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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|---|---|---|---|---|---|---|---|---|---|---|
constr_pow_gen (pb : power_basis A S) {y : S'}
(hy : aeval y (minpoly A pb.gen) = 0) :
pb.basis.constr A (λ i, y ^ (i : ℕ)) pb.gen = y | by { convert pb.constr_pow_aeval hy X; rw aeval_X } | lemma | power_basis.constr_pow_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constr_pow_algebra_map (pb : power_basis A S) {y : S'}
(hy : aeval y (minpoly A pb.gen) = 0) (x : A) :
pb.basis.constr A (λ i, y ^ (i : ℕ)) (algebra_map A S x) = algebra_map A S' x | by { convert pb.constr_pow_aeval hy (C x); rw aeval_C } | lemma | power_basis.constr_pow_algebra_map | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra_map",
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
constr_pow_mul (pb : power_basis A S) {y : S'}
(hy : aeval y (minpoly A pb.gen) = 0) (x x' : S) :
pb.basis.constr A (λ i, y ^ (i : ℕ)) (x * x') =
pb.basis.constr A (λ i, y ^ (i : ℕ)) x * pb.basis.constr A (λ i, y ^ (i : ℕ)) x' | begin
obtain ⟨f, rfl⟩ := pb.exists_eq_aeval' x,
obtain ⟨g, rfl⟩ := pb.exists_eq_aeval' x',
simp only [← aeval_mul, pb.constr_pow_aeval hy]
end | lemma | power_basis.constr_pow_mul | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (pb : power_basis A S) (y : S')
(hy : aeval y (minpoly A pb.gen) = 0) :
S →ₐ[A] S' | { map_one' := by { convert pb.constr_pow_algebra_map hy 1 using 2; rw ring_hom.map_one },
map_zero' := by { convert pb.constr_pow_algebra_map hy 0 using 2; rw ring_hom.map_zero },
map_mul' := pb.constr_pow_mul hy,
commutes' := pb.constr_pow_algebra_map hy,
.. pb.basis.constr A (λ i, y ^ (i : ℕ)) } | def | power_basis.lift | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"lift",
"minpoly",
"power_basis",
"ring_hom.map_one",
"ring_hom.map_zero"
] | `pb.lift y hy` is the algebra map sending `pb.gen` to `y`,
where `hy` states the higher powers of `y` are the same as the higher powers of `pb.gen`.
See `power_basis.lift_equiv` for a bundled equiv sending `⟨y, hy⟩` to the algebra map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_gen (pb : power_basis A S) (y : S')
(hy : aeval y (minpoly A pb.gen) = 0) :
pb.lift y hy pb.gen = y | pb.constr_pow_gen hy | lemma | power_basis.lift_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_aeval (pb : power_basis A S) (y : S')
(hy : aeval y (minpoly A pb.gen) = 0) (f : A[X]) :
pb.lift y hy (aeval pb.gen f) = aeval y f | pb.constr_pow_aeval hy f | lemma | power_basis.lift_aeval | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_equiv (pb : power_basis A S) :
(S →ₐ[A] S') ≃ {y : S' // aeval y (minpoly A pb.gen) = 0} | { to_fun := λ f, ⟨f pb.gen, by rw [aeval_alg_hom_apply, minpoly.aeval, f.map_zero]⟩,
inv_fun := λ y, pb.lift y y.2,
left_inv := λ f, pb.alg_hom_ext $ lift_gen _ _ _,
right_inv := λ y, subtype.ext $ lift_gen _ _ y.prop } | def | power_basis.lift_equiv | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"inv_fun",
"minpoly",
"minpoly.aeval",
"power_basis",
"subtype.ext"
] | `pb.lift_equiv` states that roots of the minimal polynomial of `pb.gen` correspond to
maps sending `pb.gen` to that root.
This is the bundled equiv version of `power_basis.lift`.
If the codomain of the `alg_hom`s is an integral domain, then the roots form a multiset,
see `lift_equiv'` for the corresponding statement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_equiv' (pb : power_basis A S) :
(S →ₐ[A] B) ≃ {y : B // y ∈ ((minpoly A pb.gen).map (algebra_map A B)).roots} | pb.lift_equiv.trans ((equiv.refl _).subtype_equiv (λ x,
begin
rw [mem_roots, is_root.def, equiv.refl_apply, ← eval₂_eq_eval_map, ← aeval_def],
exact map_monic_ne_zero (minpoly.monic pb.is_integral_gen)
end)) | def | power_basis.lift_equiv' | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra_map",
"equiv.refl",
"equiv.refl_apply",
"minpoly",
"minpoly.monic",
"power_basis"
] | `pb.lift_equiv'` states that elements of the root set of the minimal
polynomial of `pb.gen` correspond to maps sending `pb.gen` to that root. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom.fintype (pb : power_basis A S) :
fintype (S →ₐ[A] B) | by letI := classical.dec_eq B; exact
fintype.of_equiv _ pb.lift_equiv'.symm | def | power_basis.alg_hom.fintype | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"classical.dec_eq",
"fintype",
"fintype.of_equiv",
"power_basis"
] | There are finitely many algebra homomorphisms `S →ₐ[A] B` if `S` is of the form `A[x]`
and `B` is an integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_root
(pb : power_basis A S) (pb' : power_basis A S')
(h₁ : aeval pb.gen (minpoly A pb'.gen) = 0) (h₂ : aeval pb'.gen (minpoly A pb.gen) = 0) :
S ≃ₐ[A] S' | alg_equiv.of_alg_hom
(pb.lift pb'.gen h₂)
(pb'.lift pb.gen h₁)
(by { ext x, obtain ⟨f, hf, rfl⟩ := pb'.exists_eq_aeval' x, simp })
(by { ext x, obtain ⟨f, hf, rfl⟩ := pb.exists_eq_aeval' x, simp }) | def | power_basis.equiv_of_root | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_equiv.of_alg_hom",
"minpoly",
"power_basis"
] | `pb.equiv_of_root pb' h₁ h₂` is an equivalence of algebras with the same power basis,
where "the same" means that `pb` is a root of `pb'`s minimal polynomial and vice versa.
See also `power_basis.equiv_of_minpoly` which takes the hypothesis that the
minimal polynomials are identical. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_root_aeval
(pb : power_basis A S) (pb' : power_basis A S')
(h₁ : aeval pb.gen (minpoly A pb'.gen) = 0) (h₂ : aeval pb'.gen (minpoly A pb.gen) = 0)
(f : A[X]) :
pb.equiv_of_root pb' h₁ h₂ (aeval pb.gen f) = aeval pb'.gen f | pb.lift_aeval _ h₂ _ | lemma | power_basis.equiv_of_root_aeval | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_root_gen
(pb : power_basis A S) (pb' : power_basis A S')
(h₁ : aeval pb.gen (minpoly A pb'.gen) = 0) (h₂ : aeval pb'.gen (minpoly A pb.gen) = 0) :
pb.equiv_of_root pb' h₁ h₂ pb.gen = pb'.gen | pb.lift_gen _ h₂ | lemma | power_basis.equiv_of_root_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_root_symm
(pb : power_basis A S) (pb' : power_basis A S')
(h₁ : aeval pb.gen (minpoly A pb'.gen) = 0) (h₂ : aeval pb'.gen (minpoly A pb.gen) = 0) :
(pb.equiv_of_root pb' h₁ h₂).symm = pb'.equiv_of_root pb h₂ h₁ | rfl | lemma | power_basis.equiv_of_root_symm | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_minpoly
(pb : power_basis A S) (pb' : power_basis A S')
(h : minpoly A pb.gen = minpoly A pb'.gen) :
S ≃ₐ[A] S' | pb.equiv_of_root pb' (h ▸ minpoly.aeval _ _) (h.symm ▸ minpoly.aeval _ _) | def | power_basis.equiv_of_minpoly | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"minpoly.aeval",
"power_basis"
] | `pb.equiv_of_minpoly pb' h` is an equivalence of algebras with the same power basis,
where "the same" means that they have identical minimal polynomials.
See also `power_basis.equiv_of_root` which takes the hypothesis that each generator is a root of the
other basis' minimal polynomial; `power_basis.equiv_root` is mor... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_of_minpoly_aeval
(pb : power_basis A S) (pb' : power_basis A S')
(h : minpoly A pb.gen = minpoly A pb'.gen)
(f : A[X]) :
pb.equiv_of_minpoly pb' h (aeval pb.gen f) = aeval pb'.gen f | pb.equiv_of_root_aeval pb' _ _ _ | lemma | power_basis.equiv_of_minpoly_aeval | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_minpoly_gen
(pb : power_basis A S) (pb' : power_basis A S')
(h : minpoly A pb.gen = minpoly A pb'.gen) :
pb.equiv_of_minpoly pb' h pb.gen = pb'.gen | pb.equiv_of_root_gen pb' _ _ | lemma | power_basis.equiv_of_minpoly_gen | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_minpoly_symm
(pb : power_basis A S) (pb' : power_basis A S')
(h : minpoly A pb.gen = minpoly A pb'.gen) :
(pb.equiv_of_minpoly pb' h).symm = pb'.equiv_of_minpoly pb h.symm | rfl | lemma | power_basis.equiv_of_minpoly_symm | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_independent_pow [algebra K S] (x : S) :
linear_independent K (λ (i : fin (minpoly K x).nat_degree), x ^ (i : ℕ)) | begin
by_cases is_integral K x, swap,
{ rw [minpoly.eq_zero h, nat_degree_zero], exact linear_independent_empty_type },
refine fintype.linear_independent_iff.2 (λ g hg i, _),
simp only at hg,
simp_rw [algebra.smul_def, ← aeval_monomial, ← map_sum] at hg,
apply (λ hn0, (minpoly.degree_le_of_ne_zero K x (mt (... | lemma | linear_independent_pow | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"algebra",
"algebra.smul_def",
"fin.coe_eq_coe",
"finset.mem_univ",
"is_integral",
"linear_independent",
"linear_independent_empty_type",
"minpoly",
"minpoly.degree_le_of_ne_zero",
"minpoly.eq_zero",
"minpoly.ne_zero"
] | Useful lemma to show `x` generates a power basis:
the powers of `x` less than the degree of `x`'s minimal polynomial are linearly independent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_integral.mem_span_pow [nontrivial R] {x y : S} (hx : is_integral R x)
(hy : ∃ f : R[X], y = aeval x f) :
y ∈ submodule.span R (set.range (λ (i : fin (minpoly R x).nat_degree), x ^ (i : ℕ))) | begin
obtain ⟨f, rfl⟩ := hy,
apply mem_span_pow'.mpr _,
have := minpoly.monic hx,
refine ⟨f %ₘ minpoly R x, (degree_mod_by_monic_lt _ this).trans_le degree_le_nat_degree, _⟩,
conv_lhs { rw ← mod_by_monic_add_div f this },
simp only [add_zero, zero_mul, minpoly.aeval, aeval_add, alg_hom.map_mul]
end | lemma | is_integral.mem_span_pow | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_hom.map_mul",
"is_integral",
"minpoly",
"minpoly.aeval",
"minpoly.monic",
"nontrivial",
"set.range",
"submodule.span",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (pb : power_basis R S) (e : S ≃ₐ[R] S') : power_basis R S' | { dim := pb.dim,
basis := pb.basis.map e.to_linear_equiv,
gen := e pb.gen,
basis_eq_pow :=
λ i, by rw [basis.map_apply, pb.basis_eq_pow, e.to_linear_equiv_apply, e.map_pow] } | def | power_basis.map | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"basis",
"basis.map_apply",
"power_basis"
] | `power_basis.map pb (e : S ≃ₐ[R] S')` is the power basis for `S'` generated by `e pb.gen`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
minpoly_gen_map (pb : power_basis A S) (e : S ≃ₐ[A] S') :
(pb.map e).minpoly_gen = pb.minpoly_gen | by { dsimp only [minpoly_gen, map_dim], -- Turn `fin (pb.map e).dim` into `fin pb.dim`
simp only [linear_equiv.trans_apply, map_basis, basis.map_repr,
map_gen, alg_equiv.to_linear_equiv_apply, e.to_linear_equiv_symm, alg_equiv.map_pow,
alg_equiv.symm_apply_apply, sub_right_inj] } | lemma | power_basis.minpoly_gen_map | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"alg_equiv.map_pow",
"alg_equiv.symm_apply_apply",
"linear_equiv.trans_apply",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_root_map (pb : power_basis A S) (e : S ≃ₐ[A] S')
(h₁ h₂) :
pb.equiv_of_root (pb.map e) h₁ h₂ = e | by { ext x, obtain ⟨f, rfl⟩ := pb.exists_eq_aeval' x, simp [aeval_alg_equiv] } | lemma | power_basis.equiv_of_root_map | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_of_minpoly_map (pb : power_basis A S) (e : S ≃ₐ[A] S')
(h : minpoly A pb.gen = minpoly A (pb.map e).gen) :
pb.equiv_of_minpoly (pb.map e) h = e | pb.equiv_of_root_map _ _ _ | lemma | power_basis.equiv_of_minpoly_map | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"minpoly",
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_gen_eq_top (B : power_basis R S) : adjoin R ({B.gen} : set S) = ⊤ | begin
rw [← to_submodule_eq_top, _root_.eq_top_iff, ← B.basis.span_eq, submodule.span_le],
rintros x ⟨i, rfl⟩,
rw [B.basis_eq_pow i],
exact subalgebra.pow_mem _ (subset_adjoin (set.mem_singleton _)) _,
end | lemma | power_basis.adjoin_gen_eq_top | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis",
"set.mem_singleton",
"subalgebra.pow_mem",
"submodule.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_eq_top_of_gen_mem_adjoin {B : power_basis R S} {x : S}
(hx : B.gen ∈ adjoin R ({x} : set S)) : adjoin R ({x} : set S) = ⊤ | begin
rw [_root_.eq_top_iff, ← B.adjoin_gen_eq_top],
refine adjoin_le _,
simp [hx],
end | lemma | power_basis.adjoin_eq_top_of_gen_mem_adjoin | ring_theory | src/ring_theory/power_basis.lean | [
"field_theory.minpoly.field"
] | [
"power_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_prime_prod {α : Type*} [decidable_eq α] {x y a : R} {s : finset α}
{p : α → R} (hp : ∀ i ∈ s, prime (p i)) (hx : x * y = a * ∏ i in s, p i) :
∃ (t u : finset α) (b c : R),
t ∪ u = s
∧ disjoint t u
∧ a = b * c
∧ x = b * ∏ i in t, p i
∧ y = c * ∏ i in u, p i | begin
induction s using finset.induction with i s his ih generalizing x y a,
{ exact ⟨∅, ∅, x, y, by simp [hx]⟩ },
{ rw [prod_insert his, ← mul_assoc] at hx,
have hpi : prime (p i), { exact hp i (mem_insert_self _ _) },
rcases ih (λ i hi, hp i (mem_insert_of_mem hi)) hx with
⟨t, u, b, c, htus, htu, ... | lemma | mul_eq_mul_prime_prod | ring_theory | src/ring_theory/prime.lean | [
"algebra.associated",
"algebra.big_operators.basic"
] | [
"disjoint",
"finset",
"finset.induction",
"ih",
"mul_assoc",
"mul_comm",
"mul_left_comm",
"mul_left_inj'",
"mul_right_comm",
"mul_right_inj'",
"prime"
] | If `x * y = a * ∏ i in s, p i` where `p i` is always prime, then
`x` and `y` can both be written as a divisor of `a` multiplied by
a product over a subset of `s` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_mul_prime_pow {x y a p : R} {n : ℕ} (hp : prime p) (hx : x * y = a * p ^ n) :
∃ (i j : ℕ) (b c : R),
i + j = n
∧ a = b * c
∧ x = b * p ^ i
∧ y = c * p ^ j | begin
rcases mul_eq_mul_prime_prod (λ _ _, hp)
(show x * y = a * (range n).prod (λ _, p), by simpa) with
⟨t, u, b, c, htus, htu, rfl, rfl, rfl⟩,
exact ⟨t.card, u.card, b, c, by rw [← card_disjoint_union htu, htus, card_range], by simp⟩,
end | lemma | mul_eq_mul_prime_pow | ring_theory | src/ring_theory/prime.lean | [
"algebra.associated",
"algebra.big_operators.basic"
] | [
"mul_eq_mul_prime_prod",
"prime"
] | If ` x * y = a * p ^ n` where `p` is prime, then `x` and `y` can both be written
as the product of a power of `p` and a divisor of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime.neg {p : α} (hp : prime p) : prime (-p) | begin
obtain ⟨h1, h2, h3⟩ := hp,
exact ⟨neg_ne_zero.mpr h1, by rwa is_unit.neg_iff, by simpa [neg_dvd] using h3⟩
end | lemma | prime.neg | ring_theory | src/ring_theory/prime.lean | [
"algebra.associated",
"algebra.big_operators.basic"
] | [
"is_unit.neg_iff",
"neg_dvd",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.abs [linear_order α] {p : α} (hp : prime p) : prime (abs p) | begin
obtain h|h := abs_choice p; rw h,
{ exact hp },
{ exact hp.neg }
end | lemma | prime.abs | ring_theory | src/ring_theory/prime.lean | [
"algebra.associated",
"algebra.big_operators.basic"
] | [
"abs_choice",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submodule.is_principal (S : submodule R M) : Prop | (principal [] : ∃ a, S = span R {a}) | class | submodule.is_principal | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | An `R`-submodule of `M` is principal if it is generated by one element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_is_principal : (⊥ : submodule R M).is_principal | ⟨⟨0, by simp⟩⟩ | instance | bot_is_principal | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_is_principal : (⊤ : submodule R R).is_principal | ⟨⟨1, ideal.span_singleton_one.symm⟩⟩ | instance | top_is_principal | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_principal_ideal_ring (R : Type u) [ring R] : Prop | (principal : ∀ (S : ideal R), S.is_principal) | class | is_principal_ideal_ring | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ring"
] | A ring is a principal ideal ring if all (left) ideals are principal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_ring.is_principal_ideal_ring (K : Type u) [division_ring K] :
is_principal_ideal_ring K | { principal := λ S, by rcases ideal.eq_bot_or_top S with (rfl|rfl); apply_instance } | instance | division_ring.is_principal_ideal_ring | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"division_ring",
"ideal.eq_bot_or_top",
"is_principal_ideal_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generator (S : submodule R M) [S.is_principal] : M | classical.some (principal S) | def | submodule.is_principal.generator | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | `generator I`, if `I` is a principal submodule, is an `x ∈ M` such that `span R {x} = I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_singleton_generator (S : submodule R M) [S.is_principal] : span R {generator S} = S | eq.symm (classical.some_spec (principal S)) | lemma | submodule.is_principal.span_singleton_generator | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ideal.span_singleton_generator (I : ideal R) [I.is_principal] :
ideal.span ({generator I} : set R) = I | eq.symm (classical.some_spec (principal I)) | lemma | ideal.span_singleton_generator | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generator_mem (S : submodule R M) [S.is_principal] : generator S ∈ S | by { conv_rhs { rw ← span_singleton_generator S }, exact subset_span (mem_singleton _) } | lemma | submodule.is_principal.generator_mem | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_eq_smul_generator (S : submodule R M) [S.is_principal] {x : M} :
x ∈ S ↔ ∃ s : R, x = s • generator S | by simp_rw [@eq_comm _ x, ← mem_span_singleton, span_singleton_generator] | lemma | submodule.is_principal.mem_iff_eq_smul_generator | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_iff_generator_eq_zero (S : submodule R M) [S.is_principal] :
S = ⊥ ↔ generator S = 0 | by rw [← @span_singleton_eq_bot R M, span_singleton_generator] | lemma | submodule.is_principal.eq_bot_iff_generator_eq_zero | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_generator_dvd (S : ideal R) [S.is_principal] {x : R} : x ∈ S ↔ generator S ∣ x | (mem_iff_eq_smul_generator S).trans (exists_congr (λ a, by simp only [mul_comm, smul_eq_mul])) | lemma | submodule.is_principal.mem_iff_generator_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"mul_comm",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_generator_of_is_prime (S : ideal R) [submodule.is_principal S] [is_prime : S.is_prime]
(ne_bot : S ≠ ⊥) :
prime (generator S) | ⟨λ h, ne_bot ((eq_bot_iff_generator_eq_zero S).2 h),
λ h, is_prime.ne_top (S.eq_top_of_is_unit_mem (generator_mem S) h),
λ _ _, by simpa only [← mem_iff_generator_dvd S] using is_prime.2⟩ | lemma | submodule.is_principal.prime_generator_of_is_prime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"prime",
"submodule.is_principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generator_map_dvd_of_mem {N : submodule R M}
(ϕ : M →ₗ[R] R) [(N.map ϕ).is_principal] {x : M} (hx : x ∈ N) :
generator (N.map ϕ) ∣ ϕ x | by { rw [← mem_iff_generator_dvd, submodule.mem_map], exact ⟨x, hx, rfl⟩ } | lemma | submodule.is_principal.generator_map_dvd_of_mem | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"submodule",
"submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generator_submodule_image_dvd_of_mem {N O : submodule R M} (hNO : N ≤ O)
(ϕ : O →ₗ[R] R) [(ϕ.submodule_image N).is_principal] {x : M} (hx : x ∈ N) :
generator (ϕ.submodule_image N) ∣ ϕ ⟨x, hNO hx⟩ | by { rw [← mem_iff_generator_dvd, linear_map.mem_submodule_image_of_le hNO], exact ⟨x, hx, rfl⟩ } | lemma | submodule.is_principal.generator_submodule_image_dvd_of_mem | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"linear_map.mem_submodule_image_of_le",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_maximal_ideal [comm_ring R] [is_domain R] [is_principal_ideal_ring R] {S : ideal R}
[hpi : is_prime S] (hS : S ≠ ⊥) : is_maximal S | is_maximal_iff.2 ⟨(ne_top_iff_one S).1 hpi.1, begin
assume T x hST hxS hxT,
cases (mem_iff_generator_dvd _).1 (hST $ generator_mem S) with z hz,
cases hpi.mem_or_mem (show generator T * z ∈ S, from hz ▸ generator_mem S),
{ have hTS : T ≤ S, rwa [← T.span_singleton_generator, ideal.span_le, singleton_subset_iff]... | lemma | is_prime.to_maximal_ideal | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"ideal",
"ideal.span_le",
"is_domain",
"is_principal_ideal_ring",
"mul_left_comm",
"mul_one",
"mul_right_inj'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_mem_iff {S : ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x ∈ S | ⟨λ hxy, div_add_mod x y ▸ S.add_mem (S.mul_mem_right _ hy) hxy,
λ hx, (mod_eq_sub_mul_div x y).symm ▸ S.sub_mem hx (S.mul_mem_right _ hy)⟩ | lemma | mod_mem_iff | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
euclidean_domain.to_principal_ideal_domain : is_principal_ideal_ring R | { principal := λ S, by exactI
⟨if h : {x : R | x ∈ S ∧ x ≠ 0}.nonempty
then
have wf : well_founded (euclidean_domain.r : R → R → Prop) := euclidean_domain.r_well_founded,
have hmin : well_founded.min wf {x : R | x ∈ S ∧ x ≠ 0} h ∈ S ∧
well_founded.min wf {x : R | x ∈ S ∧ x ≠ 0} h ≠ 0,
from... | instance | euclidean_domain.to_principal_ideal_domain | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"by_contradiction",
"dvd_add",
"dvd_mul_right",
"is_principal_ideal_ring",
"mod_mem_iff",
"not_and_distrib",
"not_ne_iff",
"submodule.bot_coe",
"submodule.ext",
"submodule.mem_bot",
"well_founded.min",
"well_founded.min_mem",
"well_founded.not_lt_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_field.is_principal_ideal_ring
{R : Type*} [comm_ring R] (h : is_field R) :
is_principal_ideal_ring R | @euclidean_domain.to_principal_ideal_domain R (@field.to_euclidean_domain R h.to_field) | lemma | is_field.is_principal_ideal_ring | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"euclidean_domain.to_principal_ideal_domain",
"field.to_euclidean_domain",
"is_field",
"is_principal_ideal_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_noetherian_ring [ring R] [is_principal_ideal_ring R] :
is_noetherian_ring R | is_noetherian_ring_iff.2 ⟨assume s : ideal R,
begin
rcases (is_principal_ideal_ring.principal s).principal with ⟨a, rfl⟩,
rw [← finset.coe_singleton],
exact ⟨{a}, set_like.coe_injective rfl⟩
end⟩ | instance | principal_ideal_ring.is_noetherian_ring | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"finset.coe_singleton",
"ideal",
"is_noetherian_ring",
"is_principal_ideal_ring",
"ring",
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_of_irreducible [comm_ring R] [is_principal_ideal_ring R]
{p : R} (hp : irreducible p) :
ideal.is_maximal (span R ({p} : set R)) | ⟨⟨mt ideal.span_singleton_eq_top.1 hp.1, λ I hI, begin
rcases principal I with ⟨a, rfl⟩,
erw ideal.span_singleton_eq_top,
unfreezingI { rcases ideal.span_singleton_le_span_singleton.1 (le_of_lt hI) with ⟨b, rfl⟩ },
refine (of_irreducible_mul hp).resolve_right (mt (λ hb, _) (not_le_of_lt hI)),
erw [ideal.span_... | lemma | principal_ideal_ring.is_maximal_of_irreducible | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"ideal.is_maximal",
"ideal.span_singleton_eq_top",
"ideal.span_singleton_le_span_singleton",
"irreducible",
"is_principal_ideal_ring",
"is_unit.mul_right_dvd",
"not_le_of_lt",
"of_irreducible_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_iff_prime {p : R} : irreducible p ↔ prime p | ⟨λ hp, (ideal.span_singleton_prime hp.ne_zero).1 $
(is_maximal_of_irreducible hp).is_prime,
prime.irreducible⟩ | lemma | principal_ideal_ring.irreducible_iff_prime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal.span_singleton_prime",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates_irreducible_iff_prime : ∀{p : associates R}, irreducible p ↔ prime p | associates.irreducible_iff_prime_iff.1 (λ _, irreducible_iff_prime) | lemma | principal_ideal_ring.associates_irreducible_iff_prime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"associates",
"irreducible",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factors (a : R) : multiset R | if h : a = 0 then ∅ else classical.some (wf_dvd_monoid.exists_factors a h) | def | principal_ideal_ring.factors | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"multiset",
"wf_dvd_monoid.exists_factors"
] | `factors a` is a multiset of irreducible elements whose product is `a`, up to units | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
factors_spec (a : R) (h : a ≠ 0) :
(∀b∈factors a, irreducible b) ∧ associated (factors a).prod a | begin
unfold factors, rw [dif_neg h],
exact classical.some_spec (wf_dvd_monoid.exists_factors a h)
end | lemma | principal_ideal_ring.factors_spec | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"associated",
"irreducible",
"wf_dvd_monoid.exists_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mem_factors
{R : Type v} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] {a b : R}
(ha : a ≠ 0) (hb : b ∈ factors a) : b ≠ 0 | irreducible.ne_zero ((factors_spec a ha).1 b hb) | lemma | principal_ideal_ring.ne_zero_of_mem_factors | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"irreducible.ne_zero",
"is_domain",
"is_principal_ideal_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_submonoid_of_factors_subset_of_units_subset (s : submonoid R)
{a : R} (ha : a ≠ 0) (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) :
a ∈ s | begin
rcases ((factors_spec a ha).2) with ⟨c, hc⟩,
rw [← hc],
exact mul_mem (multiset_prod_mem _ hfac) (hunit _)
end | lemma | principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"multiset_prod_mem",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_hom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type*}
[comm_ring R] [is_domain R] [is_principal_ideal_ring R] [semiring S]
(f : R →+* S) (s : submonoid S) (a : R) (ha : a ≠ 0)
(h : ∀ b ∈ factors a, f b ∈ s) (hf: ∀ c : Rˣ, f c ∈ s) :
f a ∈ s | mem_submonoid_of_factors_subset_of_units_subset (s.comap f.to_monoid_hom) ha h hf | lemma | principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"comm_ring",
"is_domain",
"is_principal_ideal_ring",
"semiring",
"submonoid"
] | If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
also maps `a` into that submonoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_unique_factorization_monoid : unique_factorization_monoid R | { irreducible_iff_prime := λ _, principal_ideal_ring.irreducible_iff_prime
.. (is_noetherian_ring.wf_dvd_monoid : wf_dvd_monoid R) } | instance | principal_ideal_ring.to_unique_factorization_monoid | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"is_noetherian_ring.wf_dvd_monoid",
"principal_ideal_ring.irreducible_iff_prime",
"unique_factorization_monoid",
"wf_dvd_monoid"
] | A principal ideal domain has unique factorization | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.is_principal.of_comap (f : M →ₗ[R] N) (hf : function.surjective f)
(S : submodule R N) [hI : is_principal (S.comap f)] :
is_principal S | ⟨⟨f (is_principal.generator (S.comap f)),
by rw [← set.image_singleton, ← submodule.map_span,
is_principal.span_singleton_generator, submodule.map_comap_eq_of_surjective hf]⟩⟩ | lemma | submodule.is_principal.of_comap | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"set.image_singleton",
"submodule",
"submodule.map_comap_eq_of_surjective",
"submodule.map_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_principal.of_comap (f : R →+* S) (hf : function.surjective f)
(I : ideal S) [hI : is_principal (I.comap f)] :
is_principal I | ⟨⟨f (is_principal.generator (I.comap f)),
by rw [ideal.submodule_span_eq, ← set.image_singleton, ← ideal.map_span,
ideal.span_singleton_generator, ideal.map_comap_of_surjective f hf]⟩⟩ | lemma | ideal.is_principal.of_comap | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.map_comap_of_surjective",
"ideal.map_span",
"ideal.span_singleton_generator",
"ideal.submodule_span_eq",
"set.image_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_principal_ideal_ring.of_surjective [is_principal_ideal_ring R]
(f : R →+* S) (hf : function.surjective f) :
is_principal_ideal_ring S | ⟨λ I, ideal.is_principal.of_comap f hf I⟩ | lemma | is_principal_ideal_ring.of_surjective | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal.is_principal.of_comap",
"is_principal_ideal_ring"
] | The surjective image of a principal ideal ring is again a principal ideal ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_gcd (x y : R) : span ({gcd x y} : set R) = span ({x, y} : set R) | begin
obtain ⟨d, hd⟩ := is_principal_ideal_ring.principal (span ({x, y} : set R)),
rw submodule_span_eq at hd,
rw [hd],
suffices : associated d (gcd x y),
{ obtain ⟨D, HD⟩ := this,
rw ←HD,
exact (span_singleton_mul_right_unit D.is_unit _) },
apply associated_of_dvd_dvd,
{ rw dvd_gcd_iff,
split... | theorem | span_gcd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"associated",
"associated_of_dvd_dvd",
"dvd_add",
"dvd_gcd_iff",
"dvd_mul_of_dvd_right",
"ideal.mem_span_pair",
"ideal.mem_span_singleton",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y | by simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ←ideal.mem_span_pair, ←span_gcd,
ideal.mem_span_singleton] | theorem | gcd_dvd_iff_exists | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal.mem_span_singleton",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y | by rw [←gcd_dvd_iff_exists] | theorem | exists_gcd_eq_mul_add_mul | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [] | **Bézout's lemma** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_is_unit_iff (x y : R) : is_unit (gcd x y) ↔ is_coprime x y | by rw [is_coprime, ←ideal.mem_span_pair, ←span_gcd, ←span_singleton_eq_top, eq_top_iff_one] | theorem | gcd_is_unit_iff | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"is_coprime",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_of_dvd (x y : R)
(nonzero : ¬ (x = 0 ∧ y = 0)) (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬ z ∣ y) :
is_coprime x y | begin
rw [← gcd_is_unit_iff],
by_contra h,
refine H _ h _ (gcd_dvd_left _ _) (gcd_dvd_right _ _),
rwa [ne, gcd_eq_zero_iff]
end | theorem | is_coprime_of_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"by_contra",
"gcd_eq_zero_iff",
"gcd_is_unit_iff",
"is_coprime",
"nonunits"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_or_coprime (x y : R) (h : irreducible x) : x ∣ y ∨ is_coprime x y | begin
refine or_iff_not_imp_left.2 (λ h', _),
apply is_coprime_of_dvd,
{ unfreezingI { rintro ⟨rfl, rfl⟩ }, simpa using h },
{ unfreezingI { rintro z nu nz ⟨w, rfl⟩ dy },
refine h' (dvd_trans _ dy),
simpa using mul_dvd_mul_left z (is_unit_iff_dvd_one.1 $
(of_irreducible_mul h).resolve_left nu) }
e... | theorem | dvd_or_coprime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"dvd_trans",
"irreducible",
"is_coprime",
"is_coprime_of_dvd",
"mul_dvd_mul_left",
"of_irreducible_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_of_irreducible_dvd {x y : R}
(nonzero : ¬ (x = 0 ∧ y = 0))
(H : ∀ z : R, irreducible z → z ∣ x → ¬ z ∣ y) :
is_coprime x y | begin
apply is_coprime_of_dvd x y nonzero,
intros z znu znz zx zy,
obtain ⟨i, h1, h2⟩ := wf_dvd_monoid.exists_irreducible_factor znu znz,
apply H i h1;
{ apply dvd_trans h2, assumption },
end | theorem | is_coprime_of_irreducible_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"dvd_trans",
"irreducible",
"is_coprime",
"is_coprime_of_dvd",
"wf_dvd_monoid.exists_irreducible_factor"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_coprime_of_prime_dvd {x y : R}
(nonzero : ¬ (x = 0 ∧ y = 0))
(H : ∀ z : R, prime z → z ∣ x → ¬ z ∣ y) :
is_coprime x y | is_coprime_of_irreducible_dvd nonzero $ λ z zi, H z $ gcd_monoid.prime_of_irreducible zi | theorem | is_coprime_of_prime_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"gcd_monoid.prime_of_irreducible",
"is_coprime",
"is_coprime_of_irreducible_dvd",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible.coprime_iff_not_dvd {p n : R} (pp : irreducible p) : is_coprime p n ↔ ¬ p ∣ n | begin
split,
{ intros co H,
apply pp.not_unit,
rw is_unit_iff_dvd_one,
apply is_coprime.dvd_of_dvd_mul_left co,
rw mul_one n,
exact H },
{ intro nd,
apply is_coprime_of_irreducible_dvd,
{ rintro ⟨hp, -⟩,
exact pp.ne_zero hp },
rintro z zi zp zn,
exact nd (((zi.associated_... | theorem | irreducible.coprime_iff_not_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"irreducible",
"is_coprime",
"is_coprime.dvd_of_dvd_mul_left",
"is_coprime_of_irreducible_dvd",
"is_unit_iff_dvd_one",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.coprime_iff_not_dvd {p n : R} (pp : prime p) : is_coprime p n ↔ ¬ p ∣ n | pp.irreducible.coprime_iff_not_dvd | theorem | prime.coprime_iff_not_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"is_coprime",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible.dvd_iff_not_coprime {p n : R} (hp : irreducible p) : p ∣ n ↔ ¬ is_coprime p n | iff_not_comm.2 hp.coprime_iff_not_dvd | theorem | irreducible.dvd_iff_not_coprime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"irreducible",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : irreducible p) (h : ¬ p ∣ a) :
is_coprime a (p ^ m) | (hp.coprime_iff_not_dvd.2 h).symm.pow_right | theorem | irreducible.coprime_pow_of_not_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"irreducible",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible.coprime_or_dvd {p : R} (hp : irreducible p) (i : R) :
is_coprime p i ∨ p ∣ i | (em _).imp_right hp.dvd_iff_not_coprime.2 | theorem | irreducible.coprime_or_dvd | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"em",
"irreducible",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_associated_pow_of_mul_eq_pow' {a b c : R}
(hab : is_coprime a b) {k : ℕ} (h : a * b = c ^ k) : ∃ d, associated (d ^ k) a | exists_associated_pow_of_mul_eq_pow ((gcd_is_unit_iff _ _).mpr hab) h | theorem | exists_associated_pow_of_mul_eq_pow' | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"associated",
"exists_associated_pow_of_mul_eq_pow",
"gcd_is_unit_iff",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_principals | {I : ideal R | ¬ I.is_principal} | def | non_principals | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal"
] | `non_principals R` is the set of all ideals of `R` that are not principal ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
non_principals_def {I : ideal R} : I ∈ non_principals R ↔ ¬ I.is_principal | iff.rfl | lemma | non_principals_def | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"non_principals"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_principals_eq_empty_iff : non_principals R = ∅ ↔ is_principal_ideal_ring R | by simp [set.eq_empty_iff_forall_not_mem, is_principal_ideal_ring_iff, non_principals_def] | lemma | non_principals_eq_empty_iff | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"is_principal_ideal_ring",
"non_principals",
"non_principals_def",
"set.eq_empty_iff_forall_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_principals_zorn (c : set (ideal R)) (hs : c ⊆ non_principals R) (hchain : is_chain (≤) c)
{K : ideal R} (hKmem : K ∈ c) :
∃ I ∈ non_principals R, ∀ J ∈ c, J ≤ I | begin
refine ⟨Sup c, _, λ J hJ, le_Sup hJ⟩,
rintro ⟨x, hx⟩,
have hxmem : x ∈ Sup c := (hx.symm ▸ submodule.mem_span_singleton_self x),
obtain ⟨J, hJc, hxJ⟩ := (submodule.mem_Sup_of_directed ⟨K, hKmem⟩ hchain.directed_on).1 hxmem,
have hSupJ : Sup c = J := le_antisymm (by simp [hx, ideal.span_le, hxJ]) (le_Sup... | lemma | non_principals_zorn | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"ideal",
"ideal.span_le",
"is_chain",
"le_Sup",
"non_principals",
"non_principals_def",
"submodule.mem_Sup_of_directed",
"submodule.mem_span_singleton_self"
] | Any chain in the set of non-principal ideals has an upper bound which is non-principal.
(Namely, the union of the chain is such an upper bound.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_principal_ideal_ring.of_prime (H : ∀ (P : ideal R), P.is_prime → P.is_principal) :
is_principal_ideal_ring R | begin
-- Suppose the set of `non_principals` is not empty.
rw [← non_principals_eq_empty_iff, set.eq_empty_iff_forall_not_mem],
intros J hJ,
-- We will show a maximal element `I ∈ non_principals R` (which exists by Zorn) is prime.
obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partial_order₀
(non_principals R... | theorem | is_principal_ideal_ring.of_prime | ring_theory | src/ring_theory/principal_ideal_domain.lean | [
"algebra.euclidean_domain.instances",
"ring_theory.unique_factorization_domain"
] | [
"by_contra",
"ideal",
"ideal.mem_colon_singleton",
"ideal.mem_span_singleton_self",
"ideal.mem_sup_left",
"ideal.mem_sup_right",
"ideal.span_singleton_le_iff_mem",
"ideal.submodule_span_eq",
"ideal.sup_mul",
"is_principal_ideal_ring",
"mul_assoc",
"mul_comm",
"non_principals",
"non_princip... | If all prime ideals in a commutative ring are principal, so are all other ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ideal.is_radical_iff_quotient_reduced {R : Type*} [comm_ring R] (I : ideal R) :
I.is_radical ↔ is_reduced (R ⧸ I) | by { conv_lhs { rw ← @ideal.mk_ker R _ I },
exact ring_hom.ker_is_radical_iff_reduced_of_surjective (@ideal.quotient.mk_surjective R _ I) } | lemma | ideal.is_radical_iff_quotient_reduced | ring_theory | src/ring_theory/quotient_nilpotent.lean | [
"ring_theory.nilpotent",
"ring_theory.ideal.quotient_operations"
] | [
"comm_ring",
"ideal",
"ideal.mk_ker",
"ideal.quotient.mk_surjective",
"is_reduced",
"ring_hom.ker_is_radical_iff_reduced_of_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.is_nilpotent.induction_on
(hI : is_nilpotent I)
{P : ∀ ⦃S : Type*⦄ [comm_ring S], by exactI ∀ I : ideal S, Prop}
(h₁ : ∀ ⦃S : Type*⦄ [comm_ring S], by exactI ∀ I : ideal S, I ^ 2 = ⊥ → P I)
(h₂ : ∀ ⦃S : Type*⦄ [comm_ring S], by exactI
∀ I J : ideal S, I ≤ J → P I → P (J.map (ideal.quotient.mk I)) → P ... | begin
obtain ⟨n, hI : I ^ n = ⊥⟩ := hI,
unfreezingI { revert S },
apply nat.strong_induction_on n,
clear n,
introsI n H S _ I hI,
by_cases hI' : I = ⊥,
{ subst hI', apply h₁, rw [← ideal.zero_eq_bot, zero_pow], exact zero_lt_two },
cases n,
{ rw [pow_zero, ideal.one_eq_top] at hI,
haveI := subsing... | lemma | ideal.is_nilpotent.induction_on | ring_theory | src/ring_theory/quotient_nilpotent.lean | [
"ring_theory.nilpotent",
"ring_theory.ideal.quotient_operations"
] | [
"comm_ring",
"eq_bot_iff",
"ideal",
"ideal.map_pow",
"ideal.map_quotient_self",
"ideal.one_eq_top",
"ideal.pow_le_pow",
"ideal.pow_le_self",
"ideal.quotient.mk",
"ideal.zero_eq_bot",
"is_nilpotent",
"pow_mul",
"pow_one",
"pow_zero",
"subsingleton_of_bot_eq_top",
"two_ne_zero",
"zero_... | Let `P` be a property on ideals. If `P` holds for square-zero ideals, and if
`P I → P (J ⧸ I) → P J`, then `P` holds for all nilpotent ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_nilpotent.is_unit_quotient_mk_iff {R : Type*} [comm_ring R] {I : ideal R}
(hI : is_nilpotent I) {x : R} : is_unit (ideal.quotient.mk I x) ↔ is_unit x | begin
refine ⟨_, λ h, h.map I^.quotient.mk⟩,
revert x,
apply ideal.is_nilpotent.induction_on I hI; clear hI I,
swap,
{ introv e h₁ h₂ h₃,
apply h₁,
apply h₂,
exactI h₃.map ((double_quot.quot_quot_equiv_quot_sup I J).trans
(ideal.quot_equiv_of_eq (sup_eq_right.mpr e))).symm.to_ring_hom },
{... | lemma | is_nilpotent.is_unit_quotient_mk_iff | ring_theory | src/ring_theory/quotient_nilpotent.lean | [
"ring_theory.nilpotent",
"ring_theory.ideal.quotient_operations"
] | [
"comm_ring",
"double_quot.quot_quot_equiv_quot_sup",
"ideal",
"ideal.is_nilpotent.induction_on",
"ideal.mem_bot",
"ideal.pow_mem_pow",
"ideal.quot_equiv_of_eq",
"ideal.quotient.eq",
"ideal.quotient.mk",
"ideal.quotient.mk_surjective",
"is_nilpotent",
"is_unit",
"is_unit.mul_coe_inv",
"is_u... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal.quotient.is_noetherian_ring {R : Type*} [comm_ring R] [h : is_noetherian_ring R]
(I : ideal R) : is_noetherian_ring (R ⧸ I) | is_noetherian_ring_iff.mpr $ is_noetherian_of_tower R $ submodule.quotient.is_noetherian _ | instance | ideal.quotient.is_noetherian_ring | ring_theory | src/ring_theory/quotient_noetherian.lean | [
"ring_theory.noetherian",
"ring_theory.quotient_nilpotent"
] | [
"comm_ring",
"ideal",
"is_noetherian_of_tower",
"is_noetherian_ring",
"submodule.quotient.is_noetherian"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rees_algebra : subalgebra R R[X] | { carrier := { f | ∀ i, f.coeff i ∈ I ^ i },
mul_mem' := λ f g hf hg i, begin
rw coeff_mul,
apply ideal.sum_mem,
rintros ⟨j, k⟩ e,
rw [← finset.nat.mem_antidiagonal.mp e, pow_add],
exact ideal.mul_mem_mul (hf j) (hg k)
end,
one_mem' := λ i, begin
rw coeff_one,
split_ifs,
{ subst h,... | def | rees_algebra | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"algebra_map_apply",
"ideal.add_mem",
"ideal.mul_mem_mul",
"ideal.sum_mem",
"ideal.zero_mem",
"pow_add",
"subalgebra"
] | The Rees algebra of an ideal `I`, defined as the subalgebra of `R[X]` whose `i`-th coefficient
falls in `I ^ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_rees_algebra_iff (f : R[X]) :
f ∈ rees_algebra I ↔ ∀ i, f.coeff i ∈ I ^ i | iff.rfl | lemma | mem_rees_algebra_iff | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"rees_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_rees_algebra_iff_support (f : R[X]) :
f ∈ rees_algebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i | begin
apply forall_congr,
intro a,
rw [mem_support_iff, iff.comm, imp_iff_right_iff, ne.def, ← imp_iff_not_or],
exact λ e, e.symm ▸ (I ^ a).zero_mem
end | lemma | mem_rees_algebra_iff_support | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"imp_iff_not_or",
"imp_iff_right_iff",
"rees_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rees_algebra.monomial_mem {I : ideal R} {i : ℕ} {r : R} :
monomial i r ∈ rees_algebra I ↔ r ∈ I ^ i | by simp [mem_rees_algebra_iff_support, coeff_monomial, ← imp_iff_not_or] { contextual := tt } | lemma | rees_algebra.monomial_mem | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"ideal",
"imp_iff_not_or",
"mem_rees_algebra_iff_support",
"rees_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monomial_mem_adjoin_monomial {I : ideal R} {n : ℕ} {r : R} (hr : r ∈ I ^ n) :
monomial n r ∈ algebra.adjoin R (submodule.map (monomial 1 : R →ₗ[R] R[X]) I : set R[X]) | begin
induction n with n hn generalizing r,
{ exact subalgebra.algebra_map_mem _ _ },
{ rw pow_succ at hr,
apply submodule.smul_induction_on hr,
{ intros r hr s hs,
rw [nat.succ_eq_one_add, smul_eq_mul, ← monomial_mul_monomial],
exact subalgebra.mul_mem _ (algebra.subset_adjoin (set.mem_image_... | lemma | monomial_mem_adjoin_monomial | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"algebra.adjoin",
"algebra.subset_adjoin",
"ideal",
"nat.succ_eq_one_add",
"pow_succ",
"set.mem_image_of_mem",
"smul_eq_mul",
"subalgebra.add_mem",
"subalgebra.algebra_map_mem",
"subalgebra.mul_mem",
"submodule.map",
"submodule.smul_induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adjoin_monomial_eq_rees_algebra :
algebra.adjoin R (submodule.map (monomial 1 : R →ₗ[R] R[X]) I : set R[X]) = rees_algebra I | begin
apply le_antisymm,
{ apply algebra.adjoin_le _,
rintro _ ⟨r, hr, rfl⟩,
exact rees_algebra.monomial_mem.mpr (by rwa pow_one) },
{ intros p hp,
rw p.as_sum_support,
apply subalgebra.sum_mem _ _,
rintros i -,
exact monomial_mem_adjoin_monomial (hp i) }
end | lemma | adjoin_monomial_eq_rees_algebra | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"algebra.adjoin",
"algebra.adjoin_le",
"monomial_mem_adjoin_monomial",
"pow_one",
"rees_algebra",
"subalgebra.sum_mem",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rees_algebra.fg (hI : I.fg) : (rees_algebra I).fg | begin
classical,
obtain ⟨s, hs⟩ := hI,
rw [← adjoin_monomial_eq_rees_algebra, ← hs],
use s.image (monomial 1),
rw finset.coe_image,
change _ = algebra.adjoin R (submodule.map (monomial 1 : R →ₗ[R] R[X])
(submodule.span R ↑s) : set R[X]),
rw [submodule.map_span, algebra.adjoin_span]
end | lemma | rees_algebra.fg | ring_theory | src/ring_theory/rees_algebra.lean | [
"ring_theory.finite_type"
] | [
"adjoin_monomial_eq_rees_algebra",
"algebra.adjoin",
"algebra.adjoin_span",
"finset.coe_image",
"rees_algebra",
"submodule.map",
"submodule.map_span",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso : Prop | (∀ {R S T : Type u} [comm_ring R] [comm_ring S] [comm_ring T], by exactI
∀ (f : R →+* S) (e : S ≃+* T) (hf : P f), P (e.to_ring_hom.comp f)) ∧
(∀ {R S T : Type u} [comm_ring R] [comm_ring S] [comm_ring T], by exactI
∀ (f : S →+* T) (e : R ≃+* S) (hf : P f), P (f.comp e.to_ring_hom)) | def | ring_hom.respects_iso | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"comm_ring"
] | A property `respects_iso` if it still holds when composed with an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
respects_iso.cancel_left_is_iso (hP : respects_iso @P) {R S T : CommRing}
(f : R ⟶ S) (g : S ⟶ T)
[is_iso f] : P (f ≫ g) ↔ P g | ⟨λ H, by { convert hP.2 (f ≫ g) (as_iso f).symm.CommRing_iso_to_ring_equiv H,
exact (is_iso.inv_hom_id_assoc _ _).symm }, hP.2 g (as_iso f).CommRing_iso_to_ring_equiv⟩ | lemma | ring_hom.respects_iso.cancel_left_is_iso | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.cancel_right_is_iso (hP : respects_iso @P) {R S T : CommRing}
(f : R ⟶ S) (g : S ⟶ T)
[is_iso g] : P (f ≫ g) ↔ P f | ⟨λ H, by { convert hP.1 (f ≫ g) (as_iso g).symm.CommRing_iso_to_ring_equiv H,
change f = f ≫ g ≫ (inv g), simp }, hP.1 f (as_iso g).CommRing_iso_to_ring_equiv⟩ | lemma | ring_hom.respects_iso.cancel_right_is_iso | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"CommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
respects_iso.is_localization_away_iff (hP : ring_hom.respects_iso @P) {R S : Type*}
(R' S' : Type*) [comm_ring R] [comm_ring S] [comm_ring R'] [comm_ring S'] [algebra R R']
[algebra S S'] (f : R →+* S) (r : R) [is_localization.away r R'] [is_localization.away (f r) S'] :
P (localization.away_map f r) ↔ P (is_loca... | begin
let e₁ : R' ≃+* localization.away r :=
(is_localization.alg_equiv (submonoid.powers r) _ _).to_ring_equiv,
let e₂ : localization.away (f r) ≃+* S' :=
(is_localization.alg_equiv (submonoid.powers (f r)) _ _).to_ring_equiv,
refine (hP.cancel_left_is_iso e₁.to_CommRing_iso.hom (CommRing.of_hom _)).symm... | lemma | ring_hom.respects_iso.is_localization_away_iff | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"CommRing.of",
"CommRing.of_hom",
"algebra",
"category_theory.comp_apply",
"comm_ring",
"eq_iff_iff",
"is_localization.alg_equiv",
"is_localization.away",
"is_localization.away.map",
"is_localization.map_eq",
"is_localization.ring_equiv_of_ring_equiv_eq",
"is_localization.ring_hom_ext",
"loc... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_composition : Prop | ∀ ⦃R S T⦄ [comm_ring R] [comm_ring S] [comm_ring T],
by exactI ∀ (f : R →+* S) (g : S →+* T) (hf : P f) (hg : P g), P (g.comp f) | def | ring_hom.stable_under_composition | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"comm_ring"
] | A property is `stable_under_composition` if the composition of two such morphisms
still falls in the class. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_composition.respects_iso (hP : ring_hom.stable_under_composition @P)
(hP' : ∀ {R S : Type*} [comm_ring R] [comm_ring S] (e : by exactI R ≃+* S),
by exactI P e.to_ring_hom) : ring_hom.respects_iso @P | begin
split,
{ introv H, resetI, apply hP, exacts [H, hP' e] },
{ introv H, resetI, apply hP, exacts [hP' e, H] }
end | lemma | ring_hom.stable_under_composition.respects_iso | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"comm_ring",
"ring_hom.respects_iso",
"ring_hom.stable_under_composition"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change : Prop | ∀ (R S R' S') [comm_ring R] [comm_ring S] [comm_ring R'] [comm_ring S'],
by exactI ∀ [algebra R S] [algebra R R'] [algebra R S'] [algebra S S'] [algebra R' S'],
by exactI ∀ [is_scalar_tower R S S'] [is_scalar_tower R R' S'],
by exactI ∀ [algebra.is_pushout R S R' S'], P (algebra_map R S) → P (algebra_map R'... | def | ring_hom.stable_under_base_change | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"algebra",
"algebra.is_pushout",
"algebra_map",
"comm_ring",
"is_scalar_tower"
] | A morphism property `P` is `stable_under_base_change` if `P(S →+* A)` implies
`P(B →+* A ⊗[S] B)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stable_under_base_change.mk
(h₁ : respects_iso @P)
(h₂ : ∀ ⦃R S T⦄ [comm_ring R] [comm_ring S] [comm_ring T],
by exactI ∀ [algebra R S] [algebra R T], by exactI (P (algebra_map R T) →
P (algebra.tensor_product.include_left.to_ring_hom : S →+* tensor_product R S T))) :
stable_under_base_change @P | begin
introv R h H,
resetI,
let e := h.symm.1.equiv,
let f' := algebra.tensor_product.product_map (is_scalar_tower.to_alg_hom R R' S')
(is_scalar_tower.to_alg_hom R S S'),
have : ∀ x, e x = f' x,
{ intro x,
change e.to_linear_map.restrict_scalars R x = f'.to_linear_map x,
congr' 1,
apply ten... | lemma | ring_hom.stable_under_base_change.mk | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"alg_hom.to_linear_map_apply",
"algebra",
"algebra.smul_def",
"algebra.tensor_product.product_map",
"algebra_map",
"comm_ring",
"is_base_change.equiv_tmul",
"is_scalar_tower.to_alg_hom",
"map_mul",
"map_one",
"mul_one",
"tensor_product",
"tensor_product.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stable_under_base_change.pushout_inl
(hP : ring_hom.stable_under_base_change @P) (hP' : ring_hom.respects_iso @P) {R S T : CommRing}
(f : R ⟶ S) (g : R ⟶ T) (H : P g) : P (pushout.inl : S ⟶ pushout f g) | begin
rw [← (show _ = pushout.inl, from colimit.iso_colimit_cocone_ι_inv
⟨_, CommRing.pushout_cocone_is_colimit f g⟩ walking_span.left), hP'.cancel_right_is_iso],
letI := f.to_algebra,
letI := g.to_algebra,
dsimp only [CommRing.pushout_cocone_inl, pushout_cocone.ι_app_left],
apply hP R T S (tensor_product... | lemma | ring_hom.stable_under_base_change.pushout_inl | ring_theory | src/ring_theory/ring_hom_properties.lean | [
"algebra.category.Ring.constructions",
"algebra.category.Ring.colimits",
"category_theory.isomorphism",
"ring_theory.localization.away.basic",
"ring_theory.is_tensor_product"
] | [
"CommRing",
"CommRing.pushout_cocone_inl",
"CommRing.pushout_cocone_is_colimit",
"ring_hom.respects_iso",
"ring_hom.stable_under_base_change",
"tensor_product"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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