File: Mathlib/Algebra/Order/Group/Basic.lean

full_name: one_lt_zpow'
code: @[to_additive zsmul_pos] lemma one_lt_zpow' (ha : 1 < a) (hn : 0 < n) : 1 < a ^ n 
goal: 1 < a ^ n

full_name: zsmul_pos
code: @[to_additive zsmul_pos] lemma one_lt_zpow' (ha : 1 < a) (hn : 0 < n) : 1 < a ^ n 
goal: 0 < n • a

full_name: zsmul_strictMono_left
code: @[to_additive zsmul_strictMono_left]
lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n 
goal: StrictMono fun n => n • a

full_name: zpow_right_strictMono
code: @[to_additive zsmul_strictMono_left]
lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n 
goal: StrictMono fun n => a ^ n

full_name: zsmul_mono_left
code: @[to_additive zsmul_mono_left]
lemma zpow_mono_right (ha : 1 ≤ a) : Monotone fun n : ℤ ↦ a ^ n 
goal: Monotone fun n => n • a

full_name: zpow_mono_right
code: @[to_additive zsmul_mono_left]
lemma zpow_mono_right (ha : 1 ≤ a) : Monotone fun n : ℤ ↦ a ^ n 
goal: Monotone fun n => a ^ n

full_name: zsmul_le_zsmul
code: @[to_additive (attr := gcongr)]
lemma zpow_le_zpow (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n 
goal: m • a ≤ n • a

full_name: zpow_le_zpow
code: @[to_additive (attr := gcongr)]
lemma zpow_le_zpow (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n 
goal: a ^ m ≤ a ^ n

full_name: zpow_lt_zpow
code: @[to_additive (attr := gcongr)]
lemma zpow_lt_zpow (ha : 1 < a) (h : m < n) : a ^ m < a ^ n 
goal: a ^ m < a ^ n

full_name: zsmul_lt_zsmul
code: @[to_additive (attr := gcongr)]
lemma zpow_lt_zpow (ha : 1 < a) (h : m < n) : a ^ m < a ^ n 
goal: m • a < n • a

full_name: zsmul_le_zsmul_iff
code: @[to_additive]
lemma zpow_le_zpow_iff (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n 
goal: m • a ≤ n • a ↔ m ≤ n

full_name: zpow_le_zpow_iff
code: @[to_additive]
lemma zpow_le_zpow_iff (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n 
goal: a ^ m ≤ a ^ n ↔ m ≤ n

full_name: zsmul_lt_zsmul_iff
code: @[to_additive]
lemma zpow_lt_zpow_iff (ha : 1 < a) : a ^ m < a ^ n ↔ m < n 
goal: m • a < n • a ↔ m < n

full_name: zpow_lt_zpow_iff
code: @[to_additive]
lemma zpow_lt_zpow_iff (ha : 1 < a) : a ^ m < a ^ n ↔ m < n 
goal: a ^ m < a ^ n ↔ m < n

full_name: zsmul_strictMono_right
code: @[to_additive zsmul_strictMono_right]
lemma zpow_strictMono_left (hn : 0 < n) : StrictMono ((· ^ n) : α → α) 
goal: StrictMono fun x => n • x

full_name: zpow_strictMono_left
code: @[to_additive zsmul_strictMono_right]
lemma zpow_strictMono_left (hn : 0 < n) : StrictMono ((· ^ n) : α → α) 
goal: StrictMono fun x => x ^ n

full_name: zpow_mono_left
code: @[to_additive zsmul_mono_right]
lemma zpow_mono_left (hn : 0 ≤ n) : Monotone ((· ^ n) : α → α) 
goal: Monotone fun x => x ^ n

full_name: zsmul_mono_right
code: @[to_additive zsmul_mono_right]
lemma zpow_mono_left (hn : 0 ≤ n) : Monotone ((· ^ n) : α → α) 
goal: Monotone fun x => n • x

full_name: zsmul_le_zsmul'
code: @[to_additive (attr := gcongr)]
lemma zpow_le_zpow' (hn : 0 ≤ n) (h : a ≤ b) : a ^ n ≤ b ^ n 
goal: n • a ≤ n • b

full_name: zpow_le_zpow'
code: @[to_additive (attr := gcongr)]
lemma zpow_le_zpow' (hn : 0 ≤ n) (h : a ≤ b) : a ^ n ≤ b ^ n 
goal: a ^ n ≤ b ^ n

full_name: zsmul_lt_zsmul'
code: @[to_additive (attr := gcongr)]
lemma zpow_lt_zpow' (hn : 0 < n) (h : a < b) : a ^ n < b ^ n 
goal: n • a < n • b

full_name: zpow_lt_zpow'
code: @[to_additive (attr := gcongr)]
lemma zpow_lt_zpow' (hn : 0 < n) (h : a < b) : a ^ n < b ^ n 
goal: a ^ n < b ^ n

full_name: zsmul_le_zsmul_iff'
code: @[to_additive] lemma zpow_le_zpow_iff' (hn : 0 < n) : a ^ n ≤ b ^ n ↔ a ≤ b 
goal: n • a ≤ n • b ↔ a ≤ b

full_name: zpow_le_zpow_iff'
code: @[to_additive] lemma zpow_le_zpow_iff' (hn : 0 < n) : a ^ n ≤ b ^ n ↔ a ≤ b 
goal: a ^ n ≤ b ^ n ↔ a ≤ b

full_name: zpow_lt_zpow_iff'
code: @[to_additive] lemma zpow_lt_zpow_iff' (hn : 0 < n) : a ^ n < b ^ n ↔ a < b 
goal: a ^ n < b ^ n ↔ a < b

full_name: zsmul_lt_zsmul_iff'
code: @[to_additive] lemma zpow_lt_zpow_iff' (hn : 0 < n) : a ^ n < b ^ n ↔ a < b 
goal: n • a < n • b ↔ a < b

full_name: zpow_left_injective
code: @[to_additive zsmul_right_injective
"See also `smul_right_injective`. TODO: provide a `NoZeroSMulDivisors` instance. We can't do
that here because importing that definition would create import cycles."]
lemma zpow_left_injective (hn : n ≠ 0) : Injective ((· ^ n) : α → α) 
goal: Function.Injective fun x => x ^ n

full_name: zsmul_right_injective
code: @[to_additive zsmul_right_injective
"See also `smul_right_injective`. TODO: provide a `NoZeroSMulDivisors` instance. We can't do
that here because importing that definition would create import cycles."]
lemma zpow_left_injective (hn : n ≠ 0) : Injective ((· ^ n) : α → α) 
goal: Function.Injective fun x => n • x

full_name: zpow_left_inj
code: @[to_additive zsmul_right_inj]
lemma zpow_left_inj (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b 
goal: a ^ n = b ^ n ↔ a = b

full_name: zsmul_right_inj
code: @[to_additive zsmul_right_inj]
lemma zpow_left_inj (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b 
goal: n • a = n • b ↔ a = b

full_name: zpow_eq_zpow_iff'
code: /-- Alias of `zpow_left_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`. -/
@[to_additive "Alias of `zsmul_right_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`."]
lemma zpow_eq_zpow_iff' (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b 
goal: a ^ n = b ^ n ↔ a = b

full_name: zsmul_eq_zsmul_iff'
code: /-- Alias of `zpow_left_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`. -/
@[to_additive "Alias of `zsmul_right_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`."]
lemma zpow_eq_zpow_iff' (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b 
goal: n • a = n • b ↔ a = b