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
value |
|---|---|---|---|---|---|---|---|---|---|---|
units_with_zero_equiv {α : Type*} [group α] : (with_zero α)ˣ ≃* α | { to_fun := λ a, unzero a.ne_zero,
inv_fun := λ a, units.mk0 a coe_ne_zero,
left_inv := λ _, units.ext $ by simpa only [coe_unzero],
right_inv := λ _, rfl,
map_mul' := λ _ _, coe_inj.mp $ by simpa only [coe_unzero, coe_mul] } | def | with_zero.units_with_zero_equiv | algebra.group.with_one | src/algebra/group/with_one/units.lean | [
"algebra.group.with_one.basic",
"algebra.group_with_zero.units.basic"
] | [
"group",
"inv_fun",
"units.ext",
"units.mk0"
] | Any group is isomorphic to the units of itself adjoined with `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_ite (P : Prop) [decidable P] (a : M) (b c : ℕ) :
a ^ (if P then b else c) = if P then a ^ b else a ^ c | by split_ifs; refl | lemma | pow_ite | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ite_pow (P : Prop) [decidable P] (a b : M) (c : ℕ) :
(if P then a else b) ^ c = if P then a ^ c else b ^ c | by split_ifs; refl | lemma | ite_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_one (a : M) : a^1 = a | by rw [pow_succ, pow_zero, mul_one] | theorem | pow_one | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"mul_one",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_two (a : M) : a^2 = a * a | by rw [pow_succ, pow_one] | theorem | pow_two | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_one",
"pow_succ"
] | Note that most of the lemmas about powers of two refer to it as `sq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_three' (a : M) : a^3 = a * a * a | by rw [pow_succ', pow_two] | theorem | pow_three' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_succ'",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_three (a : M) : a^3 = a * (a * a) | by rw [pow_succ, pow_two] | theorem | pow_three | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_succ",
"pow_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul_comm' (a : M) (n : ℕ) : a^n * a = a * a^n | commute.pow_self a n | theorem | pow_mul_comm' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.pow_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_add (a : M) (m n : ℕ) : a^(m + n) = a^m * a^n | by induction n with n ih; [rw [nat.add_zero, pow_zero, mul_one],
rw [pow_succ', ← mul_assoc, ← ih, ← pow_succ', nat.add_assoc]] | theorem | pow_add | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"ih",
"mul_assoc",
"mul_one",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_boole (P : Prop) [decidable P] (a : M) :
a ^ (if P then 1 else 0) = if P then a else 1 | by simp | lemma | pow_boole | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_pow (n : ℕ) : (1 : M)^n = 1 | by induction n with n ih; [exact pow_zero _, rw [pow_succ, ih, one_mul]] | theorem | one_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"ih",
"one_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul (a : M) (m n : ℕ) : a^(m * n) = (a^m)^n | begin
induction n with n ih,
{ rw [nat.mul_zero, pow_zero, pow_zero] },
{ rw [nat.mul_succ, pow_add, pow_succ', ih] }
end | theorem | pow_mul | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"ih",
"pow_add",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_right_comm (a : M) (m n : ℕ) : (a^m)^n = (a^n)^m | by rw [←pow_mul, nat.mul_comm, pow_mul] | lemma | pow_right_comm | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul' (a : M) (m n : ℕ) : a^(m * n) = (a^n)^m | by rw [nat.mul_comm, pow_mul] | theorem | pow_mul' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul_pow_sub (a : M) {m n : ℕ} (h : m ≤ n) : a ^ m * a ^ (n - m) = a ^ n | by rw [←pow_add, nat.add_comm, nat.sub_add_cancel h] | theorem | pow_mul_pow_sub | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_sub_mul_pow (a : M) {m n : ℕ} (h : m ≤ n) : a ^ (n - m) * a ^ m = a ^ n | by rw [←pow_add, nat.sub_add_cancel h] | theorem | pow_sub_mul_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_pow_mod {M : Type*} [monoid M] {x : M} (m : ℕ) {n : ℕ} (h : x ^ n = 1) :
x ^ m = x ^ (m % n) | begin
have t := congr_arg (λ a, x ^ a) ((nat.add_comm _ _).trans (nat.mod_add_div _ _)).symm,
dsimp at t,
rw [t, pow_add, pow_mul, h, one_pow, one_mul],
end | lemma | pow_eq_pow_mod | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"monoid",
"one_mul",
"one_pow",
"pow_add",
"pow_mul"
] | If `x ^ n = 1`, then `x ^ m` is the same as `x ^ (m % n)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_bit0 (a : M) (n : ℕ) : a ^ bit0 n = a^n * a^n | pow_add _ _ _ | theorem | pow_bit0 | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1 (a : M) (n : ℕ) : a ^ bit1 n = a^n * a^n * a | by rw [bit1, pow_succ', pow_bit0] | theorem | pow_bit1 | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_bit0",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul_comm (a : M) (m n : ℕ) : a^m * a^n = a^n * a^m | commute.pow_pow_self a m n | theorem | pow_mul_comm | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.pow_pow_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.mul_pow {a b : M} (h : commute a b) (n : ℕ) : (a * b) ^ n = a ^ n * b ^ n | nat.rec_on n (by simp only [pow_zero, one_mul]) $ λ n ihn,
by simp only [pow_succ, ihn, ← mul_assoc, (h.pow_left n).right_comm] | lemma | commute.mul_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute",
"mul_assoc",
"one_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit0' (a : M) (n : ℕ) : a ^ bit0 n = (a * a) ^ n | by rw [pow_bit0, (commute.refl a).mul_pow] | theorem | pow_bit0' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.refl",
"mul_pow",
"pow_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_bit1' (a : M) (n : ℕ) : a ^ bit1 n = (a * a) ^ n * a | by rw [bit1, pow_succ', pow_bit0'] | theorem | pow_bit1' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_bit0'",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mul_pow_eq_one {a b : M} (n : ℕ) (h : a * b = 1) :
a ^ n * b ^ n = 1 | begin
induction n with n hn,
{ simp },
{ calc a ^ n.succ * b ^ n.succ = a ^ n * a * (b * b ^ n) : by rw [pow_succ', pow_succ]
... = a ^ n * (a * b) * b ^ n : by simp only [mul_assoc]
... = 1 : by simp [h, hn] }
end | lemma | pow_mul_pow_eq_one | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"mul_assoc",
"pow_succ",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_pow {x y : M} (hxy : x ∣ y) :
∀ {n : ℕ} (hn : n ≠ 0), x ∣ y^n | | 0 hn := (hn rfl).elim
| (n + 1) hn := by { rw pow_succ, exact hxy.mul_right _ } | lemma | dvd_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_pow_self (a : M) {n : ℕ} (hn : n ≠ 0) : a ∣ a^n | dvd_rfl.pow hn | lemma | dvd_pow_self | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pow (a b : M) (n : ℕ) : (a * b)^n = a^n * b^n | (commute.all a b).mul_pow n | theorem | mul_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_monoid_hom (n : ℕ) : M →* M | { to_fun := (^ n),
map_one' := one_pow _,
map_mul' := λ a b, mul_pow a b n } | def | pow_monoid_hom | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"mul_pow",
"one_pow"
] | The `n`th power map on a commutative monoid for a natural `n`, considered as a morphism of
monoids. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpow_one (a : G) : a ^ (1:ℤ) = a | by { convert pow_one a using 1, exact zpow_coe_nat a 1 } | theorem | zpow_one | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_one",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_two (a : G) : a ^ (2 : ℤ) = a * a | by { convert pow_two a using 1, exact zpow_coe_nat a 2 } | theorem | zpow_two | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_two",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_neg_one (x : G) : x ^ (-1:ℤ) = x⁻¹ | (zpow_neg_succ_of_nat x 0).trans $ congr_arg has_inv.inv (pow_one x) | theorem | zpow_neg_one | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"pow_one",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_neg_coe_of_pos (a : G) : ∀ {n : ℕ}, 0 < n → a ^ -(n:ℤ) = (a ^ n)⁻¹ | | (n+1) _ := zpow_neg_succ_of_nat _ _ | theorem | zpow_neg_coe_of_pos | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow (a : α) : ∀ n : ℕ, (a⁻¹) ^ n = (a ^ n)⁻¹ | | 0 := by rw [pow_zero, pow_zero, inv_one]
| (n + 1) := by rw [pow_succ', pow_succ, inv_pow, mul_inv_rev] | lemma | inv_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_one",
"mul_inv_rev",
"pow_succ",
"pow_succ'",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_zpow : ∀ (n : ℤ), (1 : α) ^ n = 1 | | (n : ℕ) := by rw [zpow_coe_nat, one_pow]
| -[1+ n] := by rw [zpow_neg_succ_of_nat, one_pow, inv_one] | lemma | one_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_one",
"one_pow",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_neg (a : α) : ∀ (n : ℤ), a ^ -n = (a ^ n)⁻¹ | | (n+1:ℕ) := div_inv_monoid.zpow_neg' _ _
| 0 := by { change a ^ (0 : ℤ) = (a ^ (0 : ℤ))⁻¹, simp }
| -[1+ n] := by { rw [zpow_neg_succ_of_nat, inv_inv, ← zpow_coe_nat], refl } | lemma | zpow_neg | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_inv",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zpow_neg_one (a b : α) : (a * b) ^ (-1 : ℤ) = b ^ (-1 : ℤ) * a ^ (-1 : ℤ) | by simp_rw [zpow_neg_one, mul_inv_rev] | lemma | mul_zpow_neg_one | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"mul_inv_rev",
"zpow_neg_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zpow (a : α) : ∀ n : ℤ, a⁻¹ ^ n = (a ^ n)⁻¹ | | (n : ℕ) := by rw [zpow_coe_nat, zpow_coe_nat, inv_pow]
| -[1+ n] := by rw [zpow_neg_succ_of_nat, zpow_neg_succ_of_nat, inv_pow] | lemma | inv_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_pow",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_zpow' (a : α) (n : ℤ) : a⁻¹ ^ n = a ^ (-n) | by rw [inv_zpow, zpow_neg] | lemma | inv_zpow' | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_zpow",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_pow (a : α) (n : ℕ) : (1 / a) ^ n = 1 / a ^ n | by simp_rw [one_div, inv_pow] | lemma | one_div_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_pow",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_zpow (a : α) (n : ℤ) : (1 / a) ^ n = 1 / a ^ n | by simp_rw [one_div, inv_zpow] | lemma | one_div_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_zpow",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.mul_zpow (h : commute a b) : ∀ (i : ℤ), (a * b) ^ i = a ^ i * b ^ i | | (n : ℕ) := by simp [h.mul_pow n]
| -[1+n] := by simp [h.mul_pow, (h.pow_pow _ _).eq, mul_inv_rev] | lemma | commute.mul_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zpow (a b : α) : ∀ n : ℤ, (a * b) ^ n = a ^ n * b ^ n | (commute.all a b).mul_zpow | lemma | mul_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pow (a b : α) (n : ℕ) : (a / b) ^ n = a ^ n / b ^ n | by simp only [div_eq_mul_inv, mul_pow, inv_pow] | lemma | div_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"div_eq_mul_inv",
"inv_pow",
"mul_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_zpow (a b : α) (n : ℤ) : (a / b) ^ n = a ^ n / b ^ n | by simp only [div_eq_mul_inv, mul_zpow, inv_zpow] | lemma | div_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"div_eq_mul_inv",
"inv_zpow",
"mul_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_group_hom (n : ℤ) : α →* α | { to_fun := (^ n),
map_one' := one_zpow n,
map_mul' := λ a b, mul_zpow a b n } | def | zpow_group_hom | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"mul_zpow",
"one_zpow"
] | The `n`-th power map (for an integer `n`) on a commutative group, considered as a group
homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_sub (a : G) {m n : ℕ} (h : n ≤ m) : a^(m - n) = a^m * (a^n)⁻¹ | eq_mul_inv_of_mul_eq $ by rw [←pow_add, nat.sub_add_cancel h] | lemma | pow_sub | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"eq_mul_inv_of_mul_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_inv_comm (a : G) (m n : ℕ) : (a⁻¹)^m * a^n = a^n * (a⁻¹)^m | (commute.refl a).inv_left.pow_pow _ _ | lemma | pow_inv_comm | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_pow_sub (a : G) {m n : ℕ} (h : n ≤ m) : a⁻¹^(m - n) = (a^m)⁻¹ * a^n | by rw [pow_sub a⁻¹ h, inv_pow, inv_pow, inv_inv] | lemma | inv_pow_sub | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"inv_inv",
"inv_pow",
"pow_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_pow [monoid R] (a : R) {m n : ℕ} (h : m ≤ n) :
a ^ m ∣ a ^ n | ⟨a ^ (n - m), by rw [← pow_add, nat.add_comm, nat.sub_add_cancel h]⟩ | lemma | pow_dvd_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"monoid",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_nsmul [add_monoid A] (x : A) (n : ℕ) :
multiplicative.of_add (n • x) = (multiplicative.of_add x)^n | rfl | lemma | of_add_nsmul | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"add_monoid",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_zsmul [sub_neg_monoid A] (x : A) (n : ℤ) :
multiplicative.of_add (n • x) = (multiplicative.of_add x)^n | rfl | lemma | of_add_zsmul | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"multiplicative.of_add",
"sub_neg_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_pow [monoid A] (x : A) (n : ℕ) :
additive.of_mul (x ^ n) = n • (additive.of_mul x) | rfl | lemma | of_mul_pow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"additive.of_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_zpow [div_inv_monoid G] (x : G) (n : ℤ) :
additive.of_mul (x ^ n) = n • additive.of_mul x | rfl | lemma | of_mul_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"additive.of_mul",
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiconj_by.zpow_right [group G] {a x y : G} (h : semiconj_by a x y) :
∀ m : ℤ, semiconj_by a (x^m) (y^m) | | (n : ℕ) := by simp [zpow_coe_nat, h.pow_right n]
| -[1+n] := by simp [(h.pow_right n.succ).inv_right] | lemma | semiconj_by.zpow_right | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"group",
"semiconj_by",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_right (h : commute a b) (m : ℤ) : commute a (b^m) | h.zpow_right m | lemma | commute.zpow_right | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_left (h : commute a b) (m : ℤ) : commute (a^m) b | (h.symm.zpow_right m).symm | lemma | commute.zpow_left | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_zpow (h : commute a b) (m n : ℤ) : commute (a^m) (b^n) | (h.zpow_left m).zpow_right n | lemma | commute.zpow_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_zpow : commute a (a ^ n) | (commute.refl a).zpow_right n | lemma | commute.self_zpow | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute",
"commute.refl",
"self_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_self : commute (a ^ n) a | (commute.refl a).zpow_left n | lemma | commute.zpow_self | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_zpow_self : commute (a ^ m) (a ^ n) | (commute.refl a).zpow_zpow m n | lemma | commute.zpow_zpow_self | algebra.group_power | src/algebra/group_power/basic.lean | [
"algebra.divisibility.basic",
"algebra.group.commute",
"algebra.group.type_tags"
] | [
"commute",
"commute.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_add_sq_mul_sq_add_sq :
(x₁^2 + x₂^2) * (y₁^2 + y₂^2) = (x₁*y₁ - x₂*y₂)^2 + (x₁*y₂ + x₂*y₁)^2 | by ring | theorem | sq_add_sq_mul_sq_add_sq | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Brahmagupta-Fibonacci identity or Diophantus identity, see
<https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity>.
This sign choice here corresponds to the signs obtained by multiplying two complex numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_add_mul_sq_mul_sq_add_mul_sq :
(x₁^2 + n*x₂^2) * (y₁^2 + n*y₂^2) = (x₁*y₁ - n*x₂*y₂)^2 + n*(x₁*y₂ + x₂*y₁)^2 | by ring | theorem | sq_add_mul_sq_mul_sq_add_mul_sq | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Brahmagupta's identity, see <https://en.wikipedia.org/wiki/Brahmagupta%27s_identity> | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_four_add_four_mul_pow_four : a^4 + 4*b^4 = ((a - b)^2 + b^2) * ((a + b)^2 + b^2) | by ring | theorem | pow_four_add_four_mul_pow_four | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Sophie Germain's identity, see <https://www.cut-the-knot.org/blue/SophieGermainIdentity.shtml>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_four_add_four_mul_pow_four' :
a^4 + 4*b^4 = (a^2 - 2*a*b + 2*b^2) * (a^2 + 2*a*b + 2*b^2) | by ring | theorem | pow_four_add_four_mul_pow_four' | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Sophie Germain's identity, see <https://www.cut-the-knot.org/blue/SophieGermainIdentity.shtml>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_four_sq_mul_sum_four_sq : (x₁^2 + x₂^2 + x₃^2 + x₄^2) * (y₁^2 + y₂^2 + y₃^2 + y₄^2) =
(x₁*y₁ - x₂*y₂ - x₃*y₃ - x₄*y₄)^2 + (x₁*y₂ + x₂*y₁ + x₃*y₄ - x₄*y₃)^2 +
(x₁*y₃ - x₂*y₄ + x₃*y₁ + x₄*y₂)^2 + (x₁*y₄ + x₂*y₃ - x₃*y₂ + x₄*y₁)^2 | by ring | theorem | sum_four_sq_mul_sum_four_sq | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Euler's four-square identity, see <https://en.wikipedia.org/wiki/Euler%27s_four-square_identity>.
This sign choice here corresponds to the signs obtained by multiplying two quaternions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_eight_sq_mul_sum_eight_sq : (x₁^2 + x₂^2 + x₃^2 + x₄^2 + x₅^2 + x₆^2 + x₇^2 + x₈^2) *
(y₁^2 + y₂^2 + y₃^2 + y₄^2 + y₅^2 + y₆^2 + y₇^2 + y₈^2) =
(x₁*y₁ - x₂*y₂ - x₃*y₃ - x₄*y₄ - x₅*y₅ - x₆*y₆ - x₇*y₇ - x₈*y₈)^2 +
(x₁*y₂ + x₂*y₁ + x₃*y₄ - x₄*y₃ + x₅*y₆ - x₆*y₅ - x₇*y₈ + x₈*y₇)^2 +
(x₁*y₃ - x₂*y₄ + x₃*y₁ + x₄*... | by ring | theorem | sum_eight_sq_mul_sum_eight_sq | algebra.group_power | src/algebra/group_power/identities.lean | [
"tactic.ring"
] | [
"ring"
] | Degen's eight squares identity, see <https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity>.
This sign choice here corresponds to the signs obtained by multiplying two octonions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nsmul_one [add_monoid_with_one A] : ∀ n : ℕ, n • (1 : A) = n | begin
refine eq_nat_cast' (⟨_, _, _⟩ : ℕ →+ A) _,
{ show 0 • (1 : A) = 0, simp [zero_nsmul] },
{ show ∀ x y : ℕ, (x + y) • (1 : A) = x • 1 + y • 1, simp [add_nsmul] },
{ show 1 • (1 : A) = 1, simp }
end | theorem | nsmul_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"
] | [
"add_monoid_with_one",
"eq_nat_cast'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_pow (m : M) [invertible m] (n : ℕ) : invertible (m ^ n) | { inv_of := ⅟ m ^ n,
inv_of_mul_self := by rw [← (commute_inv_of m).symm.mul_pow, inv_of_mul_self, one_pow],
mul_inv_of_self := by rw [← (commute_inv_of m).mul_pow, mul_inv_of_self, one_pow] } | instance | invertible_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"
] | [
"commute_inv_of",
"inv_of_mul_self",
"invertible",
"mul_inv_of_self",
"mul_pow",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_of_pow (m : M) [invertible m] (n : ℕ) [invertible (m ^ n)] :
⅟(m ^ n) = ⅟m ^ n | @invertible_unique M _ (m ^ n) (m ^ n) _ (invertible_pow m n) rfl | lemma | inv_of_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"
] | [
"invertible",
"invertible_pow",
"invertible_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.pow {m : M} (n : ℕ) : is_unit m → is_unit (m ^ n) | λ ⟨u, hu⟩, ⟨u ^ n, hu ▸ u.coe_pow _⟩ | lemma | is_unit.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"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.of_pow (u : Mˣ) (x : M) {n : ℕ} (hn : n ≠ 0) (hu : x ^ n = u) : Mˣ | u.left_of_mul x (x ^ (n - 1))
(by rwa [← pow_succ, nat.sub_add_cancel (nat.succ_le_of_lt $ nat.pos_of_ne_zero hn)])
(commute.self_pow _ _) | def | units.of_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"
] | [
"commute.self_pow",
"pow_succ"
] | If a natural power of `x` is a unit, then `x` is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_pow_iff {a : M} {n : ℕ} (hn : n ≠ 0) :
is_unit (a ^ n) ↔ is_unit a | ⟨λ ⟨u, hu⟩, (u.of_pow a hn hu.symm).is_unit, λ h, h.pow n⟩ | lemma | is_unit_pow_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"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_pow_succ_iff {m : M} {n : ℕ} : is_unit (m ^ (n + 1)) ↔ is_unit m | is_unit_pow_iff n.succ_ne_zero | lemma | is_unit_pow_succ_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"
] | [
"is_unit",
"is_unit_pow_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : n ≠ 0) : Mˣ | units.of_pow 1 x hn hx | def | units.of_pow_eq_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"
] | [
"units.of_pow"
] | If `x ^ n = 1`, `n ≠ 0`, then `x` is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.pow_of_pow_eq_one {x : M} {n : ℕ} (hx : x ^ n = 1) (hn : n ≠ 0) :
units.of_pow_eq_one x n hx hn ^ n = 1 | units.ext $ by rwa [units.coe_pow, units.coe_of_pow_eq_one, units.coe_one] | lemma | units.pow_of_pow_eq_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"
] | [
"units.coe_one",
"units.coe_pow",
"units.ext",
"units.of_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_pow_eq_one {x : M} {n : ℕ} (hx : x ^ n = 1) (hn : n ≠ 0) :
is_unit x | (units.of_pow_eq_one x n hx hn).is_unit | lemma | is_unit_of_pow_eq_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"
] | [
"is_unit",
"units.of_pow_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible_of_pow_eq_one (x : M) (n : ℕ) (hx : x ^ n = 1) (hn : n ≠ 0) :
invertible x | (units.of_pow_eq_one x n hx hn).invertible | def | invertible_of_pow_eq_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"
] | [
"invertible",
"units.of_pow_eq_one"
] | If `x ^ n = 1` then `x` has an inverse, `x^(n - 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_pow [mul_action M N] [is_scalar_tower M N N] [smul_comm_class M N N]
(k : M) (x : N) (p : ℕ) :
(k • x) ^ p = k ^ p • x ^ p | begin
induction p with p IH,
{ simp },
{ rw [pow_succ', IH, smul_mul_smul, ←pow_succ', ←pow_succ'] }
end | lemma | smul_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"
] | [
"is_scalar_tower",
"mul_action",
"pow_succ'",
"smul_comm_class",
"smul_mul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_pow' [mul_distrib_mul_action M N] (x : M) (m : N) (n : ℕ) :
x • m ^ n = (x • m) ^ n | begin
induction n with n ih,
{ rw [pow_zero, pow_zero], exact smul_one x },
{ rw [pow_succ, pow_succ], exact (smul_mul' x m (m ^ n)).trans (congr_arg _ ih) }
end | lemma | smul_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",
"mul_distrib_mul_action",
"pow_succ",
"pow_zero",
"smul_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_one [add_group_with_one A] (n : ℤ) : n • (1 : A) = n | by cases n; simp | lemma | zsmul_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"
] | [
"add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_mul (a : α) : ∀ m n : ℤ, a ^ (m * n) = (a ^ m) ^ n | | (m : ℕ) (n : ℕ) := by { rw [zpow_coe_nat, zpow_coe_nat, ← pow_mul, ← zpow_coe_nat], refl }
| (m : ℕ) -[1+ n] := by { rw [zpow_coe_nat, zpow_neg_succ_of_nat, ← pow_mul, coe_nat_mul_neg_succ,
zpow_neg, inv_inj, ← zpow_coe_nat], refl }
| -[1+ m] (n : ℕ) := by { rw [zpow_coe_nat, zpow_neg_succ_of_nat, ← inv_pow, ← po... | lemma | zpow_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"
] | [
"inv_inj",
"inv_inv",
"inv_pow",
"pow_mul",
"zpow_coe_nat",
"zpow_neg",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_mul' (a : α) (m n : ℤ) : a ^ (m * n) = (a ^ n) ^ m | by rw [mul_comm, zpow_mul] | lemma | zpow_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"
] | [
"mul_comm",
"zpow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0 (a : α) : ∀ n : ℤ, a ^ bit0 n = a ^ n * a ^ n | | (n : ℕ) := by simp only [zpow_coe_nat, ←int.coe_nat_bit0, pow_bit0]
| -[1+n] := by { simp [←mul_inv_rev, ←pow_bit0], rw [neg_succ_of_nat_eq, bit0_neg, zpow_neg],
norm_cast } | lemma | zpow_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"
] | [
"bit0_neg",
"pow_bit0",
"zpow_coe_nat",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0' (a : α) (n : ℤ) : a ^ bit0 n = (a * a) ^ n | (zpow_bit0 a n).trans ((commute.refl a).mul_zpow n).symm | lemma | zpow_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"
] | [
"commute.refl",
"mul_zpow",
"zpow_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit0_neg [has_distrib_neg α] (x : α) (n : ℤ) : (-x) ^ (bit0 n) = x ^ bit0 n | by rw [zpow_bit0', zpow_bit0', neg_mul_neg] | lemma | zpow_bit0_neg | 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"
] | [
"has_distrib_neg",
"neg_mul_neg",
"zpow_bit0'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_add_one (a : G) : ∀ n : ℤ, a ^ (n + 1) = a ^ n * a | | (n : ℕ) := by simp only [← int.coe_nat_succ, zpow_coe_nat, pow_succ']
| -[1+ 0] := by erw [zpow_zero, zpow_neg_succ_of_nat, pow_one, mul_left_inv]
| -[1+ n+1] := begin
rw [zpow_neg_succ_of_nat, pow_succ, mul_inv_rev, inv_mul_cancel_right],
rw [int.neg_succ_of_nat_eq, neg_add, add_assoc, neg_add_self, add_zero],
... | lemma | zpow_add_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"
] | [
"inv_mul_cancel_right",
"mul_inv_rev",
"mul_left_inv",
"pow_one",
"pow_succ",
"pow_succ'",
"zpow_coe_nat",
"zpow_neg_succ_of_nat",
"zpow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_sub_one (a : G) (n : ℤ) : a ^ (n - 1) = a ^ n * a⁻¹ | calc a ^ (n - 1) = a ^ (n - 1) * a * a⁻¹ : (mul_inv_cancel_right _ _).symm
... = a^n * a⁻¹ : by rw [← zpow_add_one, sub_add_cancel] | lemma | zpow_sub_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"
] | [
"mul_inv_cancel_right",
"zpow_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_add (a : G) (m n : ℤ) : a ^ (m + n) = a ^ m * a ^ n | begin
induction n using int.induction_on with n ihn n ihn,
case hz : { simp },
{ simp only [← add_assoc, zpow_add_one, ihn, mul_assoc] },
{ rw [zpow_sub_one, ← mul_assoc, ← ihn, ← zpow_sub_one, add_sub_assoc] }
end | lemma | zpow_add | 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.induction_on",
"mul_assoc",
"zpow_add_one",
"zpow_sub_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_zpow (b : G) (m : ℤ) : b*b^m = b^(m+1) | by { conv_lhs {congr, rw ← zpow_one b }, rw [← zpow_add, add_comm] } | lemma | mul_self_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_add",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zpow_self (b : G) (m : ℤ) : b^m*b = b^(m+1) | by { conv_lhs {congr, skip, rw ← zpow_one b }, rw [← zpow_add, add_comm] } | lemma | mul_zpow_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"
] | [
"zpow_add",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_sub (a : G) (m n : ℤ) : a ^ (m - n) = a ^ m * (a ^ n)⁻¹ | by rw [sub_eq_add_neg, zpow_add, zpow_neg] | lemma | zpow_sub | 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_add",
"zpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_one_add (a : G) (i : ℤ) : a ^ (1 + i) = a * a ^ i | by rw [zpow_add, zpow_one] | theorem | zpow_one_add | 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_add",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_mul_comm (a : G) (i j : ℤ) : a ^ i * a ^ j = a ^ j * a ^ i | (commute.refl _).zpow_zpow _ _ | lemma | zpow_mul_comm | 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.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_bit1 (a : G) (n : ℤ) : a ^ bit1 n = a ^ n * a ^ n * a | by rw [bit1, zpow_add, zpow_bit0, zpow_one] | theorem | zpow_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"
] | [
"zpow_add",
"zpow_bit0",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_induction_left {g : G} {P : G → Prop} (h_one : P (1 : G))
(h_mul : ∀ a, P a → P (g * a)) (h_inv : ∀ a, P a → P (g⁻¹ * a)) (n : ℤ) : P (g ^ n) | begin
induction n using int.induction_on with n ih n ih,
{ rwa zpow_zero },
{ rw [add_comm, zpow_add, zpow_one],
exact h_mul _ ih },
{ rw [sub_eq_add_neg, add_comm, zpow_add, zpow_neg_one],
exact h_inv _ ih }
end | lemma | zpow_induction_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"
] | [
"ih",
"int.induction_on",
"zpow_add",
"zpow_neg_one",
"zpow_one",
"zpow_zero"
] | To show a property of all powers of `g` it suffices to show it is closed under multiplication
by `g` and `g⁻¹` on the left. For subgroups generated by more than one element, see
`subgroup.closure_induction_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpow_induction_right {g : G} {P : G → Prop} (h_one : P (1 : G))
(h_mul : ∀ a, P a → P (a * g)) (h_inv : ∀ a, P a → P (a * g⁻¹)) (n : ℤ) : P (g ^ n) | begin
induction n using int.induction_on with n ih n ih,
{ rwa zpow_zero },
{ rw zpow_add_one,
exact h_mul _ ih },
{ rw zpow_sub_one,
exact h_inv _ ih }
end | lemma | zpow_induction_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"
] | [
"ih",
"int.induction_on",
"zpow_add_one",
"zpow_sub_one",
"zpow_zero"
] | To show a property of all powers of `g` it suffices to show it is closed under multiplication
by `g` and `g⁻¹` on the right. For subgroups generated by more than one element, see
`subgroup.closure_induction_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_lt_zpow' (ha : 1 < a) {k : ℤ} (hk : (0:ℤ) < k) : 1 < a^k | begin
lift k to ℕ using int.le_of_lt hk,
rw zpow_coe_nat,
exact one_lt_pow' ha (coe_nat_pos.mp hk).ne',
end | lemma | one_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"
] | [
"lift",
"one_lt_pow'",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_strict_mono_right (ha : 1 < a) : strict_mono (λ n : ℤ, a ^ n) | λ m n h,
calc a ^ m = a ^ m * 1 : (mul_one _).symm
... < a ^ m * a ^ (n - m) : mul_lt_mul_left' (one_lt_zpow' ha $ sub_pos_of_lt h) _
... = a ^ n : by { rw ←zpow_add, simp } | lemma | zpow_strict_mono_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"
] | [
"mul_lt_mul_left'",
"mul_one",
"one_lt_zpow'",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_mono_right (ha : 1 ≤ a) : monotone (λ n : ℤ, a ^ n) | λ m n h,
calc a ^ m = a ^ m * 1 : (mul_one _).symm
... ≤ a ^ m * a ^ (n - m) : mul_le_mul_left' (one_le_zpow ha $ sub_nonneg_of_le h) _
... = a ^ n : by { rw ←zpow_add, simp } | lemma | zpow_mono_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"
] | [
"monotone",
"mul_le_mul_left'",
"mul_one",
"one_le_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_le_zpow (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n | zpow_mono_right ha 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_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.