Documentation

Mathlib.Algebra.GroupWithZero.Commute

Lemmas about commuting elements in a MonoidWithZero or a GroupWithZero. #

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theorem Ring.mul_inverse_rev' {M₀ : Type u_1} [MonoidWithZero M₀] {a b : M₀} (h : Commute a b) :
inverse (a * b) = inverse b * inverse a
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theorem Ring.mul_inverse_rev {M₀ : Type u_3} [CommMonoidWithZero M₀] (a b : M₀) :
inverse (a * b) = inverse b * inverse a
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theorem Ring.inverse_pow {M₀ : Type u_1} [MonoidWithZero M₀] (r : M₀) (n : ℕ) :
inverse r ^ n = inverse (r ^ n)
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theorem Ring.inverse_pow_mul_eq_iff_eq_mul {M₀ : Type u_1} [MonoidWithZero M₀] {a : M₀} (b c : M₀) (ha : IsUnit a) {k : ℕ} :
inverse a ^ k * b = c ↔ b = a ^ k * c
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theorem Commute.ringInverse_ringInverse {M₀ : Type u_1} [MonoidWithZero M₀] {a b : M₀} (h : Commute a b) :
Commute (Ring.inverse a) (Ring.inverse b)
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@[deprecated Commute.ringInverse_ringInverse (since := "2025-04-22")]
theorem Commute.ring_inverse_ring_inverse {M₀ : Type u_1} [MonoidWithZero M₀] {a b : M₀} (h : Commute a b) :
Commute (Ring.inverse a) (Ring.inverse b)

Alias of Commute.ringInverse_ringInverse.

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@[simp]
theorem Commute.zero_right {G₀ : Type u_2} [MulZeroClass G₀] (a : G₀) :
Commute a 0
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@[simp]
theorem Commute.zero_left {G₀ : Type u_2} [MulZeroClass G₀] (a : G₀) :
Commute 0 a
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@[simp]
theorem Commute.inv_left_iff₀ {G₀ : Type u_2} [GroupWithZero G₀] {a b : G₀} :
Commute a⁻¹ b ↔ Commute a b
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theorem Commute.inv_left₀ {G₀ : Type u_2} [GroupWithZero G₀] {a b : G₀} (h : Commute a b) :
Commute a⁻¹ b
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@[simp]
theorem Commute.inv_right_iff₀ {G₀ : Type u_2} [GroupWithZero G₀] {a b : G₀} :
Commute a b⁻¹ ↔ Commute a b
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theorem Commute.inv_right₀ {G₀ : Type u_2} [GroupWithZero G₀] {a b : G₀} (h : Commute a b) :
Commute a b⁻¹
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@[simp]
theorem Commute.div_right {G₀ : Type u_2} [GroupWithZero G₀] {a b c : G₀} (hab : Commute a b) (hac : Commute a c) :
Commute a (b / c)
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@[simp]
theorem Commute.div_left {G₀ : Type u_2} [GroupWithZero G₀] {a b c : G₀} (hac : Commute a c) (hbc : Commute b c) :
Commute (a / b) c
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theorem pow_inv_comm₀ {G₀ : Type u_2} [GroupWithZero G₀] (a : G₀) (m n : ℕ) :
a⁻¹ ^ m * a ^ n = a ^ n * a⁻¹ ^ m