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
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nat_degree_multiset_prod_of_monic (h : ∀ f ∈ t, monic f) :
t.prod.nat_degree = (t.map nat_degree).sum | begin
nontriviality R,
apply nat_degree_multiset_prod',
suffices : (t.map (λ f, leading_coeff f)).prod = 1, { rw this, simp },
convert prod_replicate t.card (1 : R),
{ simp only [eq_replicate, multiset.card_map, eq_self_iff_true, true_and],
rintros i hi,
obtain ⟨i, hi, rfl⟩ := multiset.mem_map.mp hi,
... | lemma | polynomial.nat_degree_multiset_prod_of_monic | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset.card_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_degree_prod_of_monic (h : ∀ i ∈ s, (f i).monic) :
(∏ i in s, f i).nat_degree = ∑ i in s, (f i).nat_degree | by simpa using nat_degree_multiset_prod_of_monic (s.1.map f) (by simpa using h) | lemma | polynomial.nat_degree_prod_of_monic | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_multiset_prod_of_nat_degree_le (n : ℕ)
(hl : ∀ p ∈ t, nat_degree p ≤ n) :
coeff t.prod (t.card * n) = (t.map (λ p, coeff p n)).prod | begin
induction t using quotient.induction_on,
simpa using coeff_list_prod_of_nat_degree_le _ _ hl
end | lemma | polynomial.coeff_multiset_prod_of_nat_degree_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_prod_of_nat_degree_le (f : ι → R[X]) (n : ℕ)
(h : ∀ p ∈ s, nat_degree (f p) ≤ n) :
coeff (∏ i in s, f i) (s.card * n) = ∏ i in s, coeff (f i) n | begin
cases s with l hl,
convert coeff_multiset_prod_of_nat_degree_le (l.map f) _ _,
{ simp },
{ simp },
{ simpa using h }
end | lemma | polynomial.coeff_prod_of_nat_degree_le | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_multiset_prod :
t.prod.coeff 0 = (t.map (λ f, coeff f 0)).prod | begin
refine multiset.induction_on t _ (λ a t ht, _), { simp },
rw [multiset.prod_cons, multiset.map_cons, multiset.prod_cons, polynomial.mul_coeff_zero, ht]
end | lemma | polynomial.coeff_zero_multiset_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset.induction_on",
"multiset.map_cons",
"multiset.prod_cons",
"polynomial.mul_coeff_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_zero_prod :
(∏ i in s, f i).coeff 0 = ∏ i in s, (f i).coeff 0 | by simpa using coeff_zero_multiset_prod (s.1.map f) | lemma | polynomial.coeff_zero_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_X_sub_C_next_coeff (t : multiset R) :
next_coeff (t.map (λ x, X - C x)).prod = -t.sum | begin
rw next_coeff_multiset_prod,
{ simp only [next_coeff_X_sub_C],
exact t.sum_hom (-add_monoid_hom.id R) },
{ intros, apply monic_X_sub_C }
end | lemma | polynomial.multiset_prod_X_sub_C_next_coeff | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_C_next_coeff {s : finset ι} (f : ι → R) :
next_coeff ∏ i in s, (X - C (f i)) = -∑ i in s, f i | by simpa using multiset_prod_X_sub_C_next_coeff (s.1.map f) | lemma | polynomial.prod_X_sub_C_next_coeff | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiset_prod_X_sub_C_coeff_card_pred (t : multiset R) (ht : 0 < t.card) :
(t.map (λ x, (X - C x))).prod.coeff (t.card - 1) = -t.sum | begin
nontriviality R,
convert multiset_prod_X_sub_C_next_coeff (by assumption),
rw next_coeff, split_ifs,
{ rw nat_degree_multiset_prod_of_monic at h; simp only [multiset.mem_map] at *,
swap, { rintros _ ⟨_, _, rfl⟩, apply monic_X_sub_C },
simp_rw [multiset.sum_eq_zero_iff, multiset.mem_map] at h,
... | lemma | polynomial.multiset_prod_X_sub_C_coeff_card_pred | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_C_coeff_card_pred (s : finset ι) (f : ι → R) (hs : 0 < s.card) :
(∏ i in s, (X - C (f i))).coeff (s.card - 1) = - ∑ i in s, f i | by simpa using multiset_prod_X_sub_C_coeff_card_pred (s.1.map f) (by simpa using hs) | lemma | polynomial.prod_X_sub_C_coeff_card_pred | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_list_prod [nontrivial R] (l : list R[X]) :
l.prod.degree = (l.map degree).sum | map_list_prod (@degree_monoid_hom R _ _ _) l | lemma | polynomial.degree_list_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"map_list_prod",
"nontrivial"
] | The degree of a product of polynomials is equal to
the sum of the degrees, where the degree of the zero polynomial is ⊥.
`[nontrivial R]` is needed, otherwise for `l = []` we have `⊥` in the LHS and `0` in the RHS. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_prod (h : ∀ i ∈ s, f i ≠ 0) :
(∏ i in s, f i).nat_degree = ∑ i in s, (f i).nat_degree | begin
nontriviality R,
apply nat_degree_prod',
rw prod_ne_zero_iff,
intros x hx, simp [h x hx]
end | lemma | polynomial.nat_degree_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | The degree of a product of polynomials is equal to
the sum of the degrees.
See `polynomial.nat_degree_prod'` (with a `'`) for a version for commutative semirings,
where additionally, the product of the leading coefficients must be nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nat_degree_multiset_prod (h : (0 : R[X]) ∉ t) :
nat_degree t.prod = (t.map nat_degree).sum | begin
nontriviality R,
rw nat_degree_multiset_prod',
simp_rw [ne.def, multiset.prod_eq_zero_iff, multiset.mem_map, leading_coeff_eq_zero],
rintro ⟨_, h, rfl⟩,
contradiction
end | lemma | polynomial.nat_degree_multiset_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"multiset.mem_map",
"multiset.prod_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
degree_multiset_prod [nontrivial R] :
t.prod.degree = (t.map (λ f, degree f)).sum | map_multiset_prod (@degree_monoid_hom R _ _ _) _ | lemma | polynomial.degree_multiset_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"map_multiset_prod",
"nontrivial"
] | The degree of a product of polynomials is equal to
the sum of the degrees, where the degree of the zero polynomial is ⊥. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
degree_prod [nontrivial R] : (∏ i in s, f i).degree = ∑ i in s, (f i).degree | map_prod (@degree_monoid_hom R _ _ _) _ _ | lemma | polynomial.degree_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"map_prod",
"nontrivial"
] | The degree of a product of polynomials is equal to
the sum of the degrees, where the degree of the zero polynomial is ⊥. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
leading_coeff_multiset_prod :
t.prod.leading_coeff = (t.map (λ f, leading_coeff f)).prod | by { rw [← leading_coeff_hom_apply, monoid_hom.map_multiset_prod], refl } | lemma | polynomial.leading_coeff_multiset_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [
"monoid_hom.map_multiset_prod"
] | The leading coefficient of a product of polynomials is equal to
the product of the leading coefficients.
See `polynomial.leading_coeff_multiset_prod'` (with a `'`) for a version for commutative semirings,
where additionally, the product of the leading coefficients must be nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
leading_coeff_prod :
(∏ i in s, f i).leading_coeff = ∏ i in s, (f i).leading_coeff | by simpa using leading_coeff_multiset_prod (s.1.map f) | lemma | polynomial.leading_coeff_prod | algebra.polynomial | src/algebra/polynomial/big_operators.lean | [
"algebra.order.with_zero",
"data.polynomial.monic"
] | [] | The leading coefficient of a product of polynomials is equal to
the product of the leading coefficients.
See `polynomial.leading_coeff_prod'` (with a `'`) for a version for commutative semirings,
where additionally, the product of the leading coefficients must be nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_eq_map [mul_semiring_action M R] (m : M) :
((•) m) = map (mul_semiring_action.to_ring_hom M R m) | begin
suffices :
distrib_mul_action.to_add_monoid_hom R[X] m =
(map_ring_hom (mul_semiring_action.to_ring_hom M R m)).to_add_monoid_hom,
{ ext1 r, exact add_monoid_hom.congr_fun this r, },
ext n r : 2,
change m • monomial n r = map (mul_semiring_action.to_ring_hom M R m) (monomial n r),
simpa only [... | lemma | polynomial.smul_eq_map | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"distrib_mul_action.to_add_monoid_hom",
"mul_semiring_action",
"mul_semiring_action.to_ring_hom",
"polynomial.map_monomial",
"polynomial.smul_monomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_X (m : M) : (m • X : R[X]) = X | (smul_eq_map R m).symm ▸ map_X _ | lemma | polynomial.smul_X | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eval_smul (m : M) (f : S[X]) (x : S) :
(m • f).eval (m • x) = m • f.eval x | polynomial.induction_on f
(λ r, by rw [smul_C, eval_C, eval_C])
(λ f g ihf ihg, by rw [smul_add, eval_add, ihf, ihg, eval_add, smul_add])
(λ n r ih, by rw [smul_mul', smul_pow', smul_C, smul_X, eval_mul, eval_C, eval_pow, eval_X,
eval_mul, eval_C, eval_pow, eval_X, smul_mul', smul_pow']) | theorem | polynomial.smul_eval_smul | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"ih",
"polynomial.induction_on",
"smul_add",
"smul_mul'",
"smul_pow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_smul' [mul_semiring_action G S] (g : G) (f : S[X]) (x : S) :
f.eval (g • x) = g • (g⁻¹ • f).eval x | by rw [← smul_eval_smul, smul_inv_smul] | theorem | polynomial.eval_smul' | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"mul_semiring_action",
"smul_inv_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eval [mul_semiring_action G S] (g : G) (f : S[X]) (x : S) :
(g • f).eval x = g • f.eval (g⁻¹ • x) | by rw [← smul_eval_smul, smul_inv_smul] | theorem | polynomial.smul_eval | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"mul_semiring_action",
"smul_inv_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_smul (x : R) : R[X] | (finset.univ : finset (G ⧸ mul_action.stabilizer G x)).prod $
λ g, polynomial.X - polynomial.C (of_quotient_stabilizer G x g) | def | prod_X_sub_smul | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"finset",
"finset.univ",
"mul_action.stabilizer",
"polynomial.C",
"polynomial.X"
] | the product of `(X - g • x)` over distinct `g • x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_X_sub_smul.monic (x : R) : (prod_X_sub_smul G R x).monic | polynomial.monic_prod_of_monic _ _ $ λ g _, polynomial.monic_X_sub_C _ | theorem | prod_X_sub_smul.monic | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"polynomial.monic_X_sub_C",
"polynomial.monic_prod_of_monic",
"prod_X_sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_smul.eval (x : R) : (prod_X_sub_smul G R x).eval x = 0 | (monoid_hom.map_prod
((polynomial.aeval x).to_ring_hom.to_monoid_hom : R[X] →* R) _ _).trans $
finset.prod_eq_zero (finset.mem_univ $ quotient_group.mk 1) $
by simp | theorem | prod_X_sub_smul.eval | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"finset.mem_univ",
"finset.prod_eq_zero",
"monoid_hom.map_prod",
"polynomial.aeval",
"prod_X_sub_smul",
"quotient_group.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_smul.smul (x : R) (g : G) :
g • prod_X_sub_smul G R x = prod_X_sub_smul G R x | finset.smul_prod.trans $ fintype.prod_bijective _ (mul_action.bijective g) _ _
(λ g', by rw [of_quotient_stabilizer_smul, smul_sub, polynomial.smul_X, polynomial.smul_C]) | theorem | prod_X_sub_smul.smul | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"fintype.prod_bijective",
"mul_action.bijective",
"polynomial.smul_C",
"polynomial.smul_X",
"prod_X_sub_smul",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_X_sub_smul.coeff (x : R) (g : G) (n : ℕ) :
g • (prod_X_sub_smul G R x).coeff n =
(prod_X_sub_smul G R x).coeff n | by rw [← polynomial.coeff_smul, prod_X_sub_smul.smul] | theorem | prod_X_sub_smul.coeff | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"polynomial.coeff_smul",
"prod_X_sub_smul",
"prod_X_sub_smul.smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial (g : P →+*[M] Q) : P[X] →+*[M] Q[X] | { to_fun := map g,
map_smul' := λ m p, polynomial.induction_on p
(λ b, by rw [smul_C, map_C, coe_fn_coe, g.map_smul, map_C, coe_fn_coe, smul_C])
(λ p q ihp ihq, by rw [smul_add, polynomial.map_add, ihp, ihq, polynomial.map_add, smul_add])
(λ n b ih, by rw [smul_mul', smul_C, smul_pow', smul_X, polynomial.... | def | mul_semiring_action_hom.polynomial | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [
"ih",
"polynomial",
"polynomial.induction_on",
"polynomial.map_add",
"polynomial.map_mul",
"polynomial.map_one",
"polynomial.map_pow",
"polynomial.map_zero",
"smul_add",
"smul_mul'",
"smul_pow'"
] | An equivariant map induces an equivariant map on polynomials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_polynomial (g : P →+*[M] Q) :
(g.polynomial : P[X] → Q[X]) = map g | rfl | theorem | mul_semiring_action_hom.coe_polynomial | algebra.polynomial | src/algebra/polynomial/group_ring_action.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.group_action",
"data.polynomial.algebra_map",
"data.polynomial.monic",
"group_theory.group_action.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_regular (c : R) | ((*) c).injective | def | is_left_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [] | A left-regular element is an element `c` such that multiplication on the left by `c`
is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular (c : R) | (* c).injective | def | is_right_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [] | A right-regular element is an element `c` such that multiplication on the right by `c`
is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_add_regular {R : Type*} [has_add R] (c : R) : Prop | (left : is_add_left_regular c)
(right : is_add_right_regular c) | structure | is_add_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [] | An add-regular element is an element `c` such that addition by `c` both on the left and
on the right is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular (c : R) : Prop | (left : is_left_regular c)
(right : is_right_regular c) | structure | is_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"is_right_regular"
] | A regular element is an element `c` such that multiplication by `c` both on the left and
on the right is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_le_cancellable.is_left_regular [partial_order R] {a : R}
(ha : mul_le_cancellable a) : is_left_regular a | ha.injective | lemma | mul_le_cancellable.is_left_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"mul_le_cancellable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_regular.right_of_commute {a : R} (ca : ∀ b, commute a b)
(h : is_left_regular a) : is_right_regular a | λ x y xy, h $ (ca x).trans $ xy.trans $ (ca y).symm | lemma | is_left_regular.right_of_commute | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"commute",
"is_left_regular",
"is_right_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.is_regular_iff {a : R} (ca : ∀ b, commute a b) :
is_regular a ↔ is_left_regular a | ⟨λ h, h.left, λ h, ⟨h, h.right_of_commute ca⟩⟩ | lemma | commute.is_regular_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"commute",
"is_left_regular",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_regular.mul (lra : is_left_regular a) (lrb : is_left_regular b) :
is_left_regular (a * b) | show function.injective ((*) (a * b)), from (comp_mul_left a b) ▸ lra.comp lrb | lemma | is_left_regular.mul | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"comp_mul_left",
"is_left_regular"
] | In a semigroup, the product of left-regular elements is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.mul (rra : is_right_regular a) (rrb : is_right_regular b) :
is_right_regular (a * b) | show function.injective (* (a * b)), from (comp_mul_right b a) ▸ rrb.comp rra | lemma | is_right_regular.mul | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"comp_mul_right",
"is_right_regular"
] | In a semigroup, the product of right-regular elements is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.of_mul (ab : is_left_regular (a * b)) :
is_left_regular b | function.injective.of_comp (by rwa comp_mul_left a b) | lemma | is_left_regular.of_mul | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"comp_mul_left",
"function.injective.of_comp",
"is_left_regular"
] | If an element `b` becomes left-regular after multiplying it on the left by a left-regular
element, then `b` is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_is_left_regular_iff (b : R) (ha : is_left_regular a) :
is_left_regular (a * b) ↔ is_left_regular b | ⟨λ ab, is_left_regular.of_mul ab, λ ab, is_left_regular.mul ha ab⟩ | lemma | mul_is_left_regular_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"is_left_regular.mul",
"is_left_regular.of_mul"
] | An element is left-regular if and only if multiplying it on the left by a left-regular element
is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.of_mul (ab : is_right_regular (b * a)) :
is_right_regular b | begin
refine λ x y xy, ab (_ : x * (b * a) = y * (b * a)),
rw [← mul_assoc, ← mul_assoc],
exact congr_fun (congr_arg (*) xy) a,
end | lemma | is_right_regular.of_mul | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"mul_assoc"
] | If an element `b` becomes right-regular after multiplying it on the right by a right-regular
element, then `b` is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_is_right_regular_iff (b : R) (ha : is_right_regular a) :
is_right_regular (b * a) ↔ is_right_regular b | ⟨λ ab, is_right_regular.of_mul ab, λ ab, is_right_regular.mul ab ha⟩ | lemma | mul_is_right_regular_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"is_right_regular.mul",
"is_right_regular.of_mul"
] | An element is right-regular if and only if multiplying it on the right with a right-regular
element is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_mul_and_mul_iff :
is_regular (a * b) ∧ is_regular (b * a) ↔ is_regular a ∧ is_regular b | begin
refine ⟨_, _⟩,
{ rintros ⟨ab, ba⟩,
exact ⟨⟨is_left_regular.of_mul ba.left, is_right_regular.of_mul ab.right⟩,
⟨is_left_regular.of_mul ab.left, is_right_regular.of_mul ba.right⟩⟩ },
{ rintros ⟨ha, hb⟩,
exact ⟨⟨(mul_is_left_regular_iff _ ha.left).mpr hb.left,
(mul_is_right_regular_iff _ ... | lemma | is_regular_mul_and_mul_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"is_right_regular.of_mul",
"mul_is_left_regular_iff",
"mul_is_right_regular_iff"
] | Two elements `a` and `b` are regular if and only if both products `a * b` and `b * a`
are regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.and_of_mul_of_mul (ab : is_regular (a * b)) (ba : is_regular (b * a)) :
is_regular a ∧ is_regular b | is_regular_mul_and_mul_iff.mp ⟨ab, ba⟩ | lemma | is_regular.and_of_mul_of_mul | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular"
] | The "most used" implication of `mul_and_mul_iff`, with split hypotheses, instead of `∧`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.subsingleton (h : is_left_regular (0 : R)) : subsingleton R | ⟨λ a b, h $ eq.trans (zero_mul a) (zero_mul b).symm⟩ | lemma | is_left_regular.subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"zero_mul"
] | The element `0` is left-regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.subsingleton (h : is_right_regular (0 : R)) : subsingleton R | ⟨λ a b, h $ eq.trans (mul_zero a) (mul_zero b).symm⟩ | lemma | is_right_regular.subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"mul_zero"
] | The element `0` is right-regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.subsingleton (h : is_regular (0 : R)) : subsingleton R | h.left.subsingleton | lemma | is_regular.subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular"
] | The element `0` is regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular_zero_iff_subsingleton : is_left_regular (0 : R) ↔ subsingleton R | ⟨λ h, h.subsingleton, λ H a b h, @subsingleton.elim _ H a b⟩ | lemma | is_left_regular_zero_iff_subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular"
] | The element `0` is left-regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_left_regular_zero_iff : ¬ is_left_regular (0 : R) ↔ nontrivial R | begin
rw [nontrivial_iff, not_iff_comm, is_left_regular_zero_iff_subsingleton, subsingleton_iff],
push_neg,
exact iff.rfl
end | lemma | not_is_left_regular_zero_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"is_left_regular_zero_iff_subsingleton",
"nontrivial",
"nontrivial_iff",
"not_iff_comm",
"subsingleton_iff"
] | In a non-trivial `mul_zero_class`, the `0` element is not left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular_zero_iff_subsingleton : is_right_regular (0 : R) ↔ subsingleton R | ⟨λ h, h.subsingleton, λ H a b h, @subsingleton.elim _ H a b⟩ | lemma | is_right_regular_zero_iff_subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular"
] | The element `0` is right-regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_right_regular_zero_iff : ¬ is_right_regular (0 : R) ↔ nontrivial R | begin
rw [nontrivial_iff, not_iff_comm, is_right_regular_zero_iff_subsingleton, subsingleton_iff],
push_neg,
exact iff.rfl
end | lemma | not_is_right_regular_zero_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"is_right_regular_zero_iff_subsingleton",
"nontrivial",
"nontrivial_iff",
"not_iff_comm",
"subsingleton_iff"
] | In a non-trivial `mul_zero_class`, the `0` element is not right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_iff_subsingleton : is_regular (0 : R) ↔ subsingleton R | ⟨λ h, h.left.subsingleton,
λ h, ⟨is_left_regular_zero_iff_subsingleton.mpr h, is_right_regular_zero_iff_subsingleton.mpr h⟩⟩ | lemma | is_regular_iff_subsingleton | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular"
] | The element `0` is regular if and only if `R` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.ne_zero [nontrivial R] (la : is_left_regular a) : a ≠ 0 | begin
rintro rfl,
rcases exists_pair_ne R with ⟨x, y, xy⟩,
refine xy (la _),
rw [zero_mul, zero_mul]
end | lemma | is_left_regular.ne_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"exists_pair_ne",
"is_left_regular",
"nontrivial",
"zero_mul"
] | A left-regular element of a `nontrivial` `mul_zero_class` is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.ne_zero [nontrivial R] (ra : is_right_regular a) : a ≠ 0 | begin
rintro rfl,
rcases exists_pair_ne R with ⟨x, y, xy⟩,
refine xy (ra (_ : x * 0 = y * 0)),
rw [mul_zero, mul_zero]
end | lemma | is_right_regular.ne_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"exists_pair_ne",
"is_right_regular",
"mul_zero",
"nontrivial"
] | A right-regular element of a `nontrivial` `mul_zero_class` is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.ne_zero [nontrivial R] (la : is_regular a) : a ≠ 0 | la.left.ne_zero | lemma | is_regular.ne_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"nontrivial"
] | A regular element of a `nontrivial` `mul_zero_class` is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_left_regular_zero [nR : nontrivial R] : ¬ is_left_regular (0 : R) | not_is_left_regular_zero_iff.mpr nR | lemma | not_is_left_regular_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"nontrivial"
] | In a non-trivial ring, the element `0` is not left-regular -- with typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_right_regular_zero [nR : nontrivial R] : ¬ is_right_regular (0 : R) | not_is_right_regular_zero_iff.mpr nR | lemma | not_is_right_regular_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"nontrivial"
] | In a non-trivial ring, the element `0` is not right-regular -- with typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_is_regular_zero [nontrivial R] : ¬ is_regular (0 : R) | λ h, is_regular.ne_zero h rfl | lemma | not_is_regular_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"is_regular.ne_zero",
"nontrivial"
] | In a non-trivial ring, the element `0` is not regular -- with typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_one : is_regular (1 : R) | ⟨λ a b ab, (one_mul a).symm.trans (eq.trans ab (one_mul b)),
λ a b ab, (mul_one a).symm.trans (eq.trans ab (mul_one b))⟩ | lemma | is_regular_one | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"mul_one",
"one_mul"
] | If multiplying by `1` on either side is the identity, `1` is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_mul_iff : is_regular (a * b) ↔ is_regular a ∧ is_regular b | begin
refine iff.trans _ is_regular_mul_and_mul_iff,
refine ⟨λ ab, ⟨ab, by rwa mul_comm⟩, λ rab, rab.1⟩
end | lemma | is_regular_mul_iff | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"is_regular_mul_and_mul_iff"
] | A product is regular if and only if the factors are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular_of_mul_eq_one (h : b * a = 1) : is_left_regular a | @is_left_regular.of_mul R _ _ _ (by { rw h, exact is_regular_one.left }) | lemma | is_left_regular_of_mul_eq_one | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"is_left_regular.of_mul"
] | An element admitting a left inverse is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular_of_mul_eq_one (h : a * b = 1) : is_right_regular a | is_right_regular.of_mul (by { rw h, exact is_regular_one.right }) | lemma | is_right_regular_of_mul_eq_one | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"is_right_regular.of_mul"
] | An element admitting a right inverse is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.is_regular (a : Rˣ) : is_regular (a : R) | ⟨is_left_regular_of_mul_eq_one a.inv_mul, is_right_regular_of_mul_eq_one a.mul_inv⟩ | lemma | units.is_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"is_right_regular_of_mul_eq_one"
] | If `R` is a monoid, an element in `Rˣ` is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.is_regular (ua : is_unit a) : is_regular a | begin
rcases ua with ⟨a, rfl⟩,
exact units.is_regular a,
end | lemma | is_unit.is_regular | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"is_unit",
"units.is_regular"
] | A unit in a monoid is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular_of_left_cancel_semigroup [left_cancel_semigroup R] (g : R) :
is_left_regular g | mul_right_injective g | lemma | is_left_regular_of_left_cancel_semigroup | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_left_regular",
"left_cancel_semigroup",
"mul_right_injective"
] | Elements of a left cancel semigroup are left regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular_of_right_cancel_semigroup [right_cancel_semigroup R] (g : R) :
is_right_regular g | mul_left_injective g | lemma | is_right_regular_of_right_cancel_semigroup | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_right_regular",
"mul_left_injective",
"right_cancel_semigroup"
] | Elements of a right cancel semigroup are right regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_of_cancel_monoid (g : R) : is_regular g | ⟨mul_right_injective g, mul_left_injective g⟩ | lemma | is_regular_of_cancel_monoid | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"mul_left_injective"
] | Elements of a cancel monoid are regular. Cancel semigroups do not appear to exist. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_of_ne_zero (a0 : a ≠ 0) : is_regular a | ⟨λ b c, (mul_right_inj' a0).mp, λ b c, (mul_left_inj' a0).mp⟩ | lemma | is_regular_of_ne_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"mul_left_inj'",
"mul_right_inj'"
] | Non-zero elements of an integral domain are regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular_iff_ne_zero [nontrivial R] : is_regular a ↔ a ≠ 0 | ⟨is_regular.ne_zero, is_regular_of_ne_zero⟩ | lemma | is_regular_iff_ne_zero | algebra.regular | src/algebra/regular/basic.lean | [
"algebra.group.commute",
"algebra.order.monoid.lemmas",
"algebra.group_with_zero.basic"
] | [
"is_regular",
"nontrivial"
] | In a non-trivial integral domain, an element is regular iff it is non-zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.pow (n : ℕ) (rla : is_left_regular a) : is_left_regular (a ^ n) | by simp only [is_left_regular, ← mul_left_iterate, rla.iterate n] | lemma | is_left_regular.pow | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_left_regular",
"mul_left_iterate"
] | Any power of a left-regular element is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.pow (n : ℕ) (rra : is_right_regular a) : is_right_regular (a ^ n) | by { rw [is_right_regular, ← mul_right_iterate], exact rra.iterate n } | lemma | is_right_regular.pow | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_right_regular",
"mul_right_iterate"
] | Any power of a right-regular element is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.pow (n : ℕ) (ra : is_regular a) : is_regular (a ^ n) | ⟨is_left_regular.pow n ra.left, is_right_regular.pow n ra.right⟩ | lemma | is_regular.pow | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_regular",
"is_right_regular.pow"
] | Any power of a regular element is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.pow_iff {n : ℕ} (n0 : 0 < n) :
is_left_regular (a ^ n) ↔ is_left_regular a | begin
refine ⟨_, is_left_regular.pow n⟩,
rw [← nat.succ_pred_eq_of_pos n0, pow_succ'],
exact is_left_regular.of_mul,
end | lemma | is_left_regular.pow_iff | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_left_regular",
"is_left_regular.of_mul",
"is_left_regular.pow",
"pow_succ'"
] | An element `a` is left-regular if and only if a positive power of `a` is left-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.pow_iff {n : ℕ} (n0 : 0 < n) :
is_right_regular (a ^ n) ↔ is_right_regular a | begin
refine ⟨_, is_right_regular.pow n⟩,
rw [← nat.succ_pred_eq_of_pos n0, pow_succ],
exact is_right_regular.of_mul,
end | lemma | is_right_regular.pow_iff | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_right_regular",
"is_right_regular.of_mul",
"is_right_regular.pow",
"pow_succ"
] | An element `a` is right-regular if and only if a positive power of `a` is right-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.pow_iff {n : ℕ} (n0 : 0 < n) :
is_regular (a ^ n) ↔ is_regular a | ⟨λ h, ⟨(is_left_regular.pow_iff n0).mp h.left, (is_right_regular.pow_iff n0).mp h.right⟩,
λ h, ⟨is_left_regular.pow n h.left, is_right_regular.pow n h.right⟩⟩ | lemma | is_regular.pow_iff | algebra.regular | src/algebra/regular/pow.lean | [
"algebra.hom.iterate",
"algebra.regular.basic"
] | [
"is_left_regular.pow_iff",
"is_regular",
"is_right_regular.pow",
"is_right_regular.pow_iff"
] | An element `a` is regular if and only if a positive power of `a` is regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_smul_regular [has_smul R M] (c : R) | function.injective ((•) c : M → M) | def | is_smul_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"has_smul"
] | An `M`-regular element is an element `c` such that multiplication on the left by `c` is an
injective map `M → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular.is_smul_regular [has_mul R] {c : R} (h : is_left_regular c) :
is_smul_regular R c | h | lemma | is_left_regular.is_smul_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_left_regular",
"is_smul_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_left_regular_iff [has_mul R] {a : R} :
is_left_regular a ↔ is_smul_regular R a | iff.rfl | lemma | is_left_regular_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_left_regular",
"is_smul_regular"
] | Left-regular multiplication on `R` is equivalent to `R`-regularity of `R` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_right_regular.is_smul_regular [has_mul R] {c : R} (h : is_right_regular c) :
is_smul_regular R (mul_opposite.op c) | h | lemma | is_right_regular.is_smul_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_right_regular",
"is_smul_regular",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_regular_iff [has_mul R] {a : R} :
is_right_regular a ↔ is_smul_regular R (mul_opposite.op a) | iff.rfl | lemma | is_right_regular_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_right_regular",
"is_smul_regular",
"mul_opposite.op"
] | Right-regular multiplication on `R` is equivalent to `Rᵐᵒᵖ`-regularity of `R` itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul (ra : is_smul_regular M a) (rs : is_smul_regular M s) :
is_smul_regular M (a • s) | λ a b ab, rs (ra ((smul_assoc _ _ _).symm.trans (ab.trans (smul_assoc _ _ _)))) | lemma | is_smul_regular.smul | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"smul_assoc"
] | The product of `M`-regular elements is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_smul (a : R) (ab : is_smul_regular M (a • s)) :
is_smul_regular M s | @function.injective.of_comp _ _ _ (λ m : M, a • m) _ (λ c d cd, ab
(by rwa [smul_assoc, smul_assoc])) | lemma | is_smul_regular.of_smul | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"function.injective.of_comp",
"is_smul_regular",
"smul_assoc"
] | If an element `b` becomes `M`-regular after multiplying it on the left by an `M`-regular
element, then `b` is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_iff (b : S) (ha : is_smul_regular M a) :
is_smul_regular M (a • b) ↔ is_smul_regular M b | ⟨of_smul _, ha.smul⟩ | lemma | is_smul_regular.smul_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | An element is `M`-regular if and only if multiplying it on the left by an `M`-regular element
is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_left_regular [has_mul R] {a : R} (h : is_smul_regular R a) :
is_left_regular a | h | lemma | is_smul_regular.is_left_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_left_regular",
"is_smul_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_right_regular [has_mul R] {a : R} (h : is_smul_regular R (mul_opposite.op a)) :
is_right_regular a | h | lemma | is_smul_regular.is_right_regular | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_right_regular",
"is_smul_regular",
"mul_opposite.op"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul [has_mul R] [is_scalar_tower R R M]
(ra : is_smul_regular M a) (rb : is_smul_regular M b) : is_smul_regular M (a * b) | ra.smul rb | lemma | is_smul_regular.mul | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_scalar_tower",
"is_smul_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul [has_mul R] [is_scalar_tower R R M] (ab : is_smul_regular M (a * b)) :
is_smul_regular M b | by { rw ← smul_eq_mul at ab, exact ab.of_smul _ } | lemma | is_smul_regular.of_mul | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_scalar_tower",
"is_smul_regular",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_iff_right [has_mul R] [is_scalar_tower R R M] (ha : is_smul_regular M a) :
is_smul_regular M (a * b) ↔ is_smul_regular M b | ⟨of_mul, ha.mul⟩ | lemma | is_smul_regular.mul_iff_right | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_scalar_tower",
"is_smul_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_and_mul_iff [has_mul R] [is_scalar_tower R R M] :
is_smul_regular M (a * b) ∧ is_smul_regular M (b * a) ↔
is_smul_regular M a ∧ is_smul_regular M b | begin
refine ⟨_, _⟩,
{ rintros ⟨ab, ba⟩,
refine ⟨ba.of_mul, ab.of_mul⟩ },
{ rintros ⟨ha, hb⟩,
exact ⟨ha.mul hb, hb.mul ha⟩ }
end | lemma | is_smul_regular.mul_and_mul_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_scalar_tower",
"is_smul_regular"
] | Two elements `a` and `b` are `M`-regular if and only if both products `a * b` and `b * a`
are `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one : is_smul_regular M (1 : R) | λ a b ab, by rwa [one_smul, one_smul] at ab | lemma | is_smul_regular.one | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"one_smul"
] | One is `M`-regular always. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_eq_one (h : a * b = 1) : is_smul_regular M b | of_mul (by { rw h, exact one M }) | lemma | is_smul_regular.of_mul_eq_one | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | An element of `R` admitting a left inverse is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow (n : ℕ) (ra : is_smul_regular M a) : is_smul_regular M (a ^ n) | begin
induction n with n hn,
{ simp only [one, pow_zero] },
{ rw pow_succ, exact (ra.smul_iff (a ^ n)).mpr hn }
end | lemma | is_smul_regular.pow | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"pow_succ",
"pow_zero"
] | Any power of an `M`-regular element is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_iff {n : ℕ} (n0 : 0 < n) :
is_smul_regular M (a ^ n) ↔ is_smul_regular M a | begin
refine ⟨_, pow n⟩,
rw [← nat.succ_pred_eq_of_pos n0, pow_succ', ← smul_eq_mul],
exact of_smul _,
end | lemma | is_smul_regular.pow_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"pow_succ'",
"smul_eq_mul"
] | An element `a` is `M`-regular if and only if a positive power of `a` is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_smul_eq_one (h : a • s = 1) : is_smul_regular M s | of_smul a (by { rw h, exact one M }) | lemma | is_smul_regular.of_smul_eq_one | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | An element of `S` admitting a left inverse in `R` is `M`-regular. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton (h : is_smul_regular M (0 : R)) : subsingleton M | ⟨λ a b, h (by repeat { rw mul_action_with_zero.zero_smul })⟩ | lemma | is_smul_regular.subsingleton | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | The element `0` is `M`-regular if and only if `M` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_iff_subsingleton : is_smul_regular M (0 : R) ↔ subsingleton M | ⟨λ h, h.subsingleton, λ H a b h, @subsingleton.elim _ H a b⟩ | lemma | is_smul_regular.zero_iff_subsingleton | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | The element `0` is `M`-regular if and only if `M` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_zero_iff : ¬ is_smul_regular M (0 : R) ↔ nontrivial M | begin
rw [nontrivial_iff, not_iff_comm, zero_iff_subsingleton, subsingleton_iff],
push_neg,
exact iff.rfl
end | lemma | is_smul_regular.not_zero_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"nontrivial",
"nontrivial_iff",
"not_iff_comm",
"subsingleton_iff"
] | The `0` element is not `M`-regular, on a non-trivial module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero [sM : subsingleton M] : is_smul_regular M (0 : R) | zero_iff_subsingleton.mpr sM | lemma | is_smul_regular.zero | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | The element `0` is `M`-regular when `M` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_zero [nM : nontrivial M] : ¬ is_smul_regular M (0 : R) | not_zero_iff.mpr nM | lemma | is_smul_regular.not_zero | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular",
"nontrivial"
] | The `0` element is not `M`-regular, on a non-trivial module. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_iff : is_smul_regular M (a * b) ↔
is_smul_regular M a ∧ is_smul_regular M b | begin
rw ← mul_and_mul_iff,
exact ⟨λ ab, ⟨ab, by rwa mul_comm⟩, λ rab, rab.1⟩
end | lemma | is_smul_regular.mul_iff | algebra.regular | src/algebra/regular/smul.lean | [
"algebra.smul_with_zero",
"algebra.regular.basic"
] | [
"is_smul_regular"
] | A product is `M`-regular if and only if the factors are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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