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algebra.group_power.basic

Power operations on monoids and groups #

The power operation on monoids and groups. We separate this from group, because it depends on , which in turn depends on other parts of algebra.

This module contains lemmas about a ^ n and n • a, where n : ℕ or n : ℤ. Further lemmas can be found in algebra.group_power.lemmas.

The analogous results for groups with zero can be found in algebra.group_with_zero.power.

Notation #

Implementation details #

We adopt the convention that 0^0 = 1.

Commutativity #

First we prove some facts about semiconj_by and commute. They do not require any theory about pow and/or nsmul and will be useful later in this file.

@[simp]
theorem pow_ite {M : Type u} [has_pow M ] (P : Prop) [decidable P] (a : M) (b c : ) :
a ^ ite P b c = ite P (a ^ b) (a ^ c)
@[simp]
theorem ite_pow {M : Type u} [has_pow M ] (P : Prop) [decidable P] (a b : M) (c : ) :
ite P a b ^ c = ite P (a ^ c) (b ^ c)
@[simp]
theorem pow_one {M : Type u} [monoid M] (a : M) :
a ^ 1 = a
@[simp]
theorem one_nsmul {M : Type u} [add_monoid M] (a : M) :
1 a = a
@[nolint]
theorem pow_two {M : Type u} [monoid M] (a : M) :
a ^ 2 = a * a

Note that most of the lemmas about powers of two refer to it as sq.

theorem two_nsmul {M : Type u} [add_monoid M] (a : M) :
2 a = a + a
theorem sq {M : Type u} [monoid M] (a : M) :
a ^ 2 = a * a

Alias of pow_two.

theorem pow_three' {M : Type u} [monoid M] (a : M) :
a ^ 3 = a * a * a
theorem pow_three {M : Type u} [monoid M] (a : M) :
a ^ 3 = a * (a * a)
theorem nsmul_add_comm' {M : Type u} [add_monoid M] (a : M) (n : ) :
n a + a = a + n a
theorem pow_mul_comm' {M : Type u} [monoid M] (a : M) (n : ) :
a ^ n * a = a * a ^ n
theorem pow_add {M : Type u} [monoid M] (a : M) (m n : ) :
a ^ (m + n) = a ^ m * a ^ n
theorem add_nsmul {M : Type u} [add_monoid M] (a : M) (m n : ) :
(m + n) a = m a + n a
@[simp]
theorem pow_boole {M : Type u} [monoid M] (P : Prop) [decidable P] (a : M) :
a ^ ite P 1 0 = ite P a 1
@[simp]
theorem one_pow {M : Type u} [monoid M] (n : ) :
1 ^ n = 1
theorem nsmul_zero {M : Type u} [add_monoid M] (n : ) :
n 0 = 0
theorem pow_mul {M : Type u} [monoid M] (a : M) (m n : ) :
a ^ (m * n) = (a ^ m) ^ n
theorem mul_nsmul' {M : Type u} [add_monoid M] (a : M) (m n : ) :
(m * n) a = n m a
theorem pow_right_comm {M : Type u} [monoid M] (a : M) (m n : ) :
(a ^ m) ^ n = (a ^ n) ^ m
theorem nsmul_left_comm {M : Type u} [add_monoid M] (a : M) (m n : ) :
n m a = m n a
theorem pow_mul' {M : Type u} [monoid M] (a : M) (m n : ) :
a ^ (m * n) = (a ^ n) ^ m
theorem mul_nsmul {M : Type u} [add_monoid M] (a : M) (m n : ) :
(m * n) a = m n a
theorem nsmul_add_sub_nsmul {M : Type u} [add_monoid M] (a : M) {m n : } (h : m n) :
m a + (n - m) a = n a
theorem pow_mul_pow_sub {M : Type u} [monoid M] (a : M) {m n : } (h : m n) :
a ^ m * a ^ (n - m) = a ^ n
theorem sub_nsmul_nsmul_add {M : Type u} [add_monoid M] (a : M) {m n : } (h : m n) :
(n - m) a + m a = n a
theorem pow_sub_mul_pow {M : Type u} [monoid M] (a : M) {m n : } (h : m n) :
a ^ (n - m) * a ^ m = a ^ n
theorem pow_eq_pow_mod {M : Type u_1} [monoid M] {x : M} (m : ) {n : } (h : x ^ n = 1) :
x ^ m = x ^ (m % n)

If x ^ n = 1, then x ^ m is the same as x ^ (m % n)

theorem nsmul_eq_mod_nsmul {M : Type u_1} [add_monoid M] {x : M} (m : ) {n : } (h : n x = 0) :
m x = (m % n) x

If n • x = 0, then m • x is the same as (m % n) • x

theorem bit0_nsmul {M : Type u} [add_monoid M] (a : M) (n : ) :
bit0 n a = n a + n a
theorem pow_bit0 {M : Type u} [monoid M] (a : M) (n : ) :
a ^ bit0 n = a ^ n * a ^ n
theorem bit1_nsmul {M : Type u} [add_monoid M] (a : M) (n : ) :
bit1 n a = n a + n a + a
theorem pow_bit1 {M : Type u} [monoid M] (a : M) (n : ) :
a ^ bit1 n = a ^ n * a ^ n * a
theorem nsmul_add_comm {M : Type u} [add_monoid M] (a : M) (m n : ) :
m a + n a = n a + m a
theorem pow_mul_comm {M : Type u} [monoid M] (a : M) (m n : ) :
a ^ m * a ^ n = a ^ n * a ^ m
theorem add_commute.add_nsmul {M : Type u} [add_monoid M] {a b : M} (h : add_commute a b) (n : ) :
n (a + b) = n a + n b
theorem commute.mul_pow {M : Type u} [monoid M] {a b : M} (h : commute a b) (n : ) :
(a * b) ^ n = a ^ n * b ^ n
theorem bit0_nsmul' {M : Type u} [add_monoid M] (a : M) (n : ) :
bit0 n a = n (a + a)
theorem pow_bit0' {M : Type u} [monoid M] (a : M) (n : ) :
a ^ bit0 n = (a * a) ^ n
theorem bit1_nsmul' {M : Type u} [add_monoid M] (a : M) (n : ) :
bit1 n a = n (a + a) + a
theorem pow_bit1' {M : Type u} [monoid M] (a : M) (n : ) :
a ^ bit1 n = (a * a) ^ n * a
theorem nsmul_add_nsmul_eq_zero {M : Type u} [add_monoid M] {a b : M} (n : ) (h : a + b = 0) :
n a + n b = 0
theorem pow_mul_pow_eq_one {M : Type u} [monoid M] {a b : M} (n : ) (h : a * b = 1) :
a ^ n * b ^ n = 1
theorem dvd_pow {M : Type u} [monoid M] {x y : M} (hxy : x y) {n : } (hn : n 0) :
x y ^ n
theorem has_dvd.dvd.pow {M : Type u} [monoid M] {x y : M} (hxy : x y) {n : } (hn : n 0) :
x y ^ n

Alias of dvd_pow.

theorem dvd_pow_self {M : Type u} [monoid M] (a : M) {n : } (hn : n 0) :
a a ^ n

Commutative (additive) monoid #

theorem nsmul_add {M : Type u} [add_comm_monoid M] (a b : M) (n : ) :
n (a + b) = n a + n b
theorem mul_pow {M : Type u} [comm_monoid M] (a b : M) (n : ) :
(a * b) ^ n = a ^ n * b ^ n
@[simp]
theorem nsmul_add_monoid_hom_apply {M : Type u} [add_comm_monoid M] (n : ) (_x : M) :
def nsmul_add_monoid_hom {M : Type u} [add_comm_monoid M] (n : ) :
M →+ M

Multiplication by a natural n on a commutative additive monoid, considered as a morphism of additive monoids.

Equations
def pow_monoid_hom {M : Type u} [comm_monoid M] (n : ) :
M →* M

The nth power map on a commutative monoid for a natural n, considered as a morphism of monoids.

Equations
@[simp]
theorem pow_monoid_hom_apply {M : Type u} [comm_monoid M] (n : ) (_x : M) :
(pow_monoid_hom n) _x = _x ^ n
@[simp]
theorem zpow_one {G : Type w} [div_inv_monoid G] (a : G) :
a ^ 1 = a
@[simp]
theorem one_zsmul {G : Type w} [sub_neg_monoid G] (a : G) :
1 a = a
theorem zpow_two {G : Type w} [div_inv_monoid G] (a : G) :
a ^ 2 = a * a
theorem two_zsmul {G : Type w} [sub_neg_monoid G] (a : G) :
2 a = a + a
theorem zpow_neg_one {G : Type w} [div_inv_monoid G] (x : G) :
x ^ -1 = x⁻¹
theorem neg_one_zsmul {G : Type w} [sub_neg_monoid G] (x : G) :
(-1) x = -x
theorem zsmul_neg_coe_of_pos {G : Type w} [sub_neg_monoid G] (a : G) {n : } :
0 < n -n a = -(n a)
theorem zpow_neg_coe_of_pos {G : Type w} [div_inv_monoid G] (a : G) {n : } :
0 < n a ^ -n = (a ^ n)⁻¹
@[simp]
theorem inv_pow {α : Type u_1} [division_monoid α] (a : α) (n : ) :
a⁻¹ ^ n = (a ^ n)⁻¹
@[simp]
theorem neg_nsmul {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
n -a = -(n a)
theorem zsmul_zero {α : Type u_1} [subtraction_monoid α] (n : ) :
n 0 = 0
@[simp]
theorem one_zpow {α : Type u_1} [division_monoid α] (n : ) :
1 ^ n = 1
@[simp]
theorem neg_zsmul {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
-n a = -(n a)
@[simp]
theorem zpow_neg {α : Type u_1} [division_monoid α] (a : α) (n : ) :
a ^ -n = (a ^ n)⁻¹
theorem mul_zpow_neg_one {α : Type u_1} [division_monoid α] (a b : α) :
(a * b) ^ -1 = b ^ -1 * a ^ -1
theorem neg_one_zsmul_add {α : Type u_1} [subtraction_monoid α] (a b : α) :
(-1) (a + b) = (-1) b + (-1) a
theorem zsmul_neg {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
n -a = -(n a)
theorem inv_zpow {α : Type u_1} [division_monoid α] (a : α) (n : ) :
a⁻¹ ^ n = (a ^ n)⁻¹
@[simp]
theorem zsmul_neg' {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
n -a = -n a
@[simp]
theorem inv_zpow' {α : Type u_1} [division_monoid α] (a : α) (n : ) :
a⁻¹ ^ n = a ^ -n
theorem nsmul_zero_sub {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
n (0 - a) = 0 - n a
theorem one_div_pow {α : Type u_1} [division_monoid α] (a : α) (n : ) :
(1 / a) ^ n = 1 / a ^ n
theorem zsmul_zero_sub {α : Type u_1} [subtraction_monoid α] (a : α) (n : ) :
n (0 - a) = 0 - n a
theorem one_div_zpow {α : Type u_1} [division_monoid α] (a : α) (n : ) :
(1 / a) ^ n = 1 / a ^ n
@[protected]
theorem commute.mul_zpow {α : Type u_1} [division_monoid α] {a b : α} (h : commute a b) (i : ) :
(a * b) ^ i = a ^ i * b ^ i
@[protected]
theorem add_commute.zsmul_add {α : Type u_1} [subtraction_monoid α] {a b : α} (h : add_commute a b) (i : ) :
i (a + b) = i a + i b
theorem zsmul_add {α : Type u_1} [subtraction_comm_monoid α] (a b : α) (n : ) :
n (a + b) = n a + n b
theorem mul_zpow {α : Type u_1} [division_comm_monoid α] (a b : α) (n : ) :
(a * b) ^ n = a ^ n * b ^ n
@[simp]
theorem div_pow {α : Type u_1} [division_comm_monoid α] (a b : α) (n : ) :
(a / b) ^ n = a ^ n / b ^ n
@[simp]
theorem nsmul_sub {α : Type u_1} [subtraction_comm_monoid α] (a b : α) (n : ) :
n (a - b) = n a - n b
@[simp]
theorem zsmul_sub {α : Type u_1} [subtraction_comm_monoid α] (a b : α) (n : ) :
n (a - b) = n a - n b
@[simp]
theorem div_zpow {α : Type u_1} [division_comm_monoid α] (a b : α) (n : ) :
(a / b) ^ n = a ^ n / b ^ n
@[simp]
theorem zpow_group_hom_apply {α : Type u_1} [division_comm_monoid α] (n : ) (_x : α) :
(zpow_group_hom n) _x = _x ^ n
def zpow_group_hom {α : Type u_1} [division_comm_monoid α] (n : ) :
α →* α

The n-th power map (for an integer n) on a commutative group, considered as a group homomorphism.

Equations
def zsmul_add_group_hom {α : Type u_1} [subtraction_comm_monoid α] (n : ) :
α →+ α

Multiplication by an integer n on a commutative additive group, considered as an additive group homomorphism.

Equations
@[simp]
theorem zsmul_add_group_hom_apply {α : Type u_1} [subtraction_comm_monoid α] (n : ) (_x : α) :
theorem sub_nsmul {G : Type w} [add_group G] (a : G) {m n : } (h : n m) :
(m - n) a = m a + -(n a)
theorem pow_sub {G : Type w} [group G] (a : G) {m n : } (h : n m) :
a ^ (m - n) = a ^ m * (a ^ n)⁻¹
theorem pow_inv_comm {G : Type w} [group G] (a : G) (m n : ) :
a⁻¹ ^ m * a ^ n = a ^ n * a⁻¹ ^ m
theorem nsmul_neg_comm {G : Type w} [add_group G] (a : G) (m n : ) :
m -a + n a = n a + m -a
theorem sub_nsmul_neg {G : Type w} [add_group G] (a : G) {m n : } (h : n m) :
(m - n) -a = -(m a) + n a
theorem inv_pow_sub {G : Type w} [group G] (a : G) {m n : } (h : n m) :
a⁻¹ ^ (m - n) = (a ^ m)⁻¹ * a ^ n
theorem pow_dvd_pow {R : Type u₁} [monoid R] (a : R) {m n : } (h : m n) :
a ^ m a ^ n
theorem of_mul_pow {A : Type y} [monoid A] (x : A) (n : ) :
theorem of_mul_zpow {G : Type w} [div_inv_monoid G] (x : G) (n : ) :
@[simp]
theorem add_semiconj_by.zsmul_right {G : Type w} [add_group G] {a x y : G} (h : add_semiconj_by a x y) (m : ) :
add_semiconj_by a (m x) (m y)
@[simp]
theorem semiconj_by.zpow_right {G : Type w} [group G] {a x y : G} (h : semiconj_by a x y) (m : ) :
semiconj_by a (x ^ m) (y ^ m)
@[simp]
theorem add_commute.zsmul_right {G : Type w} [add_group G] {a b : G} (h : add_commute a b) (m : ) :
@[simp]
theorem commute.zpow_right {G : Type w} [group G] {a b : G} (h : commute a b) (m : ) :
commute a (b ^ m)
@[simp]
theorem commute.zpow_left {G : Type w} [group G] {a b : G} (h : commute a b) (m : ) :
commute (a ^ m) b
@[simp]
theorem add_commute.zsmul_left {G : Type w} [add_group G] {a b : G} (h : add_commute a b) (m : ) :
theorem commute.zpow_zpow {G : Type w} [group G] {a b : G} (h : commute a b) (m n : ) :
commute (a ^ m) (b ^ n)
theorem add_commute.zsmul_zsmul {G : Type w} [add_group G] {a b : G} (h : add_commute a b) (m n : ) :
add_commute (m a) (n b)
@[simp]
theorem commute.self_zpow {G : Type w} [group G] (a : G) (n : ) :
commute a (a ^ n)
@[simp]
theorem add_commute.self_zsmul {G : Type w} [add_group G] (a : G) (n : ) :
@[simp]
theorem commute.zpow_self {G : Type w} [group G] (a : G) (n : ) :
commute (a ^ n) a
@[simp]
theorem add_commute.zsmul_self {G : Type w} [add_group G] (a : G) (n : ) :
@[simp]
theorem commute.zpow_zpow_self {G : Type w} [group G] (a : G) (m n : ) :
commute (a ^ m) (a ^ n)
@[simp]
theorem add_commute.zsmul_zsmul_self {G : Type w} [add_group G] (a : G) (m n : ) :
add_commute (m a) (n a)