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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|---|---|---|---|---|---|---|---|---|---|---|
sq_pos_iff (a : R) : 0 < a ^ 2 ↔ a ≠ 0 | pow_bit0_pos_iff a one_ne_zero | theorem | sq_pos_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"one_ne_zero",
"pow_bit0_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_abs (x : R) : |x| ^ 2 = x ^ 2 | by simpa only [sq] using abs_mul_abs_self x | theorem | sq_abs | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_mul_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sq (x : R) : |x ^ 2| = x ^ 2 | by simpa only [sq] using abs_mul_self x | theorem | abs_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_mul_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_lt_sq : x ^ 2 < y ^ 2 ↔ |x| < |y| | by simpa only [sq_abs]
using (@strict_mono_on_pow R _ _ two_pos).lt_iff_lt (abs_nonneg x) (abs_nonneg y) | theorem | sq_lt_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_nonneg",
"sq_abs",
"strict_mono_on_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_lt_sq' (h1 : -y < x) (h2 : x < y) : x ^ 2 < y ^ 2 | sq_lt_sq.2 (lt_of_lt_of_le (abs_lt.2 ⟨h1, h2⟩) (le_abs_self _)) | theorem | sq_lt_sq' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_le_sq : x ^ 2 ≤ y ^ 2 ↔ |x| ≤ |y| | by simpa only [sq_abs]
using (@strict_mono_on_pow R _ _ two_pos).le_iff_le (abs_nonneg x) (abs_nonneg y) | theorem | sq_le_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_nonneg",
"sq_abs",
"strict_mono_on_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_le_sq' (h1 : -y ≤ x) (h2 : x ≤ y) : x ^ 2 ≤ y ^ 2 | sq_le_sq.2 (le_trans (abs_le.mpr ⟨h1, h2⟩) (le_abs_self _)) | theorem | sq_le_sq' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"le_abs_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt_of_sq_lt_sq (h : x^2 < y^2) (hy : 0 ≤ y) : |x| < y | by rwa [← abs_of_nonneg hy, ← sq_lt_sq] | theorem | abs_lt_of_sq_lt_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_of_nonneg",
"sq_lt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_lt_of_sq_lt_sq' (h : x^2 < y^2) (hy : 0 ≤ y) : -y < x ∧ x < y | abs_lt.mp $ abs_lt_of_sq_lt_sq h hy | theorem | abs_lt_of_sq_lt_sq' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_lt_of_sq_lt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_of_sq_le_sq (h : x^2 ≤ y^2) (hy : 0 ≤ y) : |x| ≤ y | by rwa [← abs_of_nonneg hy, ← sq_le_sq] | theorem | abs_le_of_sq_le_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_of_nonneg",
"sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_le_of_sq_le_sq' (h : x^2 ≤ y^2) (hy : 0 ≤ y) : -y ≤ x ∧ x ≤ y | abs_le.mp $ abs_le_of_sq_le_sq h hy | theorem | abs_le_of_sq_le_sq' | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_le_of_sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_sq_iff_abs_eq_abs (x y : R) : x^2 = y^2 ↔ |x| = |y| | by simp only [le_antisymm_iff, sq_le_sq] | lemma | sq_eq_sq_iff_abs_eq_abs | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_le_one_iff_abs_le_one (x : R) : x^2 ≤ 1 ↔ |x| ≤ 1 | by simpa only [one_pow, abs_one] using @sq_le_sq _ _ x 1 | lemma | sq_le_one_iff_abs_le_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_one",
"one_pow",
"sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_lt_one_iff_abs_lt_one (x : R) : x^2 < 1 ↔ |x| < 1 | by simpa only [one_pow, abs_one] using @sq_lt_sq _ _ x 1 | lemma | sq_lt_one_iff_abs_lt_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_one",
"one_pow",
"sq_lt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_sq_iff_one_le_abs (x : R) : 1 ≤ x^2 ↔ 1 ≤ |x| | by simpa only [one_pow, abs_one] using @sq_le_sq _ _ 1 x | lemma | one_le_sq_iff_one_le_abs | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_one",
"one_pow",
"sq_le_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_sq_iff_one_lt_abs (x : R) : 1 < x^2 ↔ 1 < |x| | by simpa only [one_pow, abs_one] using @sq_lt_sq _ _ 1 x | lemma | one_lt_sq_iff_one_lt_abs | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"abs_one",
"one_pow",
"sq_lt_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_four_le_pow_two_of_pow_two_le {x y : R} (h : x^2 ≤ y) : x^4 ≤ y^2 | (pow_mul x 2 2).symm ▸ pow_le_pow_of_le_left (sq_nonneg x) h 2 | lemma | pow_four_le_pow_two_of_pow_two_le | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_le_pow_of_le_left",
"pow_mul",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_mul_le_add_sq (a b : R) : 2 * a * b ≤ a ^ 2 + b ^ 2 | sub_nonneg.mp ((sub_add_eq_add_sub _ _ _).subst ((sub_sq a b).subst (sq_nonneg _))) | lemma | two_mul_le_add_sq | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"sq_nonneg",
"sub_sq"
] | Arithmetic mean-geometric mean (AM-GM) inequality for linearly ordered commutative rings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_pos_iff (hn : 0 < n) : 0 < a ^ n ↔ 0 < a | by simp_rw [zero_lt_iff, pow_ne_zero_iff hn] | lemma | pow_pos_iff | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"pow_ne_zero_iff",
"zero_lt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow_succ (ha : 1 < a) : a ^ n < a ^ n.succ | by { rw [←one_mul (a ^ n), pow_succ],
exact mul_lt_right₀ _ ha (pow_ne_zero _ (zero_lt_one.trans ha).ne') } | lemma | pow_lt_pow_succ | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"mul_lt_right₀",
"pow_ne_zero",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_lt_pow₀ (ha : 1 < a) (hmn : m < n) : a ^ m < a ^ n | by { induction hmn with n hmn ih, exacts [pow_lt_pow_succ ha, lt_trans ih (pow_lt_pow_succ ha)] } | lemma | pow_lt_pow₀ | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"ih",
"pow_lt_pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg_one : f (-1) = 1 | (pow_eq_one_iff (nat.succ_ne_zero 1)).1 $ by rw [←map_pow, neg_one_sq, map_one] | lemma | monoid_hom.map_neg_one | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"map_one",
"neg_one_sq",
"pow_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (x : R) : f (-x) = f x | by rw [←neg_one_mul, map_mul, map_neg_one, one_mul] | lemma | monoid_hom.map_neg | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [
"map_mul",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub_swap (x y : R) : f (x - y) = f (y - x) | by rw [←map_neg, neg_sub] | lemma | monoid_hom.map_sub_swap | algebra.group_power | src/algebra/group_power/order.lean | [
"algebra.order.ring.abs",
"algebra.order.with_zero",
"algebra.group_power.ring",
"data.set.intervals.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_pow : ∀ {n : ℕ}, 0 < n → (0 : M) ^ n = 0 | | (n+1) _ := by rw [pow_succ, zero_mul] | lemma | zero_pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_succ",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_pow' : ∀ n : ℕ, n ≠ 0 → (0 : M) ^ n = 0 | | 0 h := absurd rfl h
| (k+1) h := by { rw [pow_succ], exact zero_mul _ } | lemma | zero_pow' | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_succ",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_pow_eq (n : ℕ) : (0 : M)^n = if n = 0 then 1 else 0 | begin
split_ifs with h,
{ rw [h, pow_zero], },
{ rw [zero_pow (nat.pos_of_ne_zero h)] },
end | lemma | zero_pow_eq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_zero",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_zero_of_le {x : M} {n m : ℕ}
(hn : n ≤ m) (hx : x^n = 0) : x^m = 0 | by rw [← tsub_add_cancel_of_le hn, pow_add, hx, mul_zero] | lemma | pow_eq_zero_of_le | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"mul_zero",
"pow_add",
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_zero [no_zero_divisors M] {x : M} {n : ℕ} (H : x^n = 0) :
x = 0 | begin
induction n with n ih,
{ rw pow_zero at H,
rw [← mul_one x, H, mul_zero] },
{ rw pow_succ at H,
exact or.cases_on (mul_eq_zero.1 H) id ih }
end | theorem | pow_eq_zero | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"ih",
"mul_one",
"mul_zero",
"no_zero_divisors",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_zero_iff [no_zero_divisors M]
{a : M} {n : ℕ} (hn : 0 < n) :
a ^ n = 0 ↔ a = 0 | begin
refine ⟨pow_eq_zero, _⟩,
rintros rfl,
exact zero_pow hn,
end | lemma | pow_eq_zero_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"no_zero_divisors",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq_zero_iff' [no_zero_divisors M] [nontrivial M]
{a : M} {n : ℕ} :
a ^ n = 0 ↔ a = 0 ∧ n ≠ 0 | by cases (zero_le n).eq_or_gt; simp [*, ne_of_gt] | lemma | pow_eq_zero_iff' | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"no_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_ne_zero_iff [no_zero_divisors M] {a : M} {n : ℕ} (hn : 0 < n) :
a ^ n ≠ 0 ↔ a ≠ 0 | (pow_eq_zero_iff hn).not | lemma | pow_ne_zero_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"no_zero_divisors",
"pow_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_pow {a : M} {n : ℕ} (hn : n ≠ 0) : a ^ n ≠ 0 → a ≠ 0 | by { contrapose!, rintro rfl, exact zero_pow' n hn } | lemma | ne_zero_pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"zero_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_ne_zero [no_zero_divisors M]
{a : M} (n : ℕ) (h : a ≠ 0) : a ^ n ≠ 0 | mt pow_eq_zero h | theorem | pow_ne_zero | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"no_zero_divisors",
"pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero.pow [no_zero_divisors M] {x : M} [ne_zero x] {n : ℕ} :
ne_zero (x ^ n) | ⟨pow_ne_zero n ne_zero.out⟩ | instance | ne_zero.pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"ne_zero",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_zero_iff [no_zero_divisors M] {a : M} : a ^ 2 = 0 ↔ a = 0 | pow_eq_zero_iff two_pos | theorem | sq_eq_zero_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"no_zero_divisors",
"pow_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_pow_eq_zero [nontrivial M] {n : ℕ} : (0 : M) ^ n = 0 ↔ 0 < n | begin
split; intro h,
{ rw [pos_iff_ne_zero], rintro rfl, simpa using h },
{ exact zero_pow' n h.ne.symm }
end | lemma | zero_pow_eq_zero | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"nontrivial",
"zero_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring.inverse_pow (r : M) : ∀ (n : ℕ), ring.inverse r ^ n = ring.inverse (r ^ n) | | 0 := by rw [pow_zero, pow_zero, ring.inverse_one]
| (n + 1) := by rw [pow_succ, pow_succ', ring.mul_inverse_rev' ((commute.refl r).pow_left n),
ring.inverse_pow] | lemma | ring.inverse_pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute.refl",
"pow_succ",
"pow_succ'",
"pow_zero",
"ring.inverse",
"ring.inverse_one",
"ring.mul_inverse_rev'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_monoid_with_zero_hom : M →*₀ M | { map_zero' := zero_pow hn,
..pow_monoid_hom n } | def | pow_monoid_with_zero_hom | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_monoid_hom",
"zero_pow"
] | We define `x ↦ x^n` (for positive `n : ℕ`) as a `monoid_with_zero_hom` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_pow_monoid_with_zero_hom : (pow_monoid_with_zero_hom hn : M → M) = (^ n) | rfl | lemma | coe_pow_monoid_with_zero_hom | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_monoid_with_zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_monoid_with_zero_hom_apply (a : M) : pow_monoid_with_zero_hom hn a = a ^ n | rfl | lemma | pow_monoid_with_zero_hom_apply | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"pow_monoid_with_zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_dvd_pow_iff [cancel_comm_monoid_with_zero R]
{x : R} {n m : ℕ} (h0 : x ≠ 0) (h1 : ¬ is_unit x) :
x ^ n ∣ x ^ m ↔ n ≤ m | begin
split,
{ intro h, rw [← not_lt], intro hmn, apply h1,
have : x ^ m * x ∣ x ^ m * 1,
{ rw [← pow_succ', mul_one], exact (pow_dvd_pow _ (nat.succ_le_of_lt hmn)).trans h },
rwa [mul_dvd_mul_iff_left, ← is_unit_iff_dvd_one] at this, apply pow_ne_zero m h0 },
{ apply pow_dvd_pow }
end | lemma | pow_dvd_pow_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"cancel_comm_monoid_with_zero",
"is_unit",
"is_unit_iff_dvd_one",
"mul_dvd_mul_iff_left",
"mul_one",
"pow_dvd_pow",
"pow_ne_zero",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom.map_pow (f : R →+* S) (a) :
∀ n : ℕ, f (a ^ n) = (f a) ^ n | map_pow f a | lemma | ring_hom.map_pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_pow_dvd_add {n m : ℕ} {a b c : R} (ha : c ^ n ∣ a) (hb : c ^ m ∣ b) :
c ^ (min n m) ∣ a + b | begin
replace ha := (pow_dvd_pow c (min_le_left n m)).trans ha,
replace hb := (pow_dvd_pow c (min_le_right n m)).trans hb,
exact dvd_add ha hb
end | lemma | min_pow_dvd_add | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"dvd_add",
"pow_dvd_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sq (a b : R) : (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2 | by simp only [sq, add_mul_self_eq] | lemma | add_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"add_mul_self_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sq' (a b : R) : (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b | by rw [add_sq, add_assoc, add_comm _ (b ^ 2), add_assoc] | lemma | add_sq' | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"add_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_pow_eq_or : ∀ n : ℕ, (-1 : R)^n = 1 ∨ (-1 : R)^n = -1 | | 0 := or.inl (pow_zero _)
| (n+1) := (neg_one_pow_eq_or n).swap.imp
(λ h, by rw [pow_succ, h, neg_one_mul, neg_neg])
(λ h, by rw [pow_succ, h, mul_one]) | theorem | neg_one_pow_eq_or | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"mul_one",
"neg_one_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pow (a : R) (n : ℕ) : (- a) ^ n = (-1) ^ n * a ^ n | (neg_one_mul a) ▸ (commute.neg_one_left a).mul_pow n | theorem | neg_pow | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute.neg_one_left",
"mul_pow",
"neg_one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pow_bit0 (a : R) (n : ℕ) : (- a) ^ (bit0 n) = a ^ (bit0 n) | by rw [pow_bit0', neg_mul_neg, pow_bit0'] | theorem | neg_pow_bit0 | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"neg_mul_neg",
"pow_bit0'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pow_bit1 (a : R) (n : ℕ) : (- a) ^ (bit1 n) = - a ^ (bit1 n) | by simp only [bit1, pow_succ, neg_pow_bit0, neg_mul_eq_neg_mul] | theorem | neg_pow_bit1 | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"neg_mul_eq_neg_mul",
"neg_pow_bit0",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_sq (a : R) : (-a) ^ 2 = a ^ 2 | by simp [sq] | lemma | neg_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_sq : (-1 : R) ^ 2 = 1 | by rw [neg_sq, one_pow] | lemma | neg_one_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"neg_sq",
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.sq_sub_sq (h : commute a b) : a ^ 2 - b ^ 2 = (a + b) * (a - b) | by rw [sq, sq, h.mul_self_sub_mul_self_eq] | lemma | commute.sq_sub_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_one_pow_mul_eq_zero_iff {n : ℕ} {r : R} : (-1)^n * r = 0 ↔ r = 0 | by rcases neg_one_pow_eq_or R n; simp [h] | lemma | neg_one_pow_mul_eq_zero_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"neg_one_pow_eq_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_neg_one_pow_eq_zero_iff {n : ℕ} {r : R} : r * (-1)^n = 0 ↔ r = 0 | by rcases neg_one_pow_eq_or R n; simp [h] | lemma | mul_neg_one_pow_eq_zero_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"neg_one_pow_eq_or"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.sq_eq_sq_iff_eq_or_eq_neg (h : commute a b) :
a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b | by rw [←sub_eq_zero, h.sq_sub_sq, mul_eq_zero, add_eq_zero_iff_eq_neg, sub_eq_zero, or_comm] | lemma | commute.sq_eq_sq_iff_eq_or_eq_neg | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute",
"mul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_one_iff : a^2 = 1 ↔ a = 1 ∨ a = -1 | by rw [←(commute.one_right a).sq_eq_sq_iff_eq_or_eq_neg, one_pow] | lemma | sq_eq_one_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute.one_right",
"one_pow",
"sq_eq_sq_iff_eq_or_eq_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_ne_one_iff : a^2 ≠ 1 ↔ a ≠ 1 ∧ a ≠ -1 | sq_eq_one_iff.not.trans not_or_distrib | lemma | sq_ne_one_iff | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_sub_sq (a b : R) : a ^ 2 - b ^ 2 = (a + b) * (a - b) | (commute.all a b).sq_sub_sq | lemma | sq_sub_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_sq (a b : R) : (a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2 | by rw [sub_eq_add_neg, add_sq, neg_sq, mul_neg, ← sub_eq_add_neg] | lemma | sub_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"add_sq",
"mul_neg",
"neg_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_sq' (a b : R) : (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b | by rw [sub_eq_add_neg, add_sq', neg_sq, mul_neg, ← sub_eq_add_neg] | lemma | sub_sq' | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"add_sq'",
"mul_neg",
"neg_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_sq_iff_eq_or_eq_neg : a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b | (commute.all a b).sq_eq_sq_iff_eq_or_eq_neg | lemma | sq_eq_sq_iff_eq_or_eq_neg | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"commute.all"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_or_eq_neg_of_sq_eq_sq (a b : R) : a ^ 2 = b ^ 2 → a = b ∨ a = -b | sq_eq_sq_iff_eq_or_eq_neg.1 | lemma | eq_or_eq_neg_of_sq_eq_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_eq_sq_iff_eq_or_eq_neg {a b : Rˣ} : a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b | by simp_rw [ext_iff, coe_pow, sq_eq_sq_iff_eq_or_eq_neg, units.coe_neg] | lemma | units.sq_eq_sq_iff_eq_or_eq_neg | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"sq_eq_sq_iff_eq_or_eq_neg",
"units.coe_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_or_eq_neg_of_sq_eq_sq (a b : Rˣ) (h : a ^ 2 = b ^ 2) : a = b ∨ a = -b | units.sq_eq_sq_iff_eq_or_eq_neg.1 h | lemma | units.eq_or_eq_neg_of_sq_eq_sq | algebra.group_power | src/algebra/group_power/ring.lean | [
"algebra.group_power.basic",
"algebra.group_with_zero.commute",
"algebra.hom.ring",
"algebra.ring.commute",
"algebra.group_with_zero.divisibility",
"algebra.ring.divisibility",
"data.nat.order.basic"
] | [
"eq_or_eq_neg_of_sq_eq_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_semiring_action (M : Type u) (R : Type v) [monoid M] [semiring R]
extends distrib_mul_action M R | (smul_one : ∀ (g : M), (g • 1 : R) = 1)
(smul_mul : ∀ (g : M) (x y : R), g • (x * y) = (g • x) * (g • y)) | class | mul_semiring_action | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"distrib_mul_action",
"monoid",
"semiring"
] | Typeclass for multiplicative actions by monoids on semirings.
This combines `distrib_mul_action` with `mul_distrib_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_semiring_action.to_mul_distrib_mul_action [h : mul_semiring_action M R] :
mul_distrib_mul_action M R | { ..h } | instance | mul_semiring_action.to_mul_distrib_mul_action | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"mul_distrib_mul_action",
"mul_semiring_action"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_semiring_action.to_ring_hom [mul_semiring_action M R] (x : M) : R →+* R | { .. mul_distrib_mul_action.to_monoid_hom R x,
.. distrib_mul_action.to_add_monoid_hom R x } | def | mul_semiring_action.to_ring_hom | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"distrib_mul_action.to_add_monoid_hom",
"mul_distrib_mul_action.to_monoid_hom",
"mul_semiring_action"
] | Each element of the monoid defines a semiring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_ring_hom_injective [mul_semiring_action M R] [has_faithful_smul M R] :
function.injective (mul_semiring_action.to_ring_hom M R) | λ m₁ m₂ h, eq_of_smul_eq_smul $ λ r, ring_hom.ext_iff.1 h r | theorem | to_ring_hom_injective | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"has_faithful_smul",
"mul_semiring_action",
"mul_semiring_action.to_ring_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_semiring_action.to_ring_equiv [mul_semiring_action G R] (x : G) : R ≃+* R | { .. distrib_mul_action.to_add_equiv R x,
.. mul_semiring_action.to_ring_hom G R x } | def | mul_semiring_action.to_ring_equiv | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"distrib_mul_action.to_add_equiv",
"mul_semiring_action",
"mul_semiring_action.to_ring_hom"
] | Each element of the group defines a semiring isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_semiring_action.comp_hom (f : N →* M) [mul_semiring_action M R] :
mul_semiring_action N R | { smul := has_smul.comp.smul f,
..distrib_mul_action.comp_hom R f,
..mul_distrib_mul_action.comp_hom R f } | def | mul_semiring_action.comp_hom | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"distrib_mul_action.comp_hom",
"has_smul.comp.smul",
"mul_distrib_mul_action.comp_hom",
"mul_semiring_action"
] | Compose a `mul_semiring_action` with a `monoid_hom`, with action `f r' • m`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_inv'' [mul_semiring_action M F] (x : M) (m : F) : x • m⁻¹ = (x • m)⁻¹ | map_inv₀ (mul_semiring_action.to_ring_hom M F x) _ | lemma | smul_inv'' | algebra.group_ring_action | src/algebra/group_ring_action/basic.lean | [
"algebra.ring.equiv",
"algebra.field.defs",
"group_theory.group_action.group"
] | [
"map_inv₀",
"mul_semiring_action",
"mul_semiring_action.to_ring_hom"
] | Note that `smul_inv'` refers to the group case, and `smul_inv` has an additional inverse
on `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_invariant_subring : Prop | (smul_mem : ∀ (m : M) {x : R}, x ∈ S → m • x ∈ S) | class | is_invariant_subring | algebra.group_ring_action | src/algebra/group_ring_action/invariant.lean | [
"algebra.hom.group_action",
"ring_theory.subring.pointwise"
] | [] | A typeclass for subrings invariant under a `mul_semiring_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_invariant_subring.to_mul_semiring_action [is_invariant_subring M S] :
mul_semiring_action M S | { smul := λ m x, ⟨m • x, is_invariant_subring.smul_mem m x.2⟩,
one_smul := λ s, subtype.eq $ one_smul M s,
mul_smul := λ m₁ m₂ s, subtype.eq $ mul_smul m₁ m₂ s,
smul_add := λ m s₁ s₂, subtype.eq $ smul_add m s₁ s₂,
smul_zero := λ m, subtype.eq $ smul_zero m,
smul_one := λ m, subtype.eq $ smul_one m,
smul_mu... | instance | is_invariant_subring.to_mul_semiring_action | algebra.group_ring_action | src/algebra/group_ring_action/invariant.lean | [
"algebra.hom.group_action",
"ring_theory.subring.pointwise"
] | [
"is_invariant_subring",
"mul_semiring_action",
"one_smul",
"smul_add",
"smul_mul'",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_invariant_subring.subtype_hom : U →+*[M] R' | { map_smul' := λ m s, rfl, ..U.subtype } | def | is_invariant_subring.subtype_hom | algebra.group_ring_action | src/algebra/group_ring_action/invariant.lean | [
"algebra.hom.group_action",
"ring_theory.subring.pointwise"
] | [] | The canonical inclusion from an invariant subring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_invariant_subring.coe_subtype_hom :
(is_invariant_subring.subtype_hom M U : U → R') = coe | rfl | theorem | is_invariant_subring.coe_subtype_hom | algebra.group_ring_action | src/algebra/group_ring_action/invariant.lean | [
"algebra.hom.group_action",
"ring_theory.subring.pointwise"
] | [
"is_invariant_subring.subtype_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_invariant_subring.coe_subtype_hom' :
(is_invariant_subring.subtype_hom M U : U →+* R') = U.subtype | rfl | theorem | is_invariant_subring.coe_subtype_hom' | algebra.group_ring_action | src/algebra/group_ring_action/invariant.lean | [
"algebra.hom.group_action",
"ring_theory.subring.pointwise"
] | [
"is_invariant_subring.subtype_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submonoid.mul_semiring_action [mul_semiring_action M R] (H : submonoid M) :
mul_semiring_action H R | { smul := (•),
.. H.mul_distrib_mul_action,
.. H.distrib_mul_action } | instance | submonoid.mul_semiring_action | algebra.group_ring_action | src/algebra/group_ring_action/subobjects.lean | [
"algebra.group_ring_action.basic",
"group_theory.subgroup.basic"
] | [
"mul_semiring_action",
"submonoid"
] | A stronger version of `submonoid.distrib_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subgroup.mul_semiring_action [mul_semiring_action G R] (H : subgroup G) :
mul_semiring_action H R | H.to_submonoid.mul_semiring_action | instance | subgroup.mul_semiring_action | algebra.group_ring_action | src/algebra/group_ring_action/subobjects.lean | [
"algebra.group_ring_action.basic",
"group_theory.subgroup.basic"
] | [
"mul_semiring_action",
"subgroup"
] | A stronger version of `subgroup.distrib_mul_action`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_ne_zero_of_mul : a * b ≠ 0 → a ≠ 0 | mt (λ h, mul_eq_zero_of_left h b) | lemma | left_ne_zero_of_mul | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_zero_of_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_ne_zero_of_mul : a * b ≠ 0 → b ≠ 0 | mt (mul_eq_zero_of_right a) | lemma | right_ne_zero_of_mul | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_eq_zero_of_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_and_ne_zero_of_mul (h : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 | ⟨left_ne_zero_of_mul h, right_ne_zero_of_mul h⟩ | lemma | ne_zero_and_ne_zero_of_mul | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"right_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_zero_of_ne_zero_imp_eq_zero {a b : M₀} (h : a ≠ 0 → b = 0) :
a * b = 0 | if ha : a = 0 then by rw [ha, zero_mul] else by rw [h ha, mul_zero] | lemma | mul_eq_zero_of_ne_zero_imp_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mul_eq_const : ((*) (0 : M₀)) = function.const _ 0 | funext zero_mul | lemma | zero_mul_eq_const | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"zero_mul"
] | To match `one_mul_eq_id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_zero_eq_const : (* (0 : M₀)) = function.const _ 0 | funext mul_zero | lemma | mul_zero_eq_const | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_zero"
] | To match `mul_one_eq_id`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_mul_self_eq_zero (h : a * a = 0) : a = 0 | (eq_zero_or_eq_zero_of_mul_eq_zero h).elim id id | lemma | eq_zero_of_mul_self_eq_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_zero (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 | mt eq_zero_or_eq_zero_of_mul_eq_zero $ not_or_distrib.mpr ⟨ha, hb⟩ | theorem | mul_ne_zero | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul [has_zero M₀] [has_mul M₀] [no_zero_divisors M₀] {x y : M₀}
[ne_zero x] [ne_zero y] : ne_zero (x * y) | ⟨mul_ne_zero out out⟩ | instance | ne_zero.mul | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"ne_zero",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_zero_eq_one (h : (0 : M₀) = 1) (a : M₀) : a = 0 | by rw [← mul_one a, ← h, mul_zero] | lemma | eq_zero_of_zero_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_one",
"mul_zero"
] | In a monoid with zero, if zero equals one, then zero is the only element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_of_zero_eq_one (h : (0 : M₀) = 1) : unique M₀ | { default := 0, uniq := eq_zero_of_zero_eq_one h } | def | unique_of_zero_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"eq_zero_of_zero_eq_one",
"unique"
] | In a monoid with zero, if zero equals one, then zero is the unique element.
Somewhat arbitrarily, we define the default element to be `0`.
All other elements will be provably equal to it, but not necessarily definitionally equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_iff_zero_eq_one : (0 : M₀) = 1 ↔ subsingleton M₀ | ⟨λ h, @unique.subsingleton _ (unique_of_zero_eq_one h), λ h, @subsingleton.elim _ h _ _⟩ | theorem | subsingleton_iff_zero_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"unique_of_zero_eq_one"
] | In a monoid with zero, zero equals one if and only if all elements of that semiring
are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_zero_eq_one (h : (0 : M₀) = 1) (a b : M₀) : a = b | @subsingleton.elim _ (subsingleton_of_zero_eq_one h) a b | lemma | eq_of_zero_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_one_or_forall_eq_0 : (0 : M₀) ≠ 1 ∨ (∀a:M₀, a = 0) | not_or_of_imp eq_zero_of_zero_eq_one | lemma | zero_ne_one_or_forall_eq_0 | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"eq_zero_of_zero_eq_one",
"not_or_of_imp"
] | In a monoid with zero, either zero and one are nonequal, or zero is the only element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_ne_zero_of_mul_eq_one (h : a * b = 1) : a ≠ 0 | left_ne_zero_of_mul $ ne_zero_of_eq_one h | lemma | left_ne_zero_of_mul_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"left_ne_zero_of_mul",
"ne_zero_of_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_ne_zero_of_mul_eq_one (h : a * b = 1) : b ≠ 0 | right_ne_zero_of_mul $ ne_zero_of_eq_one h | lemma | right_ne_zero_of_mul_eq_one | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"ne_zero_of_eq_one",
"right_ne_zero_of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid_with_zero.to_no_zero_divisors : no_zero_divisors M₀ | ⟨λ a b ab0, by { by_cases a = 0, { left, exact h }, right,
apply cancel_monoid_with_zero.mul_left_cancel_of_ne_zero h, rw [ab0, mul_zero], }⟩ | instance | cancel_monoid_with_zero.to_no_zero_divisors | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_zero",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_inj' (hc : c ≠ 0) : a * c = b * c ↔ a = b | (mul_left_injective₀ hc).eq_iff | lemma | mul_left_inj' | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_left_injective₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inj' (ha : a ≠ 0) : a * b = a * c ↔ b = c | (mul_right_injective₀ ha).eq_iff | lemma | mul_right_inj' | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_right_injective₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_right_iff : a * c = b * c ↔ a = b ∨ c = 0 | by by_cases hc : c = 0; [simp [hc], simp [mul_left_inj', hc]] | lemma | mul_eq_mul_right_iff | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_left_inj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_left_iff : a * b = a * c ↔ b = c ∨ a = 0 | by by_cases ha : a = 0; [simp [ha], simp [mul_right_inj', ha]] | lemma | mul_eq_mul_left_iff | algebra.group_with_zero | src/algebra/group_with_zero/basic.lean | [
"algebra.group.basic",
"algebra.group_with_zero.defs",
"algebra.group.order_synonym"
] | [
"mul_right_inj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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