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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 : ) :
1 a ^ n
theorem Left.nsmul_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : 0 a) (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 : ) :
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 : ) :
a ^ n 1
theorem Left.nsmul_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] {a : M} (ha : a 0) (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 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 : } (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 : } (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 : } (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 : ) :
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 : ) :
0 n a
theorem pow_le_one' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} (ha : a 1) (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 a 0
theorem pow_lt_one' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {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 : } (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 : ) => 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 a
theorem pow_le_pow_right' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftMono M] {a : M} {n m : } (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 m : } (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 m : } (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 m : } (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 : } (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 : } (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 : ) => a ^ x
theorem nsmul_left_strictMono {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftStrictMono M] {a : M} (ha : 0 < a) :
StrictMono fun (x : ) => x a
theorem pow_lt_pow_right' {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] {a : M} {n m : } (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 m : } (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 : } :
1 x ^ n
theorem Right.nsmul_nonneg {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : 0 x) {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 : } :
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 : } :
x ^ n 1
theorem Right.nsmul_nonpos {M : Type u_3} [AddMonoid M] [Preorder M] [AddRightMono M] {x : M} (hx : x 0) {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 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 : } {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 : } {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 : } {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 : βM} (hf : StrictMono f) {n : } :
n 0StrictMono 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 : βM} (hf : StrictMono f) {n : } :
n 0StrictMono fun (x : β) => n f x
theorem pow_left_strictMono {M : Type u_3} [Monoid M] [Preorder M] [MulLeftStrictMono M] [MulRightStrictMono M] {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 : } (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 : } (hn : n 0) {a 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 : } (hn : n 0) {a 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 b : M} (hab : a b) (i : ) :
a ^ i b ^ i
theorem nsmul_le_nsmul_right {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] [AddRightMono M] {a b : M} (hab : a b) (i : ) :
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 : βM} (hf : Monotone f) (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 : βM} (hf : Monotone f) (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 : ) :
Monotone fun (a : M) => a ^ n
theorem nsmul_right_mono {M : Type u_3} [AddMonoid M] [Preorder M] [AddLeftMono M] [AddRightMono M] (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 : } (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 b : M} (n : ) :
a ^ n < b ^ na < b
theorem lt_of_nsmul_lt_nsmul_right {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftMono M] [AddRightMono M] {a b : M} (n : ) :
n a < n ba < b
theorem min_lt_of_mul_lt_sq {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftMono M] [MulRightMono M] {a b 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 b 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 b 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 b 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 b : M} {n : } (hn : n 0) :
a ^ n b ^ na b
theorem le_of_nsmul_le_nsmul_right {M : Type u_3} [AddMonoid M] [LinearOrder M] [AddLeftStrictMono M] [AddRightStrictMono M] {a b : M} {n : } (hn : n 0) :
n a n ba b
theorem min_le_of_mul_le_sq {M : Type u_3} [Monoid M] [LinearOrder M] [MulLeftStrictMono M] [MulRightStrictMono M] {a b 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 b 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 b 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 b 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 : } {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 : } {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 : } {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 : } {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 : } {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 : } (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 : } (hn : 0 n) :
0 n x