mathlib documentation

algebra.field_power

Integer power operation on fields and division rings #

This file collects basic facts about the operation of raising an element of a division_ring to an integer power. More specialised results are provided in the case of a linearly ordered field.

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@[simp]
theorem zpow_bit1_neg {α : Type u_1} [division_ring α] (x : α) (n : ℤ) :
(-x) ^ bit1 n = -x ^ bit1 n
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@[simp, norm_cast]
theorem rat.cast_zpow {α : Type u_1} [division_ring α] [char_zero α] (q : ℚ) (n : ℤ) :
↑(q ^ n) = ↑q ^ n
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theorem zpow_nonneg {α : Type u_1} [linear_ordered_semifield α] {a : α} (ha : 0 ≤ a) (z : ℤ) :
0 ≤ a ^ z
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theorem zpow_pos_of_pos {α : Type u_1} [linear_ordered_semifield α] {a : α} (ha : 0 < a) (z : ℤ) :
0 < a ^ z
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theorem zpow_le_of_le {α : Type u_1} [linear_ordered_semifield α] {a : α} {m n : ℤ} (ha : 1 ≤ a) (h : m ≤ n) :
a ^ m ≤ a ^ n
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theorem pow_le_max_of_min_le {α : Type u_1} [linear_ordered_semifield α] {x : α} (hx : 1 ≤ x) {a b c : ℤ} (h : linear_order.min a b ≤ c) :
x ^ -c ≤ linear_order.max (x ^ -a) (x ^ -b)
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theorem zpow_le_one_of_nonpos {α : Type u_1} [linear_ordered_semifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : n ≤ 0) :
a ^ n ≤ 1
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theorem one_le_zpow_of_nonneg {α : Type u_1} [linear_ordered_semifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : 0 ≤ n) :
1 ≤ a ^ n
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@[protected]
theorem nat.zpow_pos_of_pos {α : Type u_1} [linear_ordered_semifield α] {a : ℕ} (h : 0 < a) (n : ℤ) :
0 < ↑a ^ n
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theorem nat.zpow_ne_zero_of_pos {α : Type u_1} [linear_ordered_semifield α] {a : ℕ} (h : 0 < a) (n : ℤ) :
↑a ^ n ≠ 0
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theorem one_lt_zpow {α : Type u_1} [linear_ordered_semifield α] {a : α} (ha : 1 < a) (n : ℤ) :
0 < n → 1 < a ^ n
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theorem zpow_strict_mono {α : Type u_1} [linear_ordered_semifield α] {x : α} (hx : 1 < x) :
strict_mono (has_pow.pow x)
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theorem zpow_strict_anti {α : Type u_1} [linear_ordered_semifield α] {x : α} (h₀ : 0 < x) (h₁ : x < 1) :
strict_anti (has_pow.pow x)
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@[simp]
theorem zpow_lt_iff_lt {α : Type u_1} [linear_ordered_semifield α] {x : α} {m n : ℤ} (hx : 1 < x) :
x ^ m < x ^ n ↔ m < n
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@[simp]
theorem zpow_le_iff_le {α : Type u_1} [linear_ordered_semifield α] {x : α} {m n : ℤ} (hx : 1 < x) :
x ^ m ≤ x ^ n ↔ m ≤ n
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theorem min_le_of_zpow_le_max {α : Type u_1} [linear_ordered_semifield α] {x : α} (hx : 1 < x) {a b c : ℤ} (h_max : x ^ -c ≤ linear_order.max (x ^ -a) (x ^ -b)) :
linear_order.min a b ≤ c
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@[simp]
theorem pos_div_pow_pos {α : Type u_1} [linear_ordered_semifield α] {a b : α} (ha : 0 < a) (hb : 0 < b) (k : ℕ) :
0 < a / b ^ k
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@[simp]
theorem div_pow_le {α : Type u_1} [linear_ordered_semifield α] {a b : α} (ha : 0 < a) (hb : 1 ≤ b) (k : ℕ) :
a / b ^ k ≤ a
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theorem zpow_injective {α : Type u_1} [linear_ordered_semifield α] {x : α} (h₀ : 0 < x) (h₁ : x ≠ 1) :
function.injective (has_pow.pow x)
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@[simp]
theorem zpow_inj {α : Type u_1} [linear_ordered_semifield α] {x : α} {m n : ℤ} (h₀ : 0 < x) (h₁ : x ≠ 1) :
x ^ m = x ^ n ↔ m = n
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theorem zpow_bit0_nonneg {α : Type u_1} [linear_ordered_field α] (a : α) (n : ℤ) :
0 ≤ a ^ bit0 n
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theorem zpow_two_nonneg {α : Type u_1} [linear_ordered_field α] (a : α) :
0 ≤ a ^ 2
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theorem zpow_bit0_pos {α : Type u_1} [linear_ordered_field α] {a : α} (h : a ≠ 0) (n : ℤ) :
0 < a ^ bit0 n
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theorem zpow_two_pos_of_ne_zero {α : Type u_1} [linear_ordered_field α] {a : α} (h : a ≠ 0) :
0 < a ^ 2
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@[simp]
theorem zpow_bit1_neg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : ℤ} :
a ^ bit1 n < 0 ↔ a < 0
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@[simp]
theorem zpow_bit1_nonneg_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : ℤ} :
0 ≤ a ^ bit1 n ↔ 0 ≤ a
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@[simp]
theorem zpow_bit1_nonpos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : ℤ} :
a ^ bit1 n ≤ 0 ↔ a ≤ 0
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@[simp]
theorem zpow_bit1_pos_iff {α : Type u_1} [linear_ordered_field α] {a : α} {n : ℤ} :
0 < a ^ bit1 n ↔ 0 < a