mathlib documentation

algebra.group.commute

Commuting pairs of elements in monoids #

We define the predicate commute a b := a * b = b * a and provide some operations on terms (h : commute a b). E.g., if a, b, and c are elements of a semiring, and that hb : commute a b and hc : commute a c. Then hb.pow_left 5 proves commute (a ^ 5) b and (hb.pow_right 2).add_right (hb.mul_right hc) proves commute a (b ^ 2 + b * c).

Lean does not immediately recognise these terms as equations, so for rewriting we need syntax like rw [(hb.pow_left 5).eq] rather than just rw [hb.pow_left 5].

This file defines only a few operations (mul_left, inv_right, etc). Other operations (pow_right, field inverse etc) are in the files that define corresponding notions.

Implementation details #

Most of the proofs come from the properties of semiconj_by.

def add_commute {S : Type u_1} [has_add S] (a b : S) :

Prop

Two elements additively commute if a + b = b + a

Equations

add_commute a b = add_semiconj_by a b b

def commute {S : Type u_1} [has_mul S] (a b : S) :

Prop

Two elements commute if a * b = b * a.

Equations

commute a b = semiconj_by a b b

@[protected]

theorem commute.eq {S : Type u_1} [has_mul S] {a b : S} (h : commute a b) :

a * b = b * a

Equality behind commute a b; useful for rewriting.

@[protected]

theorem add_commute.eq {S : Type u_1} [has_add S] {a b : S} (h : add_commute a b) :

a + b = b + a

@[protected, simp]

theorem add_commute.refl {S : Type u_1} [has_add S] (a : S) :

add_commute a a

@[protected, simp]

theorem commute.refl {S : Type u_1} [has_mul S] (a : S) :

Any element commutes with itself.

@[protected]

theorem add_commute.symm {S : Type u_1} [has_add S] {a b : S} (h : add_commute a b) :

add_commute b a

@[protected]

theorem commute.symm {S : Type u_1} [has_mul S] {a b : S} (h : commute a b) :

If a commutes with b, then b commutes with a.

@[protected]

theorem commute.semiconj_by {S : Type u_1} [has_mul S] {a b : S} (h : commute a b) :

semiconj_by a b b

@[protected]

theorem add_commute.semiconj_by {S : Type u_1} [has_add S] {a b : S} (h : add_commute a b) :

add_semiconj_by a b b

@[protected]

theorem add_commute.symm_iff {S : Type u_1} [has_add S] {a b : S} :

add_commute a b ↔ add_commute b a

@[protected]

theorem commute.symm_iff {S : Type u_1} [has_mul S] {a b : S} :

commute a b ↔ commute b a

@[simp]

theorem commute.mul_right {S : Type u_1} [semigroup S] {a b c : S} (hab : commute a b) (hac : commute a c) :

commute a (b * c)

If a commutes with both b and c, then it commutes with their product.

@[simp]

theorem add_commute.add_right {S : Type u_1} [add_semigroup S] {a b c : S} (hab : add_commute a b) (hac : add_commute a c) :

add_commute a (b + c)

@[simp]

theorem commute.mul_left {S : Type u_1} [semigroup S] {a b c : S} (hac : commute a c) (hbc : commute b c) :

commute (a * b) c

If both a and b commute with c, then their product commutes with c.

@[simp]

theorem add_commute.add_left {S : Type u_1} [add_semigroup S] {a b c : S} (hac : add_commute a c) (hbc : add_commute b c) :

add_commute (a + b) c

@[protected]

theorem commute.right_comm {S : Type u_1} [semigroup S] {b c : S} (h : commute b c) (a : S) :

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

@[protected]

theorem add_commute.right_comm {S : Type u_1} [add_semigroup S] {b c : S} (h : add_commute b c) (a : S) :

a + b + c = a + c + b

@[protected]

theorem add_commute.left_comm {S : Type u_1} [add_semigroup S] {a b : S} (h : add_commute a b) (c : S) :

a + (b + c) = b + (a + c)

@[protected]

theorem commute.left_comm {S : Type u_1} [semigroup S] {a b : S} (h : commute a b) (c : S) :

a * b * c = b * a * c

@[protected]

theorem commute.all {S : Type u_1} [comm_semigroup S] (a b : S) :

@[protected]

theorem add_commute.all {S : Type u_1} [add_comm_semigroup S] (a b : S) :

add_commute a b

@[simp]

theorem commute.one_right {M : Type u_1} [mul_one_class M] (a : M) :

@[simp]

theorem add_commute.zero_right {M : Type u_1} [add_zero_class M] (a : M) :

add_commute a 0

@[simp]

theorem commute.one_left {M : Type u_1} [mul_one_class M] (a : M) :

@[simp]

theorem add_commute.zero_left {M : Type u_1} [add_zero_class M] (a : M) :

add_commute 0 a

@[simp]

theorem add_commute.pow_right {M : Type u_1} [add_monoid M] {a b : M} (h : add_commute a b) (n : ℕ) :

add_commute a (n • b)

@[simp]

theorem commute.pow_right {M : Type u_1} [monoid M] {a b : M} (h : commute a b) (n : ℕ) :

commute a (b ^ n)

@[simp]

theorem commute.pow_left {M : Type u_1} [monoid M] {a b : M} (h : commute a b) (n : ℕ) :

commute (a ^ n) b

@[simp]

theorem add_commute.pow_left {M : Type u_1} [add_monoid M] {a b : M} (h : add_commute a b) (n : ℕ) :

add_commute (n • a) b

@[simp]

theorem commute.pow_pow {M : Type u_1} [monoid M] {a b : M} (h : commute a b) (m n : ℕ) :

commute (a ^ m) (b ^ n)

@[simp]

theorem add_commute.pow_pow {M : Type u_1} [add_monoid M] {a b : M} (h : add_commute a b) (m n : ℕ) :

add_commute (m • a) (n • b)

@[simp]

theorem commute.self_pow {M : Type u_1} [monoid M] (a : M) (n : ℕ) :

commute a (a ^ n)

@[simp]

theorem add_commute.self_pow {M : Type u_1} [add_monoid M] (a : M) (n : ℕ) :

add_commute a (n • a)

@[simp]

theorem commute.pow_self {M : Type u_1} [monoid M] (a : M) (n : ℕ) :

commute (a ^ n) a

@[simp]

theorem add_commute.pow_self {M : Type u_1} [add_monoid M] (a : M) (n : ℕ) :

add_commute (n • a) a

@[simp]

theorem add_commute.pow_pow_self {M : Type u_1} [add_monoid M] (a : M) (m n : ℕ) :

add_commute (m • a) (n • a)

@[simp]

theorem commute.pow_pow_self {M : Type u_1} [monoid M] (a : M) (m n : ℕ) :

commute (a ^ m) (a ^ n)

theorem pow_succ' {M : Type u_1} [monoid M] (a : M) (n : ℕ) :

a ^ (n + 1) = (a ^ n) * a

theorem succ_nsmul' {M : Type u_1} [add_monoid M] (a : M) (n : ℕ) :

(n + 1) • a = n • a + a

theorem add_commute.units_neg_right {M : Type u_1} [add_monoid M] {a : M} {u : add_units M} :

add_commute a ↑u → add_commute a (↑-u)

theorem commute.units_inv_right {M : Type u_1} [monoid M] {a : M} {u : units M} :

commute a ↑u → commute a ↑u⁻¹

@[simp]

theorem commute.units_inv_right_iff {M : Type u_1} [monoid M] {a : M} {u : units M} :

commute a ↑u⁻¹ ↔ commute a ↑u

@[simp]

theorem add_commute.units_neg_right_iff {M : Type u_1} [add_monoid M] {a : M} {u : add_units M} :

add_commute a (↑-u) ↔ add_commute a ↑u

theorem commute.units_inv_left {M : Type u_1} [monoid M] {a : M} {u : units M} :

commute ↑u a → commute ↑u⁻¹ a

theorem add_commute.units_neg_left {M : Type u_1} [add_monoid M] {a : M} {u : add_units M} :

add_commute ↑u a → add_commute (↑-u) a

@[simp]

theorem commute.units_inv_left_iff {M : Type u_1} [monoid M] {a : M} {u : units M} :

commute ↑u⁻¹ a ↔ commute ↑u a

@[simp]

theorem add_commute.units_neg_left_iff {M : Type u_1} [add_monoid M] {a : M} {u : add_units M} :

add_commute (↑-u) a ↔ add_commute ↑u a

theorem add_commute.units_coe {M : Type u_1} [add_monoid M] {u₁ u₂ : add_units M} :

add_commute u₁ u₂ → add_commute ↑u₁ ↑u₂

theorem commute.units_coe {M : Type u_1} [monoid M] {u₁ u₂ : units M} :

commute u₁ u₂ → commute ↑u₁ ↑u₂

theorem add_commute.units_of_coe {M : Type u_1} [add_monoid M] {u₁ u₂ : add_units M} :

add_commute ↑u₁ ↑u₂ → add_commute u₁ u₂

theorem commute.units_of_coe {M : Type u_1} [monoid M] {u₁ u₂ : units M} :

commute ↑u₁ ↑u₂ → commute u₁ u₂

@[simp]

theorem add_commute.units_coe_iff {M : Type u_1} [add_monoid M] {u₁ u₂ : add_units M} :

add_commute ↑u₁ ↑u₂ ↔ add_commute u₁ u₂

@[simp]

theorem commute.units_coe_iff {M : Type u_1} [monoid M] {u₁ u₂ : units M} :

commute ↑u₁ ↑u₂ ↔ commute u₁ u₂

theorem add_commute.is_unit_add_iff {M : Type u_1} [add_monoid M] {a b : M} (h : add_commute a b) :

is_add_unit (a + b) ↔ is_add_unit a ∧ is_add_unit b

theorem commute.is_unit_mul_iff {M : Type u_1} [monoid M] {a b : M} (h : commute a b) :

is_unit (a * b) ↔ is_unit a ∧ is_unit b

@[simp]

theorem is_unit_add_self_iff {M : Type u_1} [add_monoid M] {a : M} :

is_add_unit (a + a) ↔ is_add_unit a

@[simp]

theorem is_unit_mul_self_iff {M : Type u_1} [monoid M] {a : M} :

is_unit (a * a) ↔ is_unit a

theorem add_commute.neg_right {G : Type u_1} [add_group G] {a b : G} :

add_commute a b → add_commute a (-b)

theorem commute.inv_right {G : Type u_1} [group G] {a b : G} :

commute a b → commute a b⁻¹

@[simp]

theorem commute.inv_right_iff {G : Type u_1} [group G] {a b : G} :

commute a b⁻¹ ↔ commute a b

@[simp]

theorem add_commute.neg_right_iff {G : Type u_1} [add_group G] {a b : G} :

add_commute a (-b) ↔ add_commute a b

theorem add_commute.neg_left {G : Type u_1} [add_group G] {a b : G} :

add_commute a b → add_commute (-a) b

theorem commute.inv_left {G : Type u_1} [group G] {a b : G} :

commute a b → commute a⁻¹ b

@[simp]

theorem commute.inv_left_iff {G : Type u_1} [group G] {a b : G} :

commute a⁻¹ b ↔ commute a b

@[simp]

theorem add_commute.neg_left_iff {G : Type u_1} [add_group G] {a b : G} :

add_commute (-a) b ↔ add_commute a b

theorem add_commute.neg_neg {G : Type u_1} [add_group G] {a b : G} :

add_commute a b → add_commute (-a) (-b)

theorem commute.inv_inv {G : Type u_1} [group G] {a b : G} :

commute a b → commute a⁻¹ b⁻¹

@[simp]

theorem add_commute.neg_neg_iff {G : Type u_1} [add_group G] {a b : G} :

add_commute (-a) (-b) ↔ add_commute a b

@[simp]

theorem commute.inv_inv_iff {G : Type u_1} [group G] {a b : G} :

commute a⁻¹ b⁻¹ ↔ commute a b

@[protected]

theorem add_commute.neg_add_cancel {G : Type u_1} [add_group G] {a b : G} (h : add_commute a b) :

-a + b + a = b

@[protected]

theorem commute.inv_mul_cancel {G : Type u_1} [group G] {a b : G} (h : commute a b) :

(a⁻¹ * b) * a = b

theorem commute.inv_mul_cancel_assoc {G : Type u_1} [group G] {a b : G} (h : commute a b) :

a⁻¹ * b * a = b

theorem add_commute.neg_add_cancel_assoc {G : Type u_1} [add_group G] {a b : G} (h : add_commute a b) :

-a + (b + a) = b

@[protected]

theorem commute.mul_inv_cancel {G : Type u_1} [group G] {a b : G} (h : commute a b) :

(a * b) * a⁻¹ = b

@[protected]

theorem add_commute.add_neg_cancel {G : Type u_1} [add_group G] {a b : G} (h : add_commute a b) :

a + b + -a = b

theorem commute.mul_inv_cancel_assoc {G : Type u_1} [group G] {a b : G} (h : commute a b) :

a * b * a⁻¹ = b

theorem add_commute.add_neg_cancel_assoc {G : Type u_1} [add_group G] {a b : G} (h : add_commute a b) :

a + (b + -a) = b

@[simp]

theorem mul_inv_cancel_comm {G : Type u_1} [comm_group G] (a b : G) :

(a * b) * a⁻¹ = b

@[simp]

theorem add_neg_cancel_comm {G : Type u_1} [add_comm_group G] (a b : G) :

a + b + -a = b

@[simp]

theorem mul_inv_cancel_comm_assoc {G : Type u_1} [comm_group G] (a b : G) :

a * b * a⁻¹ = b

@[simp]

theorem add_neg_cancel_comm_assoc {G : Type u_1} [add_comm_group G] (a b : G) :

a + (b + -a) = b

@[simp]

theorem inv_mul_cancel_comm {G : Type u_1} [comm_group G] (a b : G) :

(a⁻¹ * b) * a = b

@[simp]

theorem neg_add_cancel_comm {G : Type u_1} [add_comm_group G] (a b : G) :

-a + b + a = b

@[simp]

theorem neg_add_cancel_comm_assoc {G : Type u_1} [add_comm_group G] (a b : G) :

-a + (b + a) = b

@[simp]

theorem inv_mul_cancel_comm_assoc {G : Type u_1} [comm_group G] (a b : G) :

a⁻¹ * b * a = b