Lemmas about powers in ordered fields.

Integer powers

theorem zpow_le_of_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n

theorem zpow_le_one_of_nonpos {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : n ≤ 0) : a ^ n ≤ 1

theorem one_le_zpow_of_nonneg {α : Type u_1} [LinearOrderedSemifield α] {a : α} {n : ℤ} (ha : 1 ≤ a) (hn : 0 ≤ n) : 1 ≤ a ^ n

theorem Nat.zpow_pos_of_pos {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ) : 0 < ↑a ^ n

theorem Nat.zpow_ne_zero_of_pos {α : Type u_1} [LinearOrderedSemifield α] {a : ℕ} (h : 0 < a) (n : ℤ) : ↑a ^ n ≠ 0

theorem one_lt_zpow {α : Type u_1} [LinearOrderedSemifield α] {a : α} (ha : 1 < a) (n : ℤ) : 0 < n → 1 < a ^ n

theorem zpow_strictMono {α : Type u_1} [LinearOrderedSemifield α] {a : α} (hx : 1 < a) : StrictMono ((fun x x_1 => x ^ x_1) a)

theorem zpow_strictAnti {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a < 1) : StrictAnti ((fun x x_1 => x ^ x_1) a)

@[simp]

theorem zpow_lt_iff_lt {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) : a ^ m < a ^ n ↔ m < n

@[simp]

theorem zpow_le_iff_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (hx : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n

@[simp]

theorem div_pow_le {α : Type u_1} [LinearOrderedSemifield α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ) : a / b ^ k ≤ a

theorem zpow_injective {α : Type u_1} [LinearOrderedSemifield α] {a : α} (h₀ : 0 < a) (h₁ : a ≠ 1) : Function.Injective ((fun x x_1 => x ^ x_1) a)

@[simp]

theorem zpow_inj {α : Type u_1} [LinearOrderedSemifield α] {a : α} {m : ℤ} {n : ℤ} (h₀ : 0 < a) (h₁ : a ≠ 1) : a ^ m = a ^ n ↔ m = n

theorem zpow_le_max_of_min_le {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 ≤ x) {a : ℤ} {b : ℤ} {c : ℤ} (h : min a b ≤ c) : x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b))

theorem zpow_le_max_iff_min_le {α : Type u_1} [LinearOrderedSemifield α] {x : α} (hx : 1 < x) {a : ℤ} {b : ℤ} {c : ℤ} : x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c

Lemmas about powers to numerals.

theorem zpow_bit0_nonneg {α : Type u_1} [LinearOrderedField α] (a : α) (n : ℤ) : 0 ≤ a ^ bit0 n

theorem zpow_two_nonneg {α : Type u_1} [LinearOrderedField α] (a : α) : 0 ≤ a ^ 2

theorem zpow_neg_two_nonneg {α : Type u_1} [LinearOrderedField α] (a : α) : 0 ≤ a ^ (-2)

theorem zpow_bit0_pos {α : Type u_1} [LinearOrderedField α] {a : α} (h : a ≠ 0) (n : ℤ) : 0 < a ^ bit0 n

theorem zpow_two_pos_of_ne_zero {α : Type u_1} [LinearOrderedField α] {a : α} (h : a ≠ 0) : 0 < a ^ 2

@[simp]

theorem zpow_bit0_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : n ≠ 0) : 0 < a ^ bit0 n ↔ a ≠ 0

@[simp]

theorem zpow_bit1_neg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} : a ^ bit1 n < 0 ↔ a < 0

@[simp]

theorem zpow_bit1_nonneg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} : 0 ≤ a ^ bit1 n ↔ 0 ≤ a

@[simp]

theorem zpow_bit1_nonpos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} : a ^ bit1 n ≤ 0 ↔ a ≤ 0

@[simp]

theorem zpow_bit1_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} : 0 < a ^ bit1 n ↔ 0 < a

theorem Even.zpow_nonneg {α : Type u_1} [LinearOrderedField α] {n : ℤ} (hn : Even n) (a : α) : 0 ≤ a ^ n

theorem Even.zpow_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (h : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0

theorem Odd.zpow_neg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : a ^ n < 0 ↔ a < 0

theorem Odd.zpow_nonneg_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : 0 ≤ a ^ n ↔ 0 ≤ a

theorem Odd.zpow_nonpos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : a ^ n ≤ 0 ↔ a ≤ 0

theorem Odd.zpow_pos_iff {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : 0 < a ^ n ↔ 0 < a

theorem Even.zpow_pos {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Even n) (h : n ≠ 0) : a ≠ 0 → 0 < a ^ n

Alias of the reverse direction of Even.zpow_pos_iff.

theorem Odd.zpow_neg {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : a < 0 → a ^ n < 0

Alias of the reverse direction of Odd.zpow_neg_iff.

theorem Odd.zpow_nonpos {α : Type u_1} [LinearOrderedField α] {a : α} {n : ℤ} (hn : Odd n) : a ≤ 0 → a ^ n ≤ 0

Alias of the reverse direction of Odd.zpow_nonpos_iff.

theorem Even.zpow_abs {α : Type u_1} [LinearOrderedField α] {p : ℤ} (hp : Even p) (a : α) : |a| ^ p = a ^ p

@[simp]

theorem zpow_bit0_abs {α : Type u_1} [LinearOrderedField α] (a : α) (p : ℤ) : |a| ^ bit0 p = a ^ bit0 p

Miscellaneous lemmmas

theorem Nat.cast_le_pow_sub_div_sub {α : Type u_1} [LinearOrderedField α] {a : α} (H : 1 < a) (n : ℕ) : ↑n ≤ (a ^ n - 1) / (a - 1)

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

theorem Nat.cast_le_pow_div_sub {α : Type u_1} [LinearOrderedField α] {a : α} (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.