg_one_pow_eq_or (R : Type u_1) [inst : Monoid R] [inst : HasDistribNeg R] (n : ℕ) :
(-1) ^ n = 1 ∨ (-1) ^ n = -1
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theorem neg_pow {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] (a : R) (n : ℕ) :
(-a) ^ n = (-1) ^ n * a ^ n
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@[simp]
theorem neg_pow_bit0 {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] (a : R) (n : ℕ) :
(-a) ^ bit0 n = a ^ bit0 n
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@[simp]
theorem neg_pow_bit1 {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] (a : R) (n : ℕ) :
(-a) ^ bit1 n = -a ^ bit1 n
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@[simp]
theorem neg_sq {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] (a : R) :
(-a) ^ 2 = a ^ 2
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theorem neg_one_sq {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] :
(-1) ^ 2 = 1
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theorem neg_pow_two {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] (a : R) :
(-a) ^ 2 = a ^ 2

Alias of neg_sq.

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theorem neg_one_pow_two {R : Type u_1} [inst : Monoid R] [inst : HasDistribNeg R] :
(-1) ^ 2 = 1

Alias of neg_one_sq.

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theorem Commute.sq_sub_sq {R : Type u_1} [inst : Ring R] {a : R} {b : R} (h : Commute a b) :
a ^ 2 - b ^ 2 = (a + b) * (a - b)
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theorem neg_one_pow_mul_eq_zero_iff {R : Type u_1} [inst : Ring R] {n : ℕ} {r : R} :
(-1) ^ n * r = 0 ↔ r = 0
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theorem mul_neg_one_pow_eq_zero_iff {R : Type u_1} [inst : Ring R] {n : ℕ} {r : R} :
r * (-1) ^ n = 0 ↔ r = 0
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theorem Commute.sq_eq_sq_iff_eq_or_eq_neg {R : Type u_1} [inst : Ring R] {a : R} {b : R} [inst : NoZeroDivisors R] (h : Commute a b) :
a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b
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theorem sq_eq_one_iff {R : Type u_1} [inst : Ring R] {a : R} [inst : NoZeroDivisors R] :
a ^ 2 = 1 ↔ a = 1 ∨ a = -1
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theorem sq_ne_one_iff {R : Type u_1} [inst : Ring R] {a : R} [inst : NoZeroDivisors R] :
a ^ 2 ≠ 1 ↔ a ≠ 1 ∧ a ≠ -1
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theorem sq_sub_sq {R : Type u_1} [inst : CommRing R] (a : R) (b : R) :
a ^ 2 - b ^ 2 = (a + b) * (a - b)
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theorem pow_two_sub_pow_two {R : Type u_1} [inst : CommRing R] (a : R) (b : R) :
a ^ 2 - b ^ 2 = (a + b) * (a - b)

Alias of sq_sub_sq.

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theorem sub_sq {R : Type u_1} [inst : CommRing R] (a : R) (b : R) :
(a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2
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theorem sub_pow_two {R : Type u_1} [inst : CommRing R] (a : R) (b : R) :
(a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2

Alias of sub_sq.

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theorem sub_sq' {R : Type u_1} [inst : CommRing R] (a : R) (b : R) :
(a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b
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theorem sq_eq_sq_iff_eq_or_eq_neg {R : Type u_1} [inst : CommRing R] [inst : NoZeroDivisors R] {a : R} {b : R} :
a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b
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theorem eq_or_eq_neg_of_sq_eq_sq {R : Type u_1} [inst : CommRing R] [inst : NoZeroDivisors R] (a : R) (b : R) :
a ^ 2 = b ^ 2 → a = b ∨ a = -b
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theorem Units.sq_eq_sq_iff_eq_or_eq_neg {R : Type u_1} [inst : CommRing R] [inst : NoZeroDivisors R] {a : Rˣ} {b : Rˣ} :
a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b
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theorem Units.eq_or_eq_neg_of_sq_eq_sq {R : Type u_1} [inst : CommRing R] [inst : NoZeroDivisors R] (a : Rˣ) (b : Rˣ) (h : a ^ 2 = b ^ 2) :
a = b ∨ a = -b