# Basic lemmas about ordered rings #

theorem MonoidHom.map_neg_one {M : Type u_2} {R : Type u_3} [Ring R] [] [] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x x_1] (f : R →* M) :
f (-1) = 1
@[simp]
theorem MonoidHom.map_neg {M : Type u_2} {R : Type u_3} [Ring R] [] [] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x x_1] (f : R →* M) (x : R) :
f (-x) = f x
theorem MonoidHom.map_sub_swap {M : Type u_2} {R : Type u_3} [Ring R] [] [] [CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x x_1] (f : R →* M) (x : R) (y : R) :
f (x - y) = f (y - x)
theorem zero_pow_le_one {R : Type u_3} [] (n : ) :
0 ^ n 1
theorem pow_add_pow_le {R : Type u_3} [] {x : R} {y : R} {n : } (hx : 0 x) (hy : 0 y) (hn : n 0) :
x ^ n + y ^ n (x + y) ^ n
theorem pow_le_one {R : Type u_3} [] {a : R} (n : ) :
0 aa 1a ^ n 1
theorem pow_lt_one {R : Type u_3} [] {a : R} (h₀ : 0 a) (h₁ : a < 1) {n : } :
n 0a ^ n < 1
theorem one_le_pow_of_one_le {R : Type u_3} [] {a : R} (H : 1 a) (n : ) :
1 a ^ n
theorem pow_right_mono {R : Type u_3} [] {a : R} (h : 1 a) :
Monotone fun (x : ) => a ^ x
theorem pow_le_pow_right {R : Type u_3} [] {a : R} {n : } {m : } (ha : 1 a) (h : n m) :
a ^ n a ^ m
theorem le_self_pow {R : Type u_3} [] {a : R} {m : } (ha : 1 a) (h : m 0) :
a a ^ m
theorem pow_le_pow_left {R : Type u_3} [] {a : R} {b : R} (ha : 0 a) (hab : a b) (n : ) :
a ^ n b ^ n
theorem one_lt_pow {R : Type u_3} [] {a : R} (ha : 1 < a) {n : } :
n 01 < a ^ n
theorem pow_add_pow_le' {R : Type u_3} [] {a : R} {b : R} {n : } (ha : 0 a) (hb : 0 b) :
a ^ n + b ^ n 2 * (a + b) ^ n
theorem pow_lt_pow_left {R : Type u_3} {x : R} {y : R} (h : x < y) (hx : 0 x) {n : } :
n 0x ^ n < y ^ n
theorem pow_left_strictMonoOn {R : Type u_3} {n : } (hn : n 0) :
StrictMonoOn (fun (x : R) => x ^ n) {a : R | 0 a}

See also pow_left_strictMono and Nat.pow_left_strictMono.

theorem pow_right_strictMono {R : Type u_3} {a : R} (h : 1 < a) :
StrictMono fun (x : ) => a ^ x

See also pow_right_strictMono'.

theorem pow_lt_pow_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) (hmn : m < n) :
a ^ m < a ^ n
theorem pow_lt_pow_iff_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n < a ^ m n < m
theorem pow_le_pow_iff_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n a ^ m n m
theorem lt_self_pow {R : Type u_3} {a : R} {m : } (h : 1 < a) (hm : 1 < m) :
a < a ^ m
theorem pow_right_strictAnti {R : Type u_3} {a : R} (h₀ : 0 < a) (h₁ : a < 1) :
StrictAnti fun (x : ) => a ^ x
theorem pow_lt_pow_iff_right_of_lt_one {R : Type u_3} {a : R} {n : } {m : } (h₀ : 0 < a) (h₁ : a < 1) :
a ^ m < a ^ n n < m
theorem pow_lt_pow_right_of_lt_one {R : Type u_3} {a : R} {n : } {m : } (h₀ : 0 < a) (h₁ : a < 1) (hmn : m < n) :
a ^ n < a ^ m
theorem pow_lt_self_of_lt_one {R : Type u_3} {a : R} {n : } (h₀ : 0 < a) (h₁ : a < 1) (hn : 1 < n) :
a ^ n < a
theorem sq_pos_of_pos {R : Type u_3} {a : R} (ha : 0 < a) :
0 < a ^ 2
theorem sq_pos_of_neg {R : Type u_3} {a : R} (ha : a < 0) :
0 < a ^ 2
theorem pow_le_pow_iff_left {R : Type u_3} {a : R} {b : R} {n : } (ha : 0 a) (hb : 0 b) (hn : n 0) :
a ^ n b ^ n a b
theorem pow_lt_pow_iff_left {R : Type u_3} {a : R} {b : R} {n : } (ha : 0 a) (hb : 0 b) (hn : n 0) :
a ^ n < b ^ n a < b
@[simp]
theorem pow_left_inj {R : Type u_3} {a : R} {b : R} {n : } (ha : 0 a) (hb : 0 b) (hn : n 0) :
a ^ n = b ^ n a = b
theorem pow_right_injective {R : Type u_3} {a : R} (ha₀ : 0 < a) (ha₁ : a 1) :
Function.Injective fun (x : ) => a ^ x
@[simp]
theorem pow_right_inj {R : Type u_3} {a : R} {m : } {n : } (ha₀ : 0 < a) (ha₁ : a 1) :
a ^ m = a ^ n m = n
theorem pow_le_one_iff_of_nonneg {R : Type u_3} {a : R} {n : } (ha : 0 a) (hn : n 0) :
a ^ n 1 a 1
theorem one_le_pow_iff_of_nonneg {R : Type u_3} {a : R} {n : } (ha : 0 a) (hn : n 0) :
1 a ^ n 1 a
theorem pow_lt_one_iff_of_nonneg {R : Type u_3} {a : R} {n : } (ha : 0 a) (hn : n 0) :
a ^ n < 1 a < 1
theorem one_lt_pow_iff_of_nonneg {R : Type u_3} {a : R} {n : } (ha : 0 a) (hn : n 0) :
1 < a ^ n 1 < a
theorem pow_eq_one_iff_of_nonneg {R : Type u_3} {a : R} {n : } (ha : 0 a) (hn : n 0) :
a ^ n = 1 a = 1
theorem sq_le_one_iff {R : Type u_3} {a : R} (ha : 0 a) :
a ^ 2 1 a 1
theorem sq_lt_one_iff {R : Type u_3} {a : R} (ha : 0 a) :
a ^ 2 < 1 a < 1
theorem one_le_sq_iff {R : Type u_3} {a : R} (ha : 0 a) :
1 a ^ 2 1 a
theorem one_lt_sq_iff {R : Type u_3} {a : R} (ha : 0 a) :
1 < a ^ 2 1 < a
theorem lt_of_pow_lt_pow_left {R : Type u_3} {a : R} {b : R} (n : ) (hb : 0 b) (h : a ^ n < b ^ n) :
a < b
theorem le_of_pow_le_pow_left {R : Type u_3} {a : R} {b : R} {n : } (hn : n 0) (hb : 0 b) (h : a ^ n b ^ n) :
a b
@[simp]
theorem sq_eq_sq {R : Type u_3} {a : R} {b : R} (ha : 0 a) (hb : 0 b) :
a ^ 2 = b ^ 2 a = b
theorem lt_of_mul_self_lt_mul_self {R : Type u_3} {a : R} {b : R} (hb : 0 b) :
a * a < b * ba < b

### Lemmas for canonically linear ordered semirings or linear ordered rings #

The slightly unusual typeclass assumptions [LinearOrderedSemiring R] [ExistsAddOfLE R] cover two more familiar settings:

• [LinearOrderedRing R], eg ℤ, ℚ or ℝ
• [CanonicallyLinearOrderedSemiring R] (although we don't actually have this typeclass), eg ℕ, ℚ≥0 or ℝ≥0
theorem add_sq_le {R : Type u_3} {a : R} {b : R} [] :
(a + b) ^ 2 2 * (a ^ 2 + b ^ 2)
theorem add_pow_le {R : Type u_3} {a : R} {b : R} [] (ha : 0 a) (hb : 0 b) (n : ) :
(a + b) ^ n 2 ^ (n - 1) * (a ^ n + b ^ n)
theorem Even.add_pow_le {R : Type u_3} {a : R} {b : R} {n : } [] (hn : Even n) :
(a + b) ^ n 2 ^ (n - 1) * (a ^ n + b ^ n)
theorem Even.pow_nonneg {R : Type u_3} {n : } [] (hn : Even n) (a : R) :
0 a ^ n
theorem Even.pow_pos {R : Type u_3} {a : R} {n : } [] (hn : Even n) (ha : a 0) :
0 < a ^ n
theorem Even.pow_pos_iff {R : Type u_3} {a : R} {n : } [] (hn : Even n) (h₀ : n 0) :
0 < a ^ n a 0
theorem Odd.pow_neg_iff {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
a ^ n < 0 a < 0
theorem Odd.pow_nonneg_iff {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
0 a ^ n 0 a
theorem Odd.pow_nonpos_iff {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
a ^ n 0 a 0
theorem Odd.pow_pos_iff {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
0 < a ^ n 0 < a
theorem Odd.pow_nonpos {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
a 0a ^ n 0

Alias of the reverse direction of Odd.pow_nonpos_iff.

theorem Odd.pow_neg {R : Type u_3} {a : R} {n : } [] (hn : Odd n) :
a < 0a ^ n < 0

Alias of the reverse direction of Odd.pow_neg_iff.

theorem Odd.strictMono_pow {R : Type u_3} {n : } [] (hn : Odd n) :
StrictMono fun (a : R) => a ^ n
theorem sq_pos_iff {R : Type u_3} [] {a : R} :
0 < a ^ 2 a 0
theorem sq_pos_of_ne_zero {R : Type u_3} [] {a : R} :
a 00 < a ^ 2

Alias of the reverse direction of sq_pos_iff.

theorem pow_two_pos_of_ne_zero {R : Type u_3} [] {a : R} :
a 00 < a ^ 2

Alias of the reverse direction of sq_pos_iff.

Alias of the reverse direction of sq_pos_iff.

theorem pow_four_le_pow_two_of_pow_two_le {R : Type u_3} {a : R} {b : R} [] (h : a ^ 2 b) :
a ^ 4 b ^ 2
@[deprecated Even.pow_nonneg]
theorem pow_bit0_nonneg {R : Type u_3} [] (a : R) (n : ) :
0 a ^ bit0 n
@[deprecated Even.pow_pos]
theorem pow_bit0_pos {R : Type u_3} [] {a : R} (h : a 0) (n : ) :
0 < a ^ bit0 n
@[deprecated Even.pow_pos_iff]
theorem pow_bit0_pos_iff {R : Type u_3} [] (a : R) {n : } (hn : n 0) :
0 < a ^ bit0 n a 0
@[simp, deprecated Odd.pow_neg_iff]
theorem pow_bit1_neg_iff {R : Type u_3} {a : R} {n : } [] :
a ^ bit1 n < 0 a < 0
@[simp, deprecated Odd.pow_nonneg_iff]
theorem pow_bit1_nonneg_iff {R : Type u_3} {a : R} {n : } [] :
0 a ^ bit1 n 0 a
@[simp, deprecated Odd.pow_nonpos_iff]
theorem pow_bit1_nonpos_iff {R : Type u_3} {a : R} {n : } [] :
a ^ bit1 n 0 a 0
@[simp, deprecated Odd.pow_pos_iff]
theorem pow_bit1_pos_iff {R : Type u_3} {a : R} {n : } [] :
0 < a ^ bit1 n 0 < a
@[deprecated Odd.strictMono_pow]
theorem strictMono_pow_bit1 {R : Type u_3} [] (n : ) :
StrictMono fun (x : R) => x ^ bit1 n

### Deprecated lemmas #

Those lemmas have been deprecated on 2023-12-23.

@[deprecated pow_right_mono]
theorem pow_mono {R : Type u_3} [] {a : R} (h : 1 a) :
Monotone fun (x : ) => a ^ x

Alias of pow_right_mono.

@[deprecated pow_le_pow_right]
theorem pow_le_pow {R : Type u_3} [] {a : R} {n : } {m : } (ha : 1 a) (h : n m) :
a ^ n a ^ m

Alias of pow_le_pow_right.

@[deprecated pow_le_pow_left]
theorem pow_le_pow_of_le_left {R : Type u_3} [] {a : R} {b : R} (ha : 0 a) (hab : a b) (n : ) :
a ^ n b ^ n

Alias of pow_le_pow_left.

@[deprecated pow_lt_pow_left]
theorem pow_lt_pow_of_lt_left {R : Type u_3} {x : R} {y : R} (h : x < y) (hx : 0 x) {n : } :
n 0x ^ n < y ^ n

Alias of pow_lt_pow_left.

@[deprecated pow_left_strictMonoOn]
theorem strictMonoOn_pow {R : Type u_3} {n : } (hn : n 0) :
StrictMonoOn (fun (x : R) => x ^ n) {a : R | 0 a}

Alias of pow_left_strictMonoOn.

See also pow_left_strictMono and Nat.pow_left_strictMono.

@[deprecated pow_right_strictMono]
theorem pow_strictMono_right {R : Type u_3} {a : R} (h : 1 < a) :
StrictMono fun (x : ) => a ^ x

Alias of pow_right_strictMono.

See also pow_right_strictMono'.

@[deprecated pow_lt_pow_right]
theorem pow_lt_pow {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) (hmn : m < n) :
a ^ m < a ^ n

Alias of pow_lt_pow_right.

@[deprecated pow_lt_pow_iff_right]
theorem pow_lt_pow_iff {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n < a ^ m n < m

Alias of pow_lt_pow_iff_right.

@[deprecated pow_le_pow_iff_right]
theorem pow_le_pow_iff {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n a ^ m n m

Alias of pow_le_pow_iff_right.

@[deprecated lt_self_pow]
theorem self_lt_pow {R : Type u_3} {a : R} {m : } (h : 1 < a) (hm : 1 < m) :
a < a ^ m

Alias of lt_self_pow.

@[deprecated pow_right_strictAnti]
theorem strictAnti_pow {R : Type u_3} {a : R} (h₀ : 0 < a) (h₁ : a < 1) :
StrictAnti fun (x : ) => a ^ x

Alias of pow_right_strictAnti.

@[deprecated pow_lt_pow_iff_right_of_lt_one]
theorem pow_lt_pow_iff_of_lt_one {R : Type u_3} {a : R} {n : } {m : } (h₀ : 0 < a) (h₁ : a < 1) :
a ^ m < a ^ n n < m

Alias of pow_lt_pow_iff_right_of_lt_one.

@[deprecated pow_lt_pow_right_of_lt_one]
theorem pow_lt_pow_of_lt_one {R : Type u_3} {a : R} {n : } {m : } (h₀ : 0 < a) (h₁ : a < 1) (hmn : m < n) :
a ^ n < a ^ m

Alias of pow_lt_pow_right_of_lt_one.

@[deprecated lt_of_pow_lt_pow_left]
theorem lt_of_pow_lt_pow {R : Type u_3} {a : R} {b : R} (n : ) (hb : 0 b) (h : a ^ n < b ^ n) :
a < b

Alias of lt_of_pow_lt_pow_left.

@[deprecated le_of_pow_le_pow_left]
theorem le_of_pow_le_pow {R : Type u_3} {a : R} {b : R} {n : } (hn : n 0) (hb : 0 b) (h : a ^ n b ^ n) :
a b

Alias of le_of_pow_le_pow_left.

@[deprecated le_self_pow]
theorem self_le_pow {R : Type u_3} [] {a : R} {m : } (ha : 1 a) (h : m 0) :
a a ^ m

Alias of le_self_pow.

@[deprecated pow_lt_pow_right]
theorem Nat.pow_lt_pow_of_lt_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) (hmn : m < n) :
a ^ m < a ^ n

Alias of pow_lt_pow_right.

@[deprecated pow_right_strictMono]
theorem Nat.pow_right_strictMono {R : Type u_3} {a : R} (h : 1 < a) :
StrictMono fun (x : ) => a ^ x

Alias of pow_right_strictMono.

See also pow_right_strictMono'.

@[deprecated pow_le_pow_iff_right]
theorem Nat.pow_le_iff_le_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n a ^ m n m

Alias of pow_le_pow_iff_right.

@[deprecated pow_lt_pow_iff_right]
theorem Nat.pow_lt_iff_lt_right {R : Type u_3} {a : R} {n : } {m : } (h : 1 < a) :
a ^ n < a ^ m n < m

Alias of pow_lt_pow_iff_right.