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Mathlib.Algebra.Order.Monoid.Unbundled.Pow

Lemmas about the interaction of power operations with order in terms of `CovariantClass` #

theorem Left.one_le_pow_of_le {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : 1 ≤ a) (n : ℕ) :

1 ≤ a ^ n

theorem Left.nsmul_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 ≤ a) (n : ℕ) :

0 ≤ n • a

@[deprecated Left.nsmul_nonneg]

theorem Left.pow_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 ≤ a) (n : ℕ) :

0 ≤ n • a

Alias of Left.nsmul_nonneg.

theorem Left.pow_le_one_of_le {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : a ≤ 1) (n : ℕ) :

a ^ n ≤ 1

theorem Left.nsmul_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : a ≤ 0) (n : ℕ) :

n • a ≤ 0

@[deprecated Left.nsmul_nonpos]

theorem Left.pow_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : a ≤ 0) (n : ℕ) :

n • a ≤ 0

Alias of Left.nsmul_nonpos.

theorem Left.pow_lt_one_of_lt {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {n : ℕ} (h : a < 1) (hn : n ≠ 0) :

a ^ n < 1

theorem Left.nsmul_neg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} {n : ℕ} (h : a < 0) (hn : n ≠ 0) :

n • a < 0

@[deprecated Left.nsmul_neg]

theorem Left.pow_neg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} {n : ℕ} (h : a < 0) (hn : n ≠ 0) :

n • a < 0

Alias of Left.nsmul_neg.

theorem one_le_pow_of_one_le' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : 1 ≤ a) (n : ℕ) :

1 ≤ a ^ n

Alias of Left.one_le_pow_of_le.

theorem nsmul_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 ≤ a) (n : ℕ) :

0 ≤ n • a

theorem pow_le_one' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : a ≤ 1) (n : ℕ) :

a ^ n ≤ 1

Alias of Left.pow_le_one_of_le.

theorem nsmul_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : a ≤ 0) (n : ℕ) :

n • a ≤ 0

theorem pow_lt_one' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {n : ℕ} (h : a < 1) (hn : n ≠ 0) :

a ^ n < 1

Alias of Left.pow_lt_one_of_lt.

theorem nsmul_neg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} {n : ℕ} (h : a < 0) (hn : n ≠ 0) :

n • a < 0

theorem pow_right_monotone {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : 1 ≤ a) :

Monotone fun (n : ℕ) => a ^ n

theorem nsmul_left_monotone {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 ≤ a) :

Monotone fun (n : ℕ) => n • a

theorem pow_le_pow_right' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {n : ℕ} {m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) :

a ^ n ≤ a ^ m

theorem nsmul_le_nsmul_left {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} {n : ℕ} {m : ℕ} (ha : 0 ≤ a) (h : n ≤ m) :

n • a ≤ m • a

theorem pow_le_pow_right_of_le_one' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {n : ℕ} {m : ℕ} (ha : a ≤ 1) (h : n ≤ m) :

a ^ m ≤ a ^ n

theorem nsmul_le_nsmul_left_of_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} {n : ℕ} {m : ℕ} (ha : a ≤ 0) (h : n ≤ m) :

m • a ≤ n • a

theorem one_lt_pow' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) :

1 < a ^ k

theorem nsmul_pos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 < a) {k : ℕ} (hk : k ≠ 0) :

0 < k • a

theorem pow_right_strictMono' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] {a : M} (ha : 1 < a) :

StrictMono fun (x : ℕ) => a ^ x

theorem nsmul_left_strictMono {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftStrictMono M] {a : M} (ha : 0 < a) :

StrictMono fun (x : ℕ) => x • a

theorem pow_lt_pow_right' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] {a : M} {n : ℕ} {m : ℕ} (ha : 1 < a) (h : n < m) :

a ^ n < a ^ m

theorem nsmul_lt_nsmul_left {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftStrictMono M] {a : M} {n : ℕ} {m : ℕ} (ha : 0 < a) (h : n < m) :

n • a < m • a

theorem Right.one_le_pow_of_le {M : Type u_3} [Monoid M] [Preorder M] [MulRightMono M] {x : M} (hx : 1 ≤ x) {n : ℕ} :

1 ≤ x ^ n

theorem Right.nsmul_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : 0 ≤ x) {n : ℕ} :

0 ≤ n • x

@[deprecated Right.nsmul_nonneg]

theorem Right.pow_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : 0 ≤ x) {n : ℕ} :

0 ≤ n • x

Alias of Right.nsmul_nonneg.

theorem Right.pow_le_one_of_le {M : Type u_3} [Monoid M] [Preorder M] [MulRightMono M] {x : M} (hx : x ≤ 1) {n : ℕ} :

x ^ n ≤ 1

theorem Right.nsmul_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : x ≤ 0) {n : ℕ} :

n • x ≤ 0

@[deprecated Right.nsmul_nonpos]

theorem Right.pow_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : x ≤ 0) {n : ℕ} :

n • x ≤ 0

Alias of Right.nsmul_nonpos.

theorem Right.pow_lt_one_of_lt {M : Type u_3} [Monoid M] [Preorder M] [MulRightMono M] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 1) :

x ^ n < 1

theorem Right.nsmul_neg {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 0) :

n • x < 0

@[deprecated Right.nsmul_neg]

theorem Right.pow_neg {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {n : ℕ} {x : M} (hn : 0 < n) (h : x < 0) :

n • x < 0

Alias of Right.nsmul_neg.

theorem StrictMono.pow_const {β : Type u_1} {M : Type u_3} [Monoid M] [Preorder M] [Preorder β] [MulLeftStrictMono M] [MulRightStrictMono M] {f : β → M} (hf : StrictMono f) {n : ℕ} :

n ≠ 0 → StrictMono fun (x : β) => f x ^ n

theorem StrictMono.const_nsmul {β : Type u_1} {M : Type u_3} [AddMonoid M] [Preorder M] [Preorder β] [AddLeftStrictMono M] [AddRightStrictMono M] {f : β → M} (hf : StrictMono f) {n : ℕ} :

n ≠ 0 → StrictMono fun (x : β) => n • f x

theorem pow_left_strictMono {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] [MulRightStrictMono M] {n : ℕ} (hn : n ≠ 0) :

StrictMono fun (x : M) => x ^ n

See also pow_left_strictMonoOn.

theorem nsmul_right_strictMono {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftStrictMono M] [AddRightStrictMono M] {n : ℕ} (hn : n ≠ 0) :

StrictMono fun (x : M) => n • x

theorem pow_lt_pow_left' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] [MulRightStrictMono M] {n : ℕ} (hn : n ≠ 0) {a : M} {b : M} (hab : a < b) :

a ^ n < b ^ n

theorem nsmul_lt_nsmul_right {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftStrictMono M] [AddRightStrictMono M] {n : ℕ} (hn : n ≠ 0) {a : M} {b : M} (hab : a < b) :

n • a < n • b

theorem pow_le_pow_left' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] [MulRightMono M] {a : M} {b : M} (hab : a ≤ b) (i : ℕ) :

a ^ i ≤ b ^ i

theorem nsmul_le_nsmul_right {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] [AddRightMono M] {a : M} {b : M} (hab : a ≤ b) (i : ℕ) :

i • a ≤ i • b

theorem Monotone.pow_const {β : Type u_1} {M : Type u_3} [Monoid M] [Preorder M] [Preorder β] [MulLeftMono M] [MulRightMono M] {f : β → M} (hf : Monotone f) (n : ℕ) :

Monotone fun (a : β) => f a ^ n

theorem Monotone.const_nsmul {β : Type u_1} {M : Type u_3} [AddMonoid M] [Preorder M] [Preorder β] [AddLeftMono M] [AddRightMono M] {f : β → M} (hf : Monotone f) (n : ℕ) :

Monotone fun (a : β) => n • f a

theorem pow_left_mono {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] [MulRightMono M] (n : ℕ) :

Monotone fun (a : M) => a ^ n

theorem nsmul_right_mono {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] [AddRightMono M] (n : ℕ) :

Monotone fun (a : M) => n • a

theorem one_le_pow_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

1 ≤ x ^ n ↔ 1 ≤ x

theorem nsmul_nonneg_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

0 ≤ n • x ↔ 0 ≤ x

theorem pow_le_one_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

x ^ n ≤ 1 ↔ x ≤ 1

theorem nsmul_nonpos_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

n • x ≤ 0 ↔ x ≤ 0

theorem one_lt_pow_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

1 < x ^ n ↔ 1 < x

theorem nsmul_pos_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

0 < n • x ↔ 0 < x

theorem pow_lt_one_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

x ^ n < 1 ↔ x < 1

theorem nsmul_neg_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

n • x < 0 ↔ x < 0

theorem pow_eq_one_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

x ^ n = 1 ↔ x = 1

theorem nsmul_eq_zero_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ} (hn : n ≠ 0) :

n • x = 0 ↔ x = 0

theorem pow_le_pow_iff_right' {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] {a : M} {m : ℕ} {n : ℕ} (ha : 1 < a) :

a ^ m ≤ a ^ n ↔ m ≤ n

theorem nsmul_le_nsmul_iff_left {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] {a : M} {m : ℕ} {n : ℕ} (ha : 0 < a) :

m • a ≤ n • a ↔ m ≤ n

theorem pow_lt_pow_iff_right' {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] {a : M} {m : ℕ} {n : ℕ} (ha : 1 < a) :

a ^ m < a ^ n ↔ m < n

theorem nsmul_lt_nsmul_iff_left {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] {a : M} {m : ℕ} {n : ℕ} (ha : 0 < a) :

m • a < n • a ↔ m < n

theorem lt_of_pow_lt_pow_left' {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] [MulRightMono M] {a : M} {b : M} (n : ℕ) :

a ^ n < b ^ n → a < b

theorem lt_of_nsmul_lt_nsmul_right {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] [AddRightMono M] {a : M} {b : M} (n : ℕ) :

n • a < n • b → a < b

theorem min_lt_of_mul_lt_sq {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] [MulRightMono M] {a : M} {b : M} {c : M} (h : a * b < c ^ 2) :

a ⊓ b < c

theorem min_lt_of_add_lt_two_nsmul {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] [AddRightMono M] {a : M} {b : M} {c : M} (h : a + b < 2 • c) :

a ⊓ b < c

theorem lt_max_of_sq_lt_mul {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] [MulRightMono M] {a : M} {b : M} {c : M} (h : a ^ 2 < b * c) :

a < b ⊔ c

theorem lt_max_of_two_nsmul_lt_add {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] [AddRightMono M] {a : M} {b : M} {c : M} (h : 2 • a < b + c) :

a < b ⊔ c

theorem le_of_pow_le_pow_left' {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] [MulRightStrictMono M] {a : M} {b : M} {n : ℕ} (hn : n ≠ 0) :

a ^ n ≤ b ^ n → a ≤ b

theorem le_of_nsmul_le_nsmul_right {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] [AddRightStrictMono M] {a : M} {b : M} {n : ℕ} (hn : n ≠ 0) :

n • a ≤ n • b → a ≤ b

theorem min_le_of_mul_le_sq {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] [MulRightStrictMono M] {a : M} {b : M} {c : M} (h : a * b ≤ c ^ 2) :

a ⊓ b ≤ c

theorem min_le_of_add_le_two_nsmul {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] [AddRightStrictMono M] {a : M} {b : M} {c : M} (h : a + b ≤ 2 • c) :

a ⊓ b ≤ c

theorem le_max_of_sq_le_mul {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] [MulRightStrictMono M] {a : M} {b : M} {c : M} (h : a ^ 2 ≤ b * c) :

a ≤ b ⊔ c

theorem le_max_of_two_nsmul_le_add {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] [AddRightStrictMono M] {a : M} {b : M} {c : M} (h : 2 • a ≤ b + c) :

a ≤ b ⊔ c

theorem Left.pow_lt_one_iff' {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] {n : ℕ} {x : M} (hn : 0 < n) :

x ^ n < 1 ↔ x < 1

theorem Left.nsmul_neg_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] {n : ℕ} {x : M} (hn : 0 < n) :

n • x < 0 ↔ x < 0

theorem Left.pow_lt_one_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] {n : ℕ} {x : M} (hn : 0 < n) :

x ^ n < 1 ↔ x < 1

theorem Right.pow_lt_one_iff {M : Type u_3} [Monoid M] [LinearOrder M] [MulRightStrictMono M] {n : ℕ} {x : M} (hn : 0 < n) :

x ^ n < 1 ↔ x < 1

theorem Right.nsmul_neg_iff {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddRightStrictMono M] {n : ℕ} {x : M} (hn : 0 < n) :

n • x < 0 ↔ x < 0

theorem one_le_zpow {G : Type u_2} [DivInvMonoid G] [Preorder G] [MulLeftMono G] {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) :

1 ≤ x ^ n

theorem zsmul_nonneg {G : Type u_2} [SubNegMonoid G] [Preorder G] [AddLeftMono G] {x : G} (H : 0 ≤ x) {n : ℤ} (hn : 0 ≤ n) :

0 ≤ n • x