statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
zpow_lt_zpow (ha : 1 < a) (h : m < n) : a ^ m < a ^ n | zpow_strict_mono_right ha h | lemma | zpow_lt_zpow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_zpow_iff (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n | (zpow_strict_mono_right ha).le_iff_le | lemma | zpow_le_zpow_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_lt_zpow_iff (ha : 1 < a) : a ^ m < a ^ n ↔ m < n | (zpow_strict_mono_right ha).lt_iff_lt | lemma | zpow_lt_zpow_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_strict_mono_left (hn : 0 < n) : strict_mono ((^ n) : α → α) | λ a b hab, by { rw [←one_lt_div', ←div_zpow], exact one_lt_zpow' (one_lt_div'.2 hab) hn } | lemma | zpow_strict_mono_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"one_lt_zpow'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_mono_left (hn : 0 ≤ n) : monotone ((^ n) : α → α) | λ a b hab, by { rw [←one_le_div', ←div_zpow], exact one_le_zpow (one_le_div'.2 hab) hn } | lemma | zpow_mono_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monotone",
"one_le_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_zpow' (hn : 0 ≤ n) (h : a ≤ b) : a ^ n ≤ b ^ n | zpow_mono_left α hn h | lemma | zpow_le_zpow' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_lt_zpow' (hn : 0 < n) (h : a < b) : a ^ n < b ^ n | zpow_strict_mono_left α hn h | lemma | zpow_lt_zpow' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_zpow_iff' (hn : 0 < n) {a b : α} : a ^ n ≤ b ^ n ↔ a ≤ b | (zpow_strict_mono_left α hn).le_iff_le | lemma | zpow_le_zpow_iff' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_lt_zpow_iff' (hn : 0 < n) {a b : α} : a ^ n < b ^ n ↔ a < b | (zpow_strict_mono_left α hn).lt_iff_lt | lemma | zpow_lt_zpow_iff' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_left_injective (hn : n ≠ 0) : function.injective ((^ n) : α → α) | begin
cases hn.symm.lt_or_lt,
{ exact (zpow_strict_mono_left α h).injective },
{ refine λ a b (hab : a ^ n = b ^ n), (zpow_strict_mono_left α (neg_pos.mpr h)).injective _,
rw [zpow_neg, zpow_neg, hab] }
end | lemma | zpow_left_injective | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_neg",
"zpow_strict_mono_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_left_inj (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b | (zpow_left_injective hn).eq_iff | lemma | zpow_left_inj | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_left_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_eq_zpow_iff' (hn : n ≠ 0) : a ^ n = b ^ n ↔ a = b | zpow_left_inj hn | lemma | zpow_eq_zpow_iff' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"zpow_left_inj"
] | Alias of `zsmul_right_inj`, for ease of discovery alongside `zsmul_le_zsmul_iff'` and
`zsmul_lt_zsmul_iff'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
abs_nsmul (n : ℕ) (a : α) : |n • a| = n • |a| | begin
cases le_total a 0 with hneg hpos,
{ rw [abs_of_nonpos hneg, ← abs_neg, ← neg_nsmul, abs_of_nonneg],
exact nsmul_nonneg (neg_nonneg.mpr hneg) n },
{ rw [abs_of_nonneg hpos, abs_of_nonneg],
exact nsmul_nonneg hpos n }
end | lemma | abs_nsmul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"abs_neg",
"abs_of_nonneg",
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_zsmul (n : ℤ) (a : α) : |n • a| = |n| • |a| | begin
obtain n0 | n0 := le_total 0 n,
{ lift n to ℕ using n0,
simp only [abs_nsmul, abs_coe_nat, coe_nat_zsmul] },
{ lift (- n) to ℕ using neg_nonneg.2 n0 with m h,
rw [← abs_neg (n • a), ← neg_zsmul, ← abs_neg n, ← h, coe_nat_zsmul, abs_coe_nat,
coe_nat_zsmul],
exact abs_nsmul m _ },
end | lemma | abs_zsmul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"abs_neg",
"abs_nsmul",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add_eq_add_abs_le (hle : a ≤ b) : |a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | begin
obtain a0 | a0 := le_or_lt 0 a; obtain b0 | b0 := le_or_lt 0 b,
{ simp [a0, b0, abs_of_nonneg, add_nonneg a0 b0] },
{ exact (lt_irrefl (0 : α) $ a0.trans_lt $ hle.trans_lt b0).elim },
any_goals { simp [a0.le, b0.le, abs_of_nonpos, add_nonpos, add_comm] },
have : (|a + b| = -a + b ↔ b ≤ 0) ↔ (|a + b| =
... | lemma | abs_add_eq_add_abs_le | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"abs_of_neg",
"abs_of_nonneg",
"abs_of_nonpos",
"abs_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_add_eq_add_abs_iff (a b : α) : |a + b| = |a| + |b| ↔ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | begin
obtain ab | ab := le_total a b,
{ exact abs_add_eq_add_abs_le ab },
{ rw [add_comm a, add_comm (abs _), abs_add_eq_add_abs_le ab, and.comm, @and.comm (b ≤ 0)] }
end | lemma | abs_add_eq_add_abs_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"abs_add_eq_add_abs_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot.coe_nsmul [add_monoid A] (a : A) (n : ℕ) :
((n • a : A) : with_bot A) = n • a | add_monoid_hom.map_nsmul ⟨(coe : A → with_bot A), with_bot.coe_zero, with_bot.coe_add⟩ a n | lemma | with_bot.coe_nsmul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_monoid",
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_eq_mul' [non_assoc_semiring R] (a : R) (n : ℕ) : n • a = a * n | by induction n with n ih; [rw [zero_nsmul, nat.cast_zero, mul_zero],
rw [succ_nsmul', ih, nat.cast_succ, mul_add, mul_one]] | theorem | nsmul_eq_mul' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"mul_one",
"mul_zero",
"nat.cast_succ",
"nat.cast_zero",
"non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_eq_mul [non_assoc_semiring R] (n : ℕ) (a : R) : n • a = n * a | by rw [nsmul_eq_mul', (n.cast_commute a).eq] | theorem | nsmul_eq_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"non_assoc_semiring",
"nsmul_eq_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_semiring.nat_smul_comm_class [non_unital_non_assoc_semiring R] :
smul_comm_class ℕ R R | ⟨λ n x y, match n with
| 0 := by simp_rw [zero_nsmul, smul_eq_mul, mul_zero]
| (n + 1) := by simp_rw [succ_nsmul, smul_eq_mul, mul_add, ←smul_eq_mul, _match n]
end⟩ | instance | non_unital_non_assoc_semiring.nat_smul_comm_class | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_zero",
"non_unital_non_assoc_semiring",
"smul_comm_class",
"smul_eq_mul"
] | Note that `add_comm_monoid.nat_smul_comm_class` requires stronger assumptions on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_non_assoc_semiring.nat_is_scalar_tower [non_unital_non_assoc_semiring R] :
is_scalar_tower ℕ R R | ⟨λ n x y, match n with
| 0 := by simp_rw [zero_nsmul, smul_eq_mul, zero_mul]
| (n + 1) := by simp_rw [succ_nsmul, ←_match n, smul_eq_mul, add_mul]
end⟩ | instance | non_unital_non_assoc_semiring.nat_is_scalar_tower | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"is_scalar_tower",
"non_unital_non_assoc_semiring",
"smul_eq_mul",
"zero_mul"
] | Note that `add_comm_monoid.nat_is_scalar_tower` requires stronger assumptions on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat.cast_pow [semiring R] (n m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m | begin
induction m with m ih,
{ rw [pow_zero, pow_zero], exact nat.cast_one },
{ rw [pow_succ', pow_succ', nat.cast_mul, ih] }
end | theorem | nat.cast_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"nat.cast_mul",
"nat.cast_one",
"pow_succ'",
"pow_zero",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.coe_nat_pow (n m : ℕ) : ((n ^ m : ℕ) : ℤ) = n ^ m | by induction m with m ih; [exact int.coe_nat_one, rw [pow_succ', pow_succ', int.coe_nat_mul, ih]] | theorem | int.coe_nat_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.nat_abs_pow (n : ℤ) (k : ℕ) : int.nat_abs (n ^ k) = (int.nat_abs n) ^ k | by induction k with k ih; [refl, rw [pow_succ', int.nat_abs_mul, pow_succ', ih]] | theorem | int.nat_abs_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"int.nat_abs_mul",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit0_mul [non_unital_non_assoc_ring R] {n r : R} : bit0 n * r = (2 : ℤ) • (n * r) | by { dsimp [bit0], rw [add_mul, add_zsmul, one_zsmul], } | lemma | bit0_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"non_unital_non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bit0 [non_unital_non_assoc_ring R] {n r : R} : r * bit0 n = (2 : ℤ) • (r * n) | by { dsimp [bit0], rw [mul_add, add_zsmul, one_zsmul], } | lemma | mul_bit0 | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"non_unital_non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bit1_mul [non_assoc_ring R] {n r : R} : bit1 n * r = (2 : ℤ) • (n * r) + r | by { dsimp [bit1], rw [add_mul, bit0_mul, one_mul], } | lemma | bit1_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"bit0_mul",
"non_assoc_ring",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_bit1 [non_assoc_ring R] {n r : R} : r * bit1 n = (2 : ℤ) • (r * n) + r | by { dsimp [bit1], rw [mul_add, mul_bit0, mul_one], } | lemma | mul_bit1 | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_bit0",
"mul_one",
"non_assoc_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.cast_mul_eq_zsmul_cast [add_comm_group_with_one α] : ∀ m n, ((m * n : ℤ) : α) = m • n | λ m, int.induction_on' m 0 (by simp) (λ k _ ih n, by simp [add_mul, add_zsmul, ih])
(λ k _ ih n, by simp [sub_mul, sub_zsmul, ih, ←sub_eq_add_neg]) | lemma | int.cast_mul_eq_zsmul_cast | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_group_with_one",
"ih",
"int.induction_on'"
] | Note this holds in marginally more generality than `int.cast_mul` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zsmul_eq_mul [ring R] (a : R) : ∀ (n : ℤ), n • a = n * a | | (n : ℕ) := by rw [coe_nat_zsmul, nsmul_eq_mul, int.cast_coe_nat]
| -[1+ n] := by simp [nat.cast_succ, neg_add_rev, int.cast_neg_succ_of_nat, add_mul] | theorem | zsmul_eq_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.cast_coe_nat",
"int.cast_neg_succ_of_nat",
"nat.cast_succ",
"nsmul_eq_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_eq_mul' [ring R] (a : R) (n : ℤ) : n • a = a * n | by rw [zsmul_eq_mul, (n.cast_commute a).eq] | theorem | zsmul_eq_mul' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ring",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_non_assoc_ring.int_smul_comm_class [non_unital_non_assoc_ring R] :
smul_comm_class ℤ R R | ⟨λ n x y, match n with
| (n : ℕ) := by simp_rw [coe_nat_zsmul, smul_comm]
| -[1+n] := by simp_rw [zsmul_neg_succ_of_nat, smul_eq_mul, mul_neg, mul_smul_comm]
end⟩ | instance | non_unital_non_assoc_ring.int_smul_comm_class | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_neg",
"mul_smul_comm",
"non_unital_non_assoc_ring",
"smul_comm_class",
"smul_eq_mul"
] | Note that `add_comm_group.int_smul_comm_class` requires stronger assumptions on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_unital_non_assoc_ring.int_is_scalar_tower [non_unital_non_assoc_ring R] :
is_scalar_tower ℤ R R | ⟨λ n x y, match n with
| (n : ℕ) := by simp_rw [coe_nat_zsmul, smul_assoc]
| -[1+n] := by simp_rw [zsmul_neg_succ_of_nat, smul_eq_mul, neg_mul, smul_mul_assoc]
end⟩ | instance | non_unital_non_assoc_ring.int_is_scalar_tower | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"is_scalar_tower",
"neg_mul",
"non_unital_non_assoc_ring",
"smul_assoc",
"smul_eq_mul",
"smul_mul_assoc"
] | Note that `add_comm_group.int_is_scalar_tower` requires stronger assumptions on `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zsmul_int_int (a b : ℤ) : a • b = a * b | by simp | lemma | zsmul_int_int | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_int_one (n : ℤ) : n • 1 = n | by simp | lemma | zsmul_int_one | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.cast_pow [ring R] (n : ℤ) (m : ℕ) : (↑(n ^ m) : R) = ↑n ^ m | begin
induction m with m ih,
{ rw [pow_zero, pow_zero, int.cast_one] },
{ rw [pow_succ, pow_succ, int.cast_mul, ih] }
end | theorem | int.cast_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"ih",
"int.cast_mul",
"int.cast_one",
"pow_succ",
"pow_zero",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_pow_eq_pow_mod_two [ring R] {n : ℕ} : (-1 : R) ^ n = (-1) ^ (n % 2) | by rw [← nat.mod_add_div n 2, pow_add, pow_mul]; simp [sq] | lemma | neg_one_pow_eq_pow_mod_two | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_add",
"pow_mul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_mul_le_pow' (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a))
(H : 0 ≤ 2 + a) :
∀ (n : ℕ), 1 + (n : R) * a ≤ (1 + a) ^ n | | 0 := by simp
| 1 := by simp
| (n+2) :=
have 0 ≤ (n : R) * (a * a * (2 + a)) + a * a,
from add_nonneg (mul_nonneg n.cast_nonneg (mul_nonneg Hsq H)) Hsq,
calc 1 + (↑(n + 2) : R) * a ≤ 1 + ↑(n + 2) * a + (n * (a * a * (2 + a)) + a * a) :
(le_add_iff_nonneg_right _).2 this
... = (1 + a) * (1 + a) * (1 + n * a... | theorem | one_add_mul_le_pow' | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"pow_succ"
] | Bernoulli's inequality. This version works for semirings but requires
additional hypotheses `0 ≤ a * a` and `0 ≤ (1 + a) * (1 + a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_le_pow_of_le_one_aux (h : 0 ≤ a) (ha : a ≤ 1) (i : ℕ) :
∀ k : ℕ, a ^ (i + k) ≤ a ^ i | | 0 := by simp
| (k+1) := by { rw [←add_assoc, ←one_mul (a^i), pow_succ],
exact mul_le_mul ha (pow_le_pow_of_le_one_aux _) (pow_nonneg h _) zero_le_one } | lemma | pow_le_pow_of_le_one_aux | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_le_mul",
"pow_nonneg",
"pow_succ",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_pow_of_le_one (h : 0 ≤ a) (ha : a ≤ 1) {i j : ℕ} (hij : i ≤ j) :
a ^ j ≤ a ^ i | let ⟨k, hk⟩ := nat.exists_eq_add_of_le hij in
by rw hk; exact pow_le_pow_of_le_one_aux h ha _ _ | lemma | pow_le_pow_of_le_one | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"nat.exists_eq_add_of_le",
"pow_le_pow_of_le_one_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_le_of_le_one (h₀ : 0 ≤ a) (h₁ : a ≤ 1) {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ a | (pow_one a).subst (pow_le_pow_of_le_one h₀ h₁ (nat.pos_of_ne_zero hn)) | lemma | pow_le_of_le_one | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_le_pow_of_le_one",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_le (h₀ : 0 ≤ a) (h₁ : a ≤ 1) : a ^ 2 ≤ a | pow_le_of_le_one h₀ h₁ two_ne_zero | lemma | sq_le | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_le_of_le_one",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_cases_of_C_mul_pow_nonneg {C r : R} (h : ∀ n : ℕ, 0 ≤ C * r ^ n) :
C = 0 ∨ (0 < C ∧ 0 ≤ r) | begin
have : 0 ≤ C, by simpa only [pow_zero, mul_one] using h 0,
refine this.eq_or_lt.elim (λ h, or.inl h.symm) (λ hC, or.inr ⟨hC, _⟩),
refine nonneg_of_mul_nonneg_right _ hC,
simpa only [pow_one] using h 1
end | lemma | sign_cases_of_C_mul_pow_nonneg | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_one",
"nonneg_of_mul_nonneg_right",
"pow_one",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_pow (a : R) (n : ℕ) : |a ^ n| = |a| ^ n | (pow_abs a n).symm | lemma | abs_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1_neg_iff : a ^ bit1 n < 0 ↔ a < 0 | ⟨λ h, not_le.1 $ λ h', not_le.2 h $ pow_nonneg h' _, λ ha, pow_bit1_neg ha n⟩ | theorem | pow_bit1_neg_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_bit1_neg",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1_nonneg_iff : 0 ≤ a ^ bit1 n ↔ 0 ≤ a | le_iff_le_iff_lt_iff_lt.2 pow_bit1_neg_iff | theorem | pow_bit1_nonneg_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"pow_bit1_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1_nonpos_iff : a ^ bit1 n ≤ 0 ↔ a ≤ 0 | by simp only [le_iff_lt_or_eq, pow_bit1_neg_iff, pow_eq_zero_iff (bit1_pos (zero_le n))] | theorem | pow_bit1_nonpos_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"bit1_pos",
"pow_bit1_neg_iff",
"pow_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1_pos_iff : 0 < a ^ bit1 n ↔ 0 < a | lt_iff_lt_of_le_iff_le pow_bit1_nonpos_iff | theorem | pow_bit1_pos_iff | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"lt_iff_lt_of_le_iff_le",
"pow_bit1_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_pow_bit1 (n : ℕ) : strict_mono (λ a : R, a ^ bit1 n) | begin
intros a b hab,
cases le_total a 0 with ha ha,
{ cases le_or_lt b 0 with hb hb,
{ rw [← neg_lt_neg_iff, ← neg_pow_bit1, ← neg_pow_bit1],
exact pow_lt_pow_of_lt_left (neg_lt_neg hab) (neg_nonneg.2 hb) (bit1_pos (zero_le n)) },
{ exact (pow_bit1_nonpos_iff.2 ha).trans_lt (pow_bit1_pos_iff.2 hb) ... | lemma | strict_mono_pow_bit1 | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"bit1_pos",
"neg_pow_bit1",
"pow_lt_pow_of_lt_left",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_mul_le_pow (H : -2 ≤ a) (n : ℕ) : 1 + (n : R) * a ≤ (1 + a) ^ n | one_add_mul_le_pow' (mul_self_nonneg _) (mul_self_nonneg _) (neg_le_iff_add_nonneg'.1 H) _ | theorem | one_add_mul_le_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"mul_self_nonneg",
"one_add_mul_le_pow'"
] | Bernoulli's inequality for `n : ℕ`, `-2 ≤ a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_add_mul_sub_le_pow (H : -1 ≤ a) (n : ℕ) : 1 + (n : R) * (a - 1) ≤ a ^ n | have -2 ≤ a - 1, by rwa [bit0, neg_add, ← sub_eq_add_neg, sub_le_sub_iff_right],
by simpa only [add_sub_cancel'_right] using one_add_mul_le_pow this n | theorem | one_add_mul_sub_le_pow | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"one_add_mul_le_pow"
] | Bernoulli's inequality reformulated to estimate `a^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_abs_sq (x : ℤ) : (x.nat_abs ^ 2 : ℤ) = x ^ 2 | by rw [sq, int.nat_abs_mul_self', sq] | lemma | int.nat_abs_sq | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.nat_abs_mul_self'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_self_sq (a : ℤ) : (int.nat_abs a : ℤ) ≤ a ^ 2 | by { rw [← int.nat_abs_sq a, sq], norm_cast, apply nat.le_mul_self } | lemma | int.abs_le_self_sq | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.nat_abs_sq",
"nat.le_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_sq (b : ℤ) : b ≤ b ^ 2 | le_trans (le_nat_abs) (abs_le_self_sq _) | lemma | int.le_self_sq | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_right_injective {x : ℤ} (h : 1 < x.nat_abs) : function.injective ((^) x : ℕ → ℤ) | begin
suffices : function.injective (nat_abs ∘ ((^) x : ℕ → ℤ)),
{ exact function.injective.of_comp this },
convert nat.pow_right_injective h,
ext n,
rw [function.comp_app, nat_abs_pow]
end | lemma | int.pow_right_injective | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"function.injective.of_comp",
"nat.pow_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers_hom [monoid M] : M ≃ (multiplicative ℕ →* M) | { to_fun := λ x, ⟨λ n, x ^ n.to_add, by { convert pow_zero x, exact to_add_one },
λ m n, pow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := pow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_pow, ← of_add_nsmul] } } | def | powers_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"inv_fun",
"monoid",
"monoid_hom.ext",
"multiplicative",
"multiplicative.of_add",
"of_add_nsmul",
"pow_add",
"pow_one",
"pow_zero",
"to_add_one"
] | Monoid homomorphisms from `multiplicative ℕ` are defined by the image
of `multiplicative.of_add 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpowers_hom [group G] : G ≃ (multiplicative ℤ →* G) | { to_fun := λ x, ⟨λ n, x ^ n.to_add, zpow_zero x, λ m n, zpow_add x m n⟩,
inv_fun := λ f, f (multiplicative.of_add 1),
left_inv := zpow_one,
right_inv := λ f, monoid_hom.ext $ λ n, by { simp [← f.map_zpow, ← of_add_zsmul ] } } | def | zpowers_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"group",
"inv_fun",
"monoid_hom.ext",
"multiplicative",
"multiplicative.of_add",
"of_add_zsmul",
"zpow_add",
"zpow_one",
"zpow_zero"
] | Monoid homomorphisms from `multiplicative ℤ` are defined by the image
of `multiplicative.of_add 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiples_hom [add_monoid A] : A ≃ (ℕ →+ A) | { to_fun := λ x, ⟨λ n, n • x, zero_nsmul x, λ m n, add_nsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_nsmul,
right_inv := λ f, add_monoid_hom.ext_nat $ one_nsmul (f 1) } | def | multiples_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_monoid",
"add_monoid_hom.ext_nat",
"inv_fun"
] | Additive homomorphisms from `ℕ` are defined by the image of `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zmultiples_hom [add_group A] : A ≃ (ℤ →+ A) | { to_fun := λ x, ⟨λ n, n • x, zero_zsmul x, λ m n, add_zsmul _ _ _⟩,
inv_fun := λ f, f 1,
left_inv := one_zsmul,
right_inv := λ f, add_monoid_hom.ext_int $ one_zsmul (f 1) } | def | zmultiples_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_group",
"add_monoid_hom.ext_int",
"inv_fun"
] | Additive homomorphisms from `ℤ` are defined by the image of `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
powers_hom_apply [monoid M] (x : M) (n : multiplicative ℕ) :
powers_hom M x n = x ^ n.to_add | rfl | lemma | powers_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monoid",
"multiplicative",
"powers_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers_hom_symm_apply [monoid M] (f : multiplicative ℕ →* M) :
(powers_hom M).symm f = f (multiplicative.of_add 1) | rfl | lemma | powers_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monoid",
"multiplicative",
"multiplicative.of_add",
"powers_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpowers_hom_apply [group G] (x : G) (n : multiplicative ℤ) :
zpowers_hom G x n = x ^ n.to_add | rfl | lemma | zpowers_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"group",
"multiplicative",
"zpowers_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpowers_hom_symm_apply [group G] (f : multiplicative ℤ →* G) :
(zpowers_hom G).symm f = f (multiplicative.of_add 1) | rfl | lemma | zpowers_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"group",
"multiplicative",
"multiplicative.of_add",
"zpowers_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiples_hom_apply [add_monoid A] (x : A) (n : ℕ) :
multiples_hom A x n = n • x | rfl | lemma | multiples_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_monoid",
"multiples_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiples_hom_symm_apply [add_monoid A] (f : ℕ →+ A) :
(multiples_hom A).symm f = f 1 | rfl | lemma | multiples_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_monoid",
"multiples_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_hom_apply [add_group A] (x : A) (n : ℤ) :
zmultiples_hom A x n = n • x | rfl | lemma | zmultiples_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_group",
"zmultiples_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_hom_symm_apply [add_group A] (f : ℤ →+ A) :
(zmultiples_hom A).symm f = f 1 | rfl | lemma | zmultiples_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_group",
"zmultiples_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.apply_mnat [monoid M] (f : multiplicative ℕ →* M) (n : multiplicative ℕ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add | by rw [← powers_hom_symm_apply, ← powers_hom_apply, equiv.apply_symm_apply] | lemma | monoid_hom.apply_mnat | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"equiv.apply_symm_apply",
"monoid",
"multiplicative",
"multiplicative.of_add",
"powers_hom_apply",
"powers_hom_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.ext_mnat [monoid M] ⦃f g : multiplicative ℕ →* M⦄
(h : f (multiplicative.of_add 1) = g (multiplicative.of_add 1)) : f = g | monoid_hom.ext $ λ n, by rw [f.apply_mnat, g.apply_mnat, h] | lemma | monoid_hom.ext_mnat | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"monoid",
"monoid_hom.ext",
"multiplicative",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.apply_mint [group M] (f : multiplicative ℤ →* M) (n : multiplicative ℤ) :
f n = (f (multiplicative.of_add 1)) ^ n.to_add | by rw [← zpowers_hom_symm_apply, ← zpowers_hom_apply, equiv.apply_symm_apply] | lemma | monoid_hom.apply_mint | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"equiv.apply_symm_apply",
"group",
"multiplicative",
"multiplicative.of_add",
"zpowers_hom_apply",
"zpowers_hom_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.apply_nat [add_monoid M] (f : ℕ →+ M) (n : ℕ) :
f n = n • (f 1) | by rw [← multiples_hom_symm_apply, ← multiples_hom_apply, equiv.apply_symm_apply] | lemma | add_monoid_hom.apply_nat | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_monoid",
"equiv.apply_symm_apply",
"multiples_hom_apply",
"multiples_hom_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.apply_int [add_group M] (f : ℤ →+ M) (n : ℤ) :
f n = n • (f 1) | by rw [← zmultiples_hom_symm_apply, ← zmultiples_hom_apply, equiv.apply_symm_apply] | lemma | add_monoid_hom.apply_int | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_group",
"equiv.apply_symm_apply",
"zmultiples_hom_apply",
"zmultiples_hom_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers_mul_hom [comm_monoid M] : M ≃* (multiplicative ℕ →* M) | { map_mul' := λ a b, monoid_hom.ext $ by simp [mul_pow],
..powers_hom M} | def | powers_mul_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_monoid",
"monoid_hom.ext",
"mul_pow",
"multiplicative",
"powers_hom"
] | If `M` is commutative, `powers_hom` is a multiplicative equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpowers_mul_hom [comm_group G] : G ≃* (multiplicative ℤ →* G) | { map_mul' := λ a b, monoid_hom.ext $ by simp [mul_zpow],
..zpowers_hom G} | def | zpowers_mul_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_group",
"monoid_hom.ext",
"mul_zpow",
"multiplicative",
"zpowers_hom"
] | If `M` is commutative, `zpowers_hom` is a multiplicative equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiples_add_hom [add_comm_monoid A] : A ≃+ (ℕ →+ A) | { map_add' := λ a b, add_monoid_hom.ext $ by simp [nsmul_add],
..multiples_hom A} | def | multiples_add_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_monoid",
"multiples_hom"
] | If `M` is commutative, `multiples_hom` is an additive equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zmultiples_add_hom [add_comm_group A] : A ≃+ (ℤ →+ A) | { map_add' := λ a b, add_monoid_hom.ext $ by simp [zsmul_add],
..zmultiples_hom A} | def | zmultiples_add_hom | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_group",
"zmultiples_hom"
] | If `M` is commutative, `zmultiples_hom` is an additive equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
powers_mul_hom_apply [comm_monoid M] (x : M) (n : multiplicative ℕ) :
powers_mul_hom M x n = x ^ n.to_add | rfl | lemma | powers_mul_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_monoid",
"multiplicative",
"powers_mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powers_mul_hom_symm_apply [comm_monoid M] (f : multiplicative ℕ →* M) :
(powers_mul_hom M).symm f = f (multiplicative.of_add 1) | rfl | lemma | powers_mul_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_monoid",
"multiplicative",
"multiplicative.of_add",
"powers_mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpowers_mul_hom_apply [comm_group G] (x : G) (n : multiplicative ℤ) :
zpowers_mul_hom G x n = x ^ n.to_add | rfl | lemma | zpowers_mul_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_group",
"multiplicative",
"zpowers_mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpowers_mul_hom_symm_apply [comm_group G] (f : multiplicative ℤ →* G) :
(zpowers_mul_hom G).symm f = f (multiplicative.of_add 1) | rfl | lemma | zpowers_mul_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"comm_group",
"multiplicative",
"multiplicative.of_add",
"zpowers_mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiples_add_hom_apply [add_comm_monoid A] (x : A) (n : ℕ) :
multiples_add_hom A x n = n • x | rfl | lemma | multiples_add_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_monoid",
"multiples_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiples_add_hom_symm_apply [add_comm_monoid A] (f : ℕ →+ A) :
(multiples_add_hom A).symm f = f 1 | rfl | lemma | multiples_add_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_monoid",
"multiples_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_add_hom_apply [add_comm_group A] (x : A) (n : ℤ) :
zmultiples_add_hom A x n = n • x | rfl | lemma | zmultiples_add_hom_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_group",
"zmultiples_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_add_hom_symm_apply [add_comm_group A] (f : ℤ →+ A) :
(zmultiples_add_hom A).symm f = f 1 | rfl | lemma | zmultiples_add_hom_symm_apply | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"add_comm_group",
"zmultiples_add_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_right (h : semiconj_by a x y) (n : ℕ) :
semiconj_by a ((n : R) * x) (n * y) | semiconj_by.mul_right (nat.commute_cast _ _) h | lemma | semiconj_by.cast_nat_mul_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"nat.commute_cast",
"semiconj_by",
"semiconj_by.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_left (h : semiconj_by a x y) (n : ℕ) : semiconj_by ((n : R) * a) x y | semiconj_by.mul_left (nat.cast_commute _ _) h | lemma | semiconj_by.cast_nat_mul_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"nat.cast_commute",
"semiconj_by",
"semiconj_by.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_cast_nat_mul (h : semiconj_by a x y) (m n : ℕ) :
semiconj_by ((m : R) * a) (n * x) (n * y) | (h.cast_nat_mul_left m).cast_nat_mul_right n | lemma | semiconj_by.cast_nat_mul_cast_nat_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_zpow_right {a : M} {x y : Mˣ} (h : semiconj_by a x y) :
∀ m : ℤ, semiconj_by a (↑(x^m)) (↑(y^m)) | | (n : ℕ) := by simp only [zpow_coe_nat, units.coe_pow, h, pow_right]
| -[1+n] := by simp only [zpow_neg_succ_of_nat, units.coe_pow, units_inv_right, h, pow_right] | lemma | semiconj_by.units_zpow_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"semiconj_by",
"units.coe_pow",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_right (h : semiconj_by a x y) (m : ℤ) :
semiconj_by a ((m : ℤ) * x) (m * y) | semiconj_by.mul_right (int.commute_cast _ _) h | lemma | semiconj_by.cast_int_mul_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.commute_cast",
"semiconj_by",
"semiconj_by.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_left (h : semiconj_by a x y) (m : ℤ) : semiconj_by ((m : R) * a) x y | semiconj_by.mul_left (int.cast_commute _ _) h | lemma | semiconj_by.cast_int_mul_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"int.cast_commute",
"semiconj_by",
"semiconj_by.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_cast_int_mul (h : semiconj_by a x y) (m n : ℤ) :
semiconj_by ((m : R) * a) (n * x) (n * y) | (h.cast_int_mul_left m).cast_int_mul_right n | lemma | semiconj_by.cast_int_mul_cast_int_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_right (h : commute a b) (n : ℕ) : commute a ((n : R) * b) | h.cast_nat_mul_right n | theorem | commute.cast_nat_mul_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_left (h : commute a b) (n : ℕ) : commute ((n : R) * a) b | h.cast_nat_mul_left n | theorem | commute.cast_nat_mul_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_cast_nat_mul (h : commute a b) (m n : ℕ) :
commute (m * a : R) (n * b : R) | h.cast_nat_mul_cast_nat_mul m n | theorem | commute.cast_nat_mul_cast_nat_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_cast_nat_mul : commute a (n * a : R) | (commute.refl a).cast_nat_mul_right n | lemma | commute.self_cast_nat_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_nat_mul_self : commute ((n : R) * a) a | (commute.refl a).cast_nat_mul_left n | lemma | commute.cast_nat_mul_self | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_cast_nat_mul_cast_nat_mul : commute (m * a : R) (n * a : R) | (commute.refl a).cast_nat_mul_cast_nat_mul m n | theorem | commute.self_cast_nat_mul_cast_nat_mul | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_zpow_right {a : M} {u : Mˣ} (h : commute a u) (m : ℤ) :
commute a (↑(u^m)) | h.units_zpow_right m | lemma | commute.units_zpow_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_zpow_left {u : Mˣ} {a : M} (h : commute ↑u a) (m : ℤ) :
commute (↑(u^m)) a | (h.symm.units_zpow_right m).symm | lemma | commute.units_zpow_left | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_int_mul_right (h : commute a b) (m : ℤ) : commute a (m * b : R) | h.cast_int_mul_right m | lemma | commute.cast_int_mul_right | algebra.group_power | src/algebra/group_power/lemmas.lean | [
"algebra.invertible",
"algebra.group_power.ring",
"algebra.order.monoid.with_top",
"data.nat.pow",
"data.int.cast.lemmas"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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