mathlib documentation

algebra.group_power.lemmas

Lemmas about power operations on monoids and groups #

This file contains lemmas about monoid.pow, group.pow, nsmul, zsmul which require additional imports besides those available in algebra.group_power.basic.

(Additive) monoid #

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@[simp]
theorem nsmul_one {A : Type y} [add_monoid A] [has_one A] (n : ℕ) :
n • 1 = ↑n
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@[simp, norm_cast]
theorem add_units.coe_pow {M : Type u} [add_monoid M] (u : add_units M) (n : ℕ) :
↑(n • u) = n • ↑u
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@[simp, norm_cast]
theorem units.coe_pow {M : Type u} [monoid M] (u : units M) (n : ℕ) :
↑(u ^ n) = ↑u ^ n
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@[protected, instance]
def invertible_pow {M : Type u} [monoid M] (m : M) [invertible m] (n : ℕ) :
invertible (m ^ n)
Equations
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theorem inv_of_pow {M : Type u} [monoid M] (m : M) [invertible m] (n : ℕ) [invertible (m ^ n)] :
⅟ (m ^ n) = ⅟ m ^ n
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theorem is_unit.pow {M : Type u} [monoid M] {m : M} (n : ℕ) :
is_unit m → is_unit (m ^ n)
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@[simp]
theorem is_unit_pow_succ_iff {M : Type u} [monoid M] {m : M} {n : ℕ} :
is_unit (m ^ (n + 1)) ↔ is_unit m
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theorem is_unit_pos_pow_iff {M : Type u} [monoid M] {m : M} {n : ℕ} (h : 0 < n) :
is_unit (m ^ n) ↔ is_unit m
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def invertible_of_pow_succ_eq_one {M : Type u} [monoid M] (x : M) (n : ℕ) (hx : x ^ n.succ = 1) :
invertible x

If x ^ n.succ = 1 then x has an inverse, x^n.

Equations
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def invertible_of_pow_eq_one {M : Type u} [monoid M] (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
invertible x

If x ^ n = 1 then x has an inverse, x^(n - 1).

Equations
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theorem is_unit_of_pow_eq_one {M : Type u} [monoid M] (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : 0 < n) :
is_unit x
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theorem smul_pow {M : Type u} {N : Type v} [monoid M] [monoid N] [mul_action M N] [is_scalar_tower M N N] [smul_comm_class M N N] (k : M) (x : N) (p : ℕ) :
(k • x) ^ p = k ^ p • x ^ p
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@[simp]
theorem smul_pow' {M : Type u} {N : Type v} [monoid M] [monoid N] [mul_distrib_mul_action M N] (x : M) (m : N) (n : ℕ) :
x • m ^ n = (x • m) ^ n
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theorem zsmul_one {A : Type y} [add_group A] [has_one A] (n : ℤ) :
n • 1 = ↑n
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theorem zpow_add_one {G : Type w} [group G] (a : G) (n : ℤ) :
a ^ (n + 1) = (a ^ n) * a
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theorem add_one_zsmul {G : Type w} [add_group G] (a : G) (n : ℤ) :
(n + 1) • a = n • a + a
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theorem zpow_sub_one {G : Type w} [group G] (a : G) (n : ℤ) :
a ^ (n - 1) = (a ^ n) * a⁻¹
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theorem zsmul_sub_one {G : Type w} [add_group G] (a : G) (n : ℤ) :
(n - 1) • a = n • a + -a
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theorem zpow_add {G : Type w} [group G] (a : G) (m n : ℤ) :
a ^ (m + n) = (a ^ m) * a ^ n
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theorem add_zsmul {G : Type w} [add_group G] (a : G) (m n : ℤ) :
(m + n) • a = m • a + n • a
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theorem mul_self_zpow {G : Type w} [group G] (b : G) (m : ℤ) :
b * b ^ m = b ^ (m + 1)
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theorem add_zsmul_self {G : Type w} [add_group G] (b : G) (m : ℤ) :
b + m • b = (m + 1) • b
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theorem add_self_zsmul {G : Type w} [add_group G] (b : G) (m : ℤ) :
m • b + b = (m + 1) • b
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theorem mul_zpow_self {G : Type w} [group G] (b : G) (m : ℤ) :
(b ^ m) * b = b ^ (m + 1)
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theorem sub_zsmul {G : Type w} [add_group G] (a : G) (m n : ℤ) :
(m - n) • a = m • a + -(n • a)
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theorem zpow_sub {G : Type w} [group G] (a : G) (m n : ℤ) :
a ^ (m - n) = (a ^ m) * (a ^ n)⁻¹
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theorem one_add_zsmul {G : Type w} [add_group G] (a : G) (i : ℤ) :
(1 + i) • a = a + i • a
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theorem zpow_one_add {G : Type w} [group G] (a : G) (i : ℤ) :
a ^ (1 + i) = a * a ^ i
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theorem zsmul_add_comm {G : Type w} [add_group G] (a : G) (i j : ℤ) :
i • a + j • a = j • a + i • a
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theorem zpow_mul_comm {G : Type w} [group G] (a : G) (i j : ℤ) :
(a ^ i) * a ^ j = (a ^ j) * a ^ i
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theorem mul_zsmul' {G : Type w} [add_group G] (a : G) (m n : ℤ) :
(m * n) • a = n • m • a
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theorem zpow_mul {G : Type w} [group G] (a : G) (m n : ℤ) :
a ^ m * n = (a ^ m) ^ n
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theorem zpow_mul' {G : Type w} [group G] (a : G) (m n : ℤ) :
a ^ m * n = (a ^ n) ^ m
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theorem mul_zsmul {G : Type w} [add_group G] (a : G) (m n : ℤ) :
(m * n) • a = m • n • a
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theorem bit0_zsmul {G : Type w} [add_group G] (a : G) (n : ℤ) :
bit0 n • a = n • a + n • a
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theorem zpow_bit0 {G : Type w} [group G] (a : G) (n : ℤ) :
a ^ bit0 n = (a ^ n) * a ^ n
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theorem zpow_bit1 {G : Type w} [group G] (a : G) (n : ℤ) :
a ^ bit1 n = ((a ^ n) * a ^ n) * a
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theorem bit1_zsmul {G : Type w} [add_group G] (a : G) (n : ℤ) :
bit1 n • a = n • a + n • a + a
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@[simp, norm_cast]
theorem units.coe_zpow {G : Type w} [group G] (u : units G) (n : ℤ) :
↑(u ^ n) = ↑u ^ n
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@[simp, norm_cast]
theorem add_units.coe_zsmul {G : Type w} [add_group G] (u : add_units G) (n : ℤ) :
↑(n • u) = n • ↑u

Lemmas about zsmul under ordering, placed here (rather than in algebra.group_power.order with their friends) because they require facts from data.int.basic

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theorem zsmul_pos {A : Type y} [ordered_add_comm_group A] {a : A} (ha : 0 < a) {k : ℤ} (hk : 0 < k) :
0 < k • a
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theorem zsmul_strict_mono_left {A : Type y} [ordered_add_comm_group A] {a : A} (ha : 0 < a) :
strict_mono (λ (n : ℤ), n • a)
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theorem zsmul_mono_left {A : Type y} [ordered_add_comm_group A] {a : A} (ha : 0 ≤ a) :
monotone (λ (n : ℤ), n • a)
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theorem zsmul_le_zsmul {A : Type y} [ordered_add_comm_group A] {a : A} {n m : ℤ} (ha : 0 ≤ a) (h : n ≤ m) :
n • a ≤ m • a
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theorem zsmul_lt_zsmul {A : Type y} [ordered_add_comm_group A] {a : A} {n m : ℤ} (ha : 0 < a) (h : n < m) :
n • a < m • a
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theorem zsmul_le_zsmul_iff {A : Type y} [ordered_add_comm_group A] {a : A} {n m : ℤ} (ha : 0 < a) :
n • a ≤ m • a ↔ n ≤ m
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theorem zsmul_lt_zsmul_iff {A : Type y} [ordered_add_comm_group A] {a : A} {n m : ℤ} (ha : 0 < a) :
n • a < m • a ↔ n < m
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theorem zsmul_strict_mono_right (A : Type y) [ordered_add_comm_group A] {n : ℤ} (hn : 0 < n) :
strict_mono (has_scalar.smul n)
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theorem zsmul_mono_right (A : Type y) [ordered_add_comm_group A] {n : ℤ} (hn : 0 ≤ n) :
monotone (has_scalar.smul n)
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theorem zsmul_le_zsmul' {A : Type y} [ordered_add_comm_group A] {n : ℤ} (hn : 0 ≤ n) {a₁ a₂ : A} (h : a₁ ≤ a₂) :
n • a₁ ≤ n • a₂
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theorem zsmul_lt_zsmul' {A : Type y} [ordered_add_comm_group A] {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} (h : a₁ < a₂) :
n • a₁ < n • a₂
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theorem abs_nsmul {α : Type u_1} [linear_ordered_add_comm_group α] (n : ℕ) (a : α) :
|n • a| = n • |a|
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theorem abs_zsmul {α : Type u_1} [linear_ordered_add_comm_group α] (n : ℤ) (a : α) :
|n • a| = |n| • |a|
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theorem abs_add_eq_add_abs_le {α : Type u_1} [linear_ordered_add_comm_group α] {a b : α} (hle : a ≤ b) :
|a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
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theorem abs_add_eq_add_abs_iff {α : Type u_1} [linear_ordered_add_comm_group α] (a b : α) :
|a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0
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theorem zsmul_le_zsmul_iff' {A : Type y} [linear_ordered_add_comm_group A] {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} :
n • a₁ ≤ n • a₂ ↔ a₁ ≤ a₂
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theorem zsmul_lt_zsmul_iff' {A : Type y} [linear_ordered_add_comm_group A] {n : ℤ} (hn : 0 < n) {a₁ a₂ : A} :
n • a₁ < n • a₂ ↔ a₁ < a₂
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theorem nsmul_le_nsmul_iff {A : Type y} [linear_ordered_add_comm_group A] {a : A} {n m : ℕ} (ha : 0 < a) :
n • a ≤ m • a ↔ n ≤ m
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theorem nsmul_lt_nsmul_iff {A : Type y} [linear_ordered_add_comm_group A] {a : A} {n m : ℕ} (ha : 0 < a) :
n • a < m • a ↔ n < m
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theorem zsmul_right_injective {A : Type y} [linear_ordered_add_comm_group A] {m : ℤ} (hm : m ≠ 0) :
function.injective (has_scalar.smul m)

See also smul_right_injective. TODO: provide a no_zero_smul_divisors instance. We can't do that here because importing that definition would create import cycles.

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theorem zsmul_right_inj {A : Type y} [linear_ordered_add_comm_group A] {a b : A} {m : ℤ} (hm : m ≠ 0) :
m • a = m • b ↔ a = b
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theorem zsmul_eq_zsmul_iff' {A : Type y} [linear_ordered_add_comm_group A] {a b : A} {m : ℤ} (hm : m ≠ 0) :
m • a = m • b ↔ a = b

Alias of zsmul_right_inj, for ease of discovery alongside zsmul_le_zsmul_iff' and zsmul_lt_zsmul_iff'.

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@[simp]
theorem with_bot.coe_nsmul {A : Type y} [add_monoid A] (a : A) (n : ℕ) :
↑(n • a) = n • ↑a
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theorem nsmul_eq_mul' {R : Type u₁} [semiring R] (a : R) (n : ℕ) :
n • a = a * ↑n
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@[simp]
theorem nsmul_eq_mul {R : Type u₁} [semiring R] (n : ℕ) (a : R) :
n • a = (↑n) * a
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theorem mul_nsmul_left {R : Type u₁} [semiring R] (a b : R) (n : ℕ) :
n • a * b = a * n • b
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theorem mul_nsmul_assoc {R : Type u₁} [semiring R] (a b : R) (n : ℕ) :
n • a * b = (n • a) * b
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@[simp, norm_cast]
theorem nat.cast_pow {R : Type u₁} [semiring R] (n m : ℕ) :
↑(n ^ m) = ↑n ^ m
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@[simp, norm_cast]
theorem int.coe_nat_pow (n m : ℕ) :
↑(n ^ m) = ↑n ^ m
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theorem int.nat_abs_pow (n : ℤ) (k : ℕ) :
(n ^ k).nat_abs = n.nat_abs ^ k
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theorem bit0_mul {R : Type u₁} [ring R] {n r : R} :
(bit0 n) * r = 2 • n * r
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theorem mul_bit0 {R : Type u₁} [ring R] {n r : R} :
r * bit0 n = 2 • r * n
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theorem bit1_mul {R : Type u₁} [ring R] {n r : R} :
(bit1 n) * r = 2 • n * r + r
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theorem mul_bit1 {R : Type u₁} [ring R] {n r : R} :
r * bit1 n = 2 • r * n + r
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@[simp]
theorem zsmul_eq_mul {R : Type u₁} [ring R] (a : R) (n : ℤ) :
n • a = (↑n) * a
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theorem zsmul_eq_mul' {R : Type u₁} [ring R] (a : R) (n : ℤ) :
n • a = a * ↑n
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theorem mul_zsmul_left {R : Type u₁} [ring R] (a b : R) (n : ℤ) :
n • a * b = a * n • b
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theorem mul_zsmul_assoc {R : Type u₁} [ring R] (a b : R) (n : ℤ) :
n • a * b = (n • a) * b
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theorem zsmul_int_int (a b : ℤ) :
a • b = a * b
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theorem zsmul_int_one (n : ℤ) :
n • 1 = n
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@[simp, norm_cast]
theorem int.cast_pow {R : Type u₁} [ring R] (n : ℤ) (m : ℕ) :
↑(n ^ m) = ↑n ^ m
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theorem neg_one_pow_eq_pow_mod_two {R : Type u₁} [ring R] {n : ℕ} :
(-1) ^ n = (-1) ^ (n % 2)
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theorem one_add_mul_le_pow' {R : Type u₁} [ordered_semiring R] {a : R} (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a)) (H : 0 ≤ 2 + a) (n : ℕ) :
1 + (↑n) * a ≤ (1 + a) ^ n

Bernoulli's inequality. This version works for semirings but requires additional hypotheses 0 ≤ a * a and 0 ≤ (1 + a) * (1 + a).

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theorem pow_le_pow_of_le_one {R : Type u₁} [ordered_semiring R] {a : R} (h : 0 ≤ a) (ha : a ≤ 1) {i j : ℕ} (hij : i ≤ j) :
a ^ j ≤ a ^ i
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theorem pow_le_of_le_one {R : Type u₁} [ordered_semiring R] {a : R} (h₀ : 0 ≤ a) (h₁ : a ≤ 1) {n : ℕ} (hn : n ≠ 0) :
a ^ n ≤ a
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theorem sq_le {R : Type u₁} [ordered_semiring R] {a : R} (h₀ : 0 ≤ a) (h₁ : a ≤ 1) :
a ^ 2 ≤ a
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theorem sign_cases_of_C_mul_pow_nonneg {R : Type u₁} [linear_ordered_semiring R] {C r : R} (h : ∀ (n : ℕ), 0 ≤ C * r ^ n) :
C = 0 ∨ 0 < C ∧ 0 ≤ r
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@[simp]
theorem abs_pow {R : Type u₁} [linear_ordered_ring R] (a : R) (n : ℕ) :
|a ^ n| = |a| ^ n
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@[simp]
theorem pow_bit1_neg_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} :
a ^ bit1 n < 0 ↔ a < 0
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@[simp]
theorem pow_bit1_nonneg_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} :
0 ≤ a ^ bit1 n ↔ 0 ≤ a
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@[simp]
theorem pow_bit1_nonpos_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} :
a ^ bit1 n ≤ 0 ↔ a ≤ 0
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@[simp]
theorem pow_bit1_pos_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} :
0 < a ^ bit1 n ↔ 0 < a
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theorem even.pow_nonneg {R : Type u₁} [linear_ordered_ring R] {n : ℕ} (hn : even n) (a : R) :
0 ≤ a ^ n
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theorem even.pow_pos {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : even n) (ha : a ≠ 0) :
0 < a ^ n
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theorem odd.pow_nonpos {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) (ha : a ≤ 0) :
a ^ n ≤ 0
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theorem odd.pow_neg {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) (ha : a < 0) :
a ^ n < 0
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theorem odd.pow_nonneg_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) :
0 ≤ a ^ n ↔ 0 ≤ a
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theorem odd.pow_nonpos_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) :
a ^ n ≤ 0 ↔ a ≤ 0
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theorem odd.pow_pos_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) :
0 < a ^ n ↔ 0 < a
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theorem odd.pow_neg_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : odd n) :
a ^ n < 0 ↔ a < 0
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theorem even.pow_pos_iff {R : Type u₁} [linear_ordered_ring R] {a : R} {n : ℕ} (hn : even n) (h₀ : 0 < n) :
0 < a ^ n ↔ a ≠ 0
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theorem even.pow_abs {R : Type u₁} [linear_ordered_ring R] {p : ℕ} (hp : even p) (a : R) :
|a| ^ p = a ^ p
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@[simp]
theorem pow_bit0_abs {R : Type u₁} [linear_ordered_ring R] (a : R) (p : ℕ) :
|a| ^ bit0 p = a ^ bit0 p
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theorem strict_mono_pow_bit1 {R : Type u₁} [linear_ordered_ring R] (n : ℕ) :
strict_mono (λ (a : R), a ^ bit1 n)
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theorem odd.strict_mono_pow {R : Type u₁} [linear_ordered_ring R] {n : ℕ} (hn : odd n) :
strict_mono (λ (a : R), a ^ n)
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theorem one_add_mul_le_pow {R : Type u₁} [linear_ordered_ring R] {a : R} (H : -2 ≤ a) (n : ℕ) :
1 + (↑n) * a ≤ (1 + a) ^ n

Bernoulli's inequality for n : ℕ, -2 ≤ a.

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theorem one_add_mul_sub_le_pow {R : Type u₁} [linear_ordered_ring R] {a : R} (H : -1 ≤ a) (n : ℕ) :
1 + (↑n) * (a - 1) ≤ a ^ n

Bernoulli's inequality reformulated to estimate a^n.

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theorem nat.cast_le_pow_sub_div_sub {K : Type u_1} [linear_ordered_field K] {a : K} (H : 1 < a) (n : ℕ) :
↑n ≤ (a ^ n - 1) / (a - 1)

Bernoulli's inequality reformulated to estimate (n : K).

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theorem nat.cast_le_pow_div_sub {K : Type u_1} [linear_ordered_field K] {a : K} (H : 1 < a) (n : ℕ) :
↑n ≤ a ^ n / (a - 1)

For any a > 1 and a natural n we have n ≤ a ^ n / (a - 1). See also nat.cast_le_pow_sub_div_sub for a stronger inequality with a ^ n - 1 in the numerator.

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theorem int.units_sq (u : units ℤ) :
u ^ 2 = 1
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theorem int.units_pow_two (u : units ℤ) :
u ^ 2 = 1

Alias of int.units_sq.

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theorem int.units_pow_eq_pow_mod_two (u : units ℤ) (n : ℕ) :
u ^ n = u ^ (n % 2)
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@[simp]
theorem int.nat_abs_sq (x : ℤ) :
↑(x.nat_abs) ^ 2 = x ^ 2
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theorem int.nat_abs_pow_two (x : ℤ) :
↑(x.nat_abs) ^ 2 = x ^ 2

Alias of int.nat_abs_sq.

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theorem int.abs_le_self_sq (a : ℤ) :
↑(a.nat_abs) ≤ a ^ 2
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theorem int.abs_le_self_pow_two (a : ℤ) :
↑(a.nat_abs) ≤ a ^ 2

Alias of int.abs_le_self_sq.

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theorem int.le_self_sq (b : ℤ) :
b ≤ b ^ 2
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theorem int.le_self_pow_two (b : ℤ) :
b ≤ b ^ 2

Alias of int.le_self_sq.

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theorem int.pow_right_injective {x : ℤ} (h : 1 < x.nat_abs) :
function.injective (pow x)
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def powers_hom (M : Type u) [monoid M] :
M ≃ (multiplicative ℕ →* M)

Monoid homomorphisms from multiplicative ℕ are defined by the image of multiplicative.of_add 1.

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def zpowers_hom (G : Type w) [group G] :
G ≃ (multiplicative ℤ →* G)

Monoid homomorphisms from multiplicative ℤ are defined by the image of multiplicative.of_add 1.

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def multiples_hom (A : Type y) [add_monoid A] :
A ≃ (ℕ →+ A)

Additive homomorphisms from ℕ are defined by the image of 1.

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def zmultiples_hom (A : Type y) [add_group A] :
A ≃ (ℤ →+ A)

Additive homomorphisms from ℤ are defined by the image of 1.

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@[simp]
theorem powers_hom_apply {M : Type u} [monoid M] (x : M) (n : multiplicative ℕ) :
⇑(⇑(powers_hom M) x) n = x ^ ⇑multiplicative.to_add n
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@[simp]
theorem powers_hom_symm_apply {M : Type u} [monoid M] (f : multiplicative ℕ →* M) :
⇑((powers_hom M).symm) f = ⇑f (⇑multiplicative.of_add 1)
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@[simp]
theorem zpowers_hom_apply {G : Type w} [group G] (x : G) (n : multiplicative ℤ) :
⇑(⇑(zpowers_hom G) x) n = x ^ ⇑multiplicative.to_add n
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@[simp]
theorem zpowers_hom_symm_apply {G : Type w} [group G] (f : multiplicative ℤ →* G) :
⇑((zpowers_hom G).symm) f = ⇑f (⇑multiplicative.of_add 1)
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@[simp]
theorem multiples_hom_apply {A : Type y} [add_monoid A] (x : A) (n : ℕ) :
⇑(⇑(multiples_hom A) x) n = n • x
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@[simp]
theorem multiples_hom_symm_apply {A : Type y} [add_monoid A] (f : ℕ →+ A) :
⇑((multiples_hom A).symm) f = ⇑f 1
source
@[simp]
theorem zmultiples_hom_apply {A : Type y} [add_group A] (x : A) (n : ℤ) :
⇑(⇑(zmultiples_hom A) x) n = n • x
source
@[simp]
theorem zmultiples_hom_symm_apply {A : Type y} [add_group A] (f : ℤ →+ A) :
⇑((zmultiples_hom A).symm) f = ⇑f 1
source
theorem monoid_hom.apply_mnat {M : Type u} [monoid M] (f : multiplicative ℕ →* M) (n : multiplicative ℕ) :
⇑f n = ⇑f (⇑multiplicative.of_add 1) ^ ⇑multiplicative.to_add n
source
@[ext]
theorem monoid_hom.ext_mnat {M : Type u} [monoid M] ⦃f g : multiplicative ℕ →* M⦄ (h : ⇑f (⇑multiplicative.of_add 1) = ⇑g (⇑multiplicative.of_add 1)) :
f = g
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theorem monoid_hom.apply_mint {M : Type u} [group M] (f : multiplicative ℤ →* M) (n : multiplicative ℤ) :
⇑f n = ⇑f (⇑multiplicative.of_add 1) ^ ⇑multiplicative.to_add n

monoid_hom.ext_mint is defined in data.int.cast

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theorem add_monoid_hom.apply_nat {M : Type u} [add_monoid M] (f : ℕ →+ M) (n : ℕ) :
⇑f n = n • ⇑f 1

add_monoid_hom.ext_nat is defined in data.nat.cast

source
theorem add_monoid_hom.apply_int {M : Type u} [add_group M] (f : ℤ →+ M) (n : ℤ) :
⇑f n = n • ⇑f 1

add_monoid_hom.ext_int is defined in data.int.cast

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def powers_mul_hom (M : Type u) [comm_monoid M] :
M ≃* (multiplicative ℕ →* M)

If M is commutative, powers_hom is a multiplicative equivalence.

Equations
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def zpowers_mul_hom (G : Type w) [comm_group G] :
G ≃* (multiplicative ℤ →* G)

If M is commutative, zpowers_hom is a multiplicative equivalence.

Equations
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def multiples_add_hom (A : Type y) [add_comm_monoid A] :
A ≃+ (ℕ →+ A)

If M is commutative, multiples_hom is an additive equivalence.

Equations
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def zmultiples_add_hom (A : Type y) [add_comm_group A] :
A ≃+ (ℤ →+ A)

If M is commutative, zmultiples_hom is an additive equivalence.

Equations
source
@[simp]
theorem powers_mul_hom_apply {M : Type u} [comm_monoid M] (x : M) (n : multiplicative ℕ) :
⇑(⇑(powers_mul_hom M) x) n = x ^ ⇑multiplicative.to_add n
source
@[simp]
theorem powers_mul_hom_symm_apply {M : Type u} [comm_monoid M] (f : multiplicative ℕ →* M) :
⇑((powers_mul_hom M).symm) f = ⇑f (⇑multiplicative.of_add 1)
source
@[simp]
theorem zpowers_mul_hom_apply {G : Type w} [comm_group G] (x : G) (n : multiplicative ℤ) :
⇑(⇑(zpowers_mul_hom G) x) n = x ^ ⇑multiplicative.to_add n
source
@[simp]
theorem zpowers_mul_hom_symm_apply {G : Type w} [comm_group G] (f : multiplicative ℤ →* G) :
⇑((zpowers_mul_hom G).symm) f = ⇑f (⇑multiplicative.of_add 1)
source
@[simp]
theorem multiples_add_hom_apply {A : Type y} [add_comm_monoid A] (x : A) (n : ℕ) :
⇑(⇑(multiples_add_hom A) x) n = n • x
source
@[simp]
theorem multiples_add_hom_symm_apply {A : Type y} [add_comm_monoid A] (f : ℕ →+ A) :
⇑((multiples_add_hom A).symm) f = ⇑f 1
source
@[simp]
theorem zmultiples_add_hom_apply {A : Type y} [add_comm_group A] (x : A) (n : ℤ) :
⇑(⇑(zmultiples_add_hom A) x) n = n • x
source
@[simp]
theorem zmultiples_add_hom_symm_apply {A : Type y} [add_comm_group A] (f : ℤ →+ A) :
⇑((zmultiples_add_hom A).symm) f = ⇑f 1

Commutativity (again) #

Facts about semiconj_by and commute that require zpow or zsmul, or the fact that integer multiplication equals semiring multiplication.

source
@[simp]
theorem semiconj_by.cast_nat_mul_right {R : Type u₁} [semiring R] {a x y : R} (h : semiconj_by a x y) (n : ℕ) :
semiconj_by a ((↑n) * x) ((↑n) * y)
source
@[simp]
theorem semiconj_by.cast_nat_mul_left {R : Type u₁} [semiring R] {a x y : R} (h : semiconj_by a x y) (n : ℕ) :
semiconj_by ((↑n) * a) x y
source
@[simp]
theorem semiconj_by.cast_nat_mul_cast_nat_mul {R : Type u₁} [semiring R] {a x y : R} (h : semiconj_by a x y) (m n : ℕ) :
semiconj_by ((↑m) * a) ((↑n) * x) ((↑n) * y)
source
@[simp]
theorem add_semiconj_by.units_zsmul_right {M : Type u} [add_monoid M] {a : M} {x y : add_units M} (h : add_semiconj_by a ↑x ↑y) (m : ℤ) :
add_semiconj_by a ↑(m • x) ↑(m • y)
source
@[simp]
theorem semiconj_by.units_zpow_right {M : Type u} [monoid M] {a : M} {x y : units M} (h : semiconj_by a ↑x ↑y) (m : ℤ) :
semiconj_by a ↑(x ^ m) ↑(y ^ m)
source
@[simp]
theorem semiconj_by.cast_int_mul_right {R : Type u₁} [ring R] {a x y : R} (h : semiconj_by a x y) (m : ℤ) :
semiconj_by a ((↑m) * x) ((↑m) * y)
source
@[simp]
theorem semiconj_by.cast_int_mul_left {R : Type u₁} [ring R] {a x y : R} (h : semiconj_by a x y) (m : ℤ) :
semiconj_by ((↑m) * a) x y
source
@[simp]
theorem semiconj_by.cast_int_mul_cast_int_mul {R : Type u₁} [ring R] {a x y : R} (h : semiconj_by a x y) (m n : ℤ) :
semiconj_by ((↑m) * a) ((↑n) * x) ((↑n) * y)
source
@[simp]
theorem commute.cast_nat_mul_right {R : Type u₁} [semiring R] {a b : R} (h : commute a b) (n : ℕ) :
commute a ((↑n) * b)
source
@[simp]
theorem commute.cast_nat_mul_left {R : Type u₁} [semiring R] {a b : R} (h : commute a b) (n : ℕ) :
commute ((↑n) * a) b
source
@[simp]
theorem commute.cast_nat_mul_cast_nat_mul {R : Type u₁} [semiring R] {a b : R} (h : commute a b) (m n : ℕ) :
commute ((↑m) * a) ((↑n) * b)
source
@[simp]
theorem commute.self_cast_nat_mul {R : Type u₁} [semiring R] {a : R} (n : ℕ) :
commute a ((↑n) * a)
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@[simp]
theorem commute.cast_nat_mul_self {R : Type u₁} [semiring R] {a : R} (n : ℕ) :
commute ((↑n) * a) a
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@[simp]
theorem commute.self_cast_nat_mul_cast_nat_mul {R : Type u₁} [semiring R] {a : R} (m n : ℕ) :
commute ((↑m) * a) ((↑n) * a)
source
@[simp]
theorem add_commute.units_zsmul_right {M : Type u} [add_monoid M] {a : M} {u : add_units M} (h : add_commute a ↑u) (m : ℤ) :
add_commute a ↑(m • u)
source
@[simp]
theorem commute.units_zpow_right {M : Type u} [monoid M] {a : M} {u : units M} (h : commute a ↑u) (m : ℤ) :
commute a ↑(u ^ m)
source
@[simp]
theorem commute.units_zpow_left {M : Type u} [monoid M] {u : units M} {a : M} (h : commute ↑u a) (m : ℤ) :
commute ↑(u ^ m) a
source
@[simp]
theorem add_commute.units_zsmul_left {M : Type u} [add_monoid M] {u : add_units M} {a : M} (h : add_commute ↑u a) (m : ℤ) :
add_commute ↑(m • u) a
source
@[simp]
theorem commute.cast_int_mul_right {R : Type u₁} [ring R] {a b : R} (h : commute a b) (m : ℤ) :
commute a ((↑m) * b)
source
@[simp]
theorem commute.cast_int_mul_left {R : Type u₁} [ring R] {a b : R} (h : commute a b) (m : ℤ) :
commute ((↑m) * a) b
source
theorem commute.cast_int_mul_cast_int_mul {R : Type u₁} [ring R] {a b : R} (h : commute a b) (m n : ℤ) :
commute ((↑m) * a) ((↑n) * b)
source
@[simp]
theorem commute.cast_int_left {R : Type u₁} [ring R] (a : R) (m : ℤ) :
commute ↑m a
source
@[simp]
theorem commute.cast_int_right {R : Type u₁} [ring R] (a : R) (m : ℤ) :
commute a ↑m
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@[simp]
theorem commute.self_cast_int_mul {R : Type u₁} [ring R] (a : R) (n : ℤ) :
commute a ((↑n) * a)
source
@[simp]
theorem commute.cast_int_mul_self {R : Type u₁} [ring R] (a : R) (n : ℤ) :
commute ((↑n) * a) a
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theorem commute.self_cast_int_mul_cast_int_mul {R : Type u₁} [ring R] (a : R) (m n : ℤ) :
commute ((↑m) * a) ((↑n) * a)
source
@[simp]
theorem nat.to_add_pow (a : multiplicative ℕ) (b : ℕ) :
⇑multiplicative.to_add (a ^ b) = (⇑multiplicative.to_add a) * b
source
@[simp]
theorem nat.of_add_mul (a b : ℕ) :
⇑multiplicative.of_add (a * b) = ⇑multiplicative.of_add a ^ b
source
@[simp]
theorem int.to_add_pow (a : multiplicative ℤ) (b : ℕ) :
⇑multiplicative.to_add (a ^ b) = (⇑multiplicative.to_add a) * ↑b
source
@[simp]
theorem int.to_add_zpow (a : multiplicative ℤ) (b : ℤ) :
⇑multiplicative.to_add (a ^ b) = (⇑multiplicative.to_add a) * b
source
@[simp]
theorem int.of_add_mul (a b : ℤ) :
⇑multiplicative.of_add (a * b) = ⇑multiplicative.of_add a ^ b
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theorem units.conj_pow {M : Type u} [monoid M] (u : units M) (x : M) (n : ℕ) :
(((↑u) * x) * ↑u⁻¹) ^ n = ((↑u) * x ^ n) * ↑u⁻¹
source
theorem units.conj_pow' {M : Type u} [monoid M] (u : units M) (x : M) (n : ℕ) :
(((↑u⁻¹) * x) * ↑u) ^ n = ((↑u⁻¹) * x ^ n) * ↑u
source
@[simp]
theorem mul_opposite.op_pow {M : Type u} [monoid M] (x : M) (n : ℕ) :
mul_opposite.op (x ^ n) = mul_opposite.op x ^ n

Moving to the opposite monoid commutes with taking powers.

source
@[simp]
theorem mul_opposite.unop_pow {M : Type u} [monoid M] (x : Mᵐᵒᵖ) (n : ℕ) :
mul_opposite.unop (x ^ n) = mul_opposite.unop x ^ n
source
@[simp]
theorem mul_opposite.op_zpow {M : Type u} [div_inv_monoid M] (x : M) (z : ℤ) :
mul_opposite.op (x ^ z) = mul_opposite.op x ^ z

Moving to the opposite group or group_with_zero commutes with taking powers.

source
@[simp]
theorem mul_opposite.unop_zpow {M : Type u} [div_inv_monoid M] (x : Mᵐᵒᵖ) (z : ℤ) :
mul_opposite.unop (x ^ z) = mul_opposite.unop x ^ z