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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|---|---|---|---|---|---|---|---|---|---|---|
multiplicity_prime_eq_multiplicity_image_by_factor_order_iso [decidable_eq (associates M)]
{m p : associates M} {n : associates N} (hn : n ≠ 0)
(hp : p ∈ normalized_factors m) (d : set.Iic m ≃o set.Iic n) :
multiplicity p m = multiplicity ↑(d ⟨p, dvd_of_mem_normalized_factors hp⟩) n | begin
refine le_antisymm (multiplicity_prime_le_multiplicity_image_by_factor_order_iso hp d) _,
suffices : multiplicity ↑(d ⟨p, dvd_of_mem_normalized_factors hp⟩) n ≤
multiplicity ↑(d.symm (d ⟨p, dvd_of_mem_normalized_factors hp⟩)) m,
{ rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalized_factors hp⟩, subtype.coe... | lemma | multiplicity_prime_eq_multiplicity_image_by_factor_order_iso | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"classical.dec_eq",
"mem_normalized_factors_factor_order_iso_of_mem_normalized_factors",
"multiplicity",
"multiplicity_prime_le_multiplicity_image_by_factor_order_iso",
"set.Iic",
"subtype.coe_eta",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_factor_order_iso_of_factor_dvd_equiv
{m : M} {n : N} {d : {l : M // l ∣ m} ≃ {l : N // l ∣ n}}
(hd : ∀ l l', ((d l) : N) ∣ (d l') ↔ (l : M) ∣ (l' : M)) :
set.Iic (associates.mk m) ≃o set.Iic (associates.mk n) | { to_fun := λ l, ⟨associates.mk (d ⟨associates_equiv_of_unique_units ↑l,
by { obtain ⟨x, hx⟩ := l, rw [subtype.coe_mk, associates_equiv_of_unique_units_apply,
out_dvd_iff], exact hx } ⟩),
mk_le_mk_iff_dvd_iff.mpr (subtype.prop (d ⟨associates_equiv_of_unique_units ↑l, _ ⟩)) ⟩,
inv_fun := λ l, ⟨associate... | def | mk_factor_order_iso_of_factor_dvd_equiv | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates.mk",
"associates.mk_le_mk_iff_dvd_iff",
"associates_equiv_of_unique_units",
"equiv.apply_symm_apply",
"equiv.coe_fn_mk",
"equiv.symm_apply_apply",
"inv_fun",
"normalize_eq",
"set.Iic",
"subtype.coe_eta",
"subtype.coe_mk",
"subtype.mk_le_mk",
"subtype.prop"
] | The order isomorphism between the factors of `mk m` and the factors of `mk n` induced by a
bijection between the factors of `m` and the factors of `n` that preserves `∣`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_normalized_factors_factor_dvd_iso_of_mem_normalized_factors [decidable_eq N] {m p : M}
{n : N} (hm : m ≠ 0) (hn : n ≠ 0) (hp : p ∈ normalized_factors m)
{d : {l : M // l ∣ m} ≃ {l : N // l ∣ n}}
(hd : ∀ l l', ((d l) : N) ∣ (d l') ↔ (l : M) ∣ (l' : M)) :
↑(d ⟨p, dvd_of_mem_normalized_factors hp⟩) ∈ normali... | begin
suffices : prime ↑(d ⟨associates_equiv_of_unique_units (associates_equiv_of_unique_units.symm p),
by simp [dvd_of_mem_normalized_factors hp]⟩),
{ simp only [associates_equiv_of_unique_units_apply, out_mk, normalize_eq,
associates_equiv_of_unique_units_symm_apply] at this,
obtain ⟨q, hq, hq'⟩ := ... | lemma | mem_normalized_factors_factor_dvd_iso_of_mem_normalized_factors | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"associates.mk",
"associates.prime_mk",
"classical.dec_eq",
"irreducible",
"map_prime_of_factor_order_iso",
"mk_factor_order_iso_of_factor_dvd_equiv",
"normalize_eq",
"prime",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalized_factor {m p : M} {n : N}
(hm : m ≠ 0) (hn : n ≠ 0) (hp : p ∈ normalized_factors m)
{d : {l : M // l ∣ m} ≃ {l : N // l ∣ n}} (hd : ∀ l l', ((d l) : N) ∣ (d l') ↔ (l : M) ∣ l') :
multiplicity ((d ⟨p, dvd_of_mem_normalized_factors hp⟩) : N) n = multipl... | begin
apply eq.symm,
suffices : multiplicity (associates.mk p) (associates.mk m) = multiplicity (associates.mk
↑(d ⟨associates_equiv_of_unique_units (associates_equiv_of_unique_units.symm p),
by simp [dvd_of_mem_normalized_factors hp]⟩))
(associates.mk n),
{ simpa only [multiplicity_mk_eq_multiplici... | lemma | multiplicity_factor_dvd_iso_eq_multiplicity_of_mem_normalized_factor | ring_theory | src/ring_theory/chain_of_divisors.lean | [
"algebra.is_prime_pow",
"algebra.squarefree",
"order.hom.bounded",
"algebra.gcd_monoid.basic"
] | [
"associates",
"associates.mk",
"classical.dec_eq",
"irreducible",
"mk_factor_order_iso_of_factor_dvd_equiv",
"multiplicity",
"multiplicity_prime_eq_multiplicity_image_by_factor_order_iso",
"normalize_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_principal_ideal : Kˣ →* (fractional_ideal R⁰ K)ˣ | { to_fun := λ x,
⟨span_singleton _ x,
span_singleton _ x⁻¹,
by simp only [span_singleton_one, units.mul_inv', span_singleton_mul_span_singleton],
by simp only [span_singleton_one, units.inv_mul', span_singleton_mul_span_singleton]⟩,
map_mul' := λ x y, ext
(by simp only [units.coe_mk, units.coe_mul, spa... | def | to_principal_ideal | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"units.coe_mk",
"units.coe_mul",
"units.coe_one",
"units.inv_mul'",
"units.mul_inv'"
] | `to_principal_ideal R K x` sends `x ≠ 0 : K` to the fractional `R`-ideal generated by `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_principal_ideal (x : Kˣ) :
(to_principal_ideal R K x : fractional_ideal R⁰ K) = span_singleton _ x | by { simp only [to_principal_ideal], refl } | lemma | coe_to_principal_ideal | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"to_principal_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_principal_ideal_eq_iff {I : (fractional_ideal R⁰ K)ˣ} {x : Kˣ} :
to_principal_ideal R K x = I ↔ span_singleton R⁰ (x : K) = I | by { simp only [to_principal_ideal], exact units.ext_iff } | lemma | to_principal_ideal_eq_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"to_principal_ideal",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_principal_ideals_iff {I : (fractional_ideal R⁰ K)ˣ} :
I ∈ (to_principal_ideal R K).range ↔ ∃ x : K, span_singleton R⁰ x = I | begin
simp only [monoid_hom.mem_range, to_principal_ideal_eq_iff],
split; rintros ⟨x, hx⟩,
{ exact ⟨x, hx⟩ },
{ refine ⟨units.mk0 x _, hx⟩,
rintro rfl,
simpa [I.ne_zero.symm] using hx }
end | lemma | mem_principal_ideals_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"monoid_hom.mem_range",
"to_principal_ideal",
"to_principal_ideal_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_ideals.normal : (to_principal_ideal R K).range.normal | subgroup.normal_of_comm _ | instance | principal_ideals.normal | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"subgroup.normal_of_comm",
"to_principal_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group | (fractional_ideal R⁰ (fraction_ring R))ˣ ⧸ (to_principal_ideal R (fraction_ring R)).range | def | class_group | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fraction_ring",
"fractional_ideal",
"to_principal_ideal"
] | The ideal class group of `R` is the group of invertible fractional ideals
modulo the principal ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
class_group.mk : (fractional_ideal R⁰ K)ˣ →* class_group R | (quotient_group.mk' (to_principal_ideal R (fraction_ring R)).range).comp
(units.map (fractional_ideal.canonical_equiv R⁰ K (fraction_ring R))) | def | class_group.mk | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group",
"fraction_ring",
"fractional_ideal",
"fractional_ideal.canonical_equiv",
"quotient_group.mk'",
"to_principal_ideal",
"units.map"
] | Send a nonzero fractional ideal to the corresponding class in the class group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
class_group.mk_eq_mk {I J : (fractional_ideal R⁰ $ fraction_ring R)ˣ} :
class_group.mk I = class_group.mk J
↔ ∃ x : (fraction_ring R)ˣ, I * to_principal_ideal R (fraction_ring R) x = J | by { erw [quotient_group.mk'_eq_mk', canonical_equiv_self, units.map_id, set.exists_range_iff],
refl } | lemma | class_group.mk_eq_mk | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.mk",
"fraction_ring",
"fractional_ideal",
"quotient_group.mk'_eq_mk'",
"set.exists_range_iff",
"to_principal_ideal",
"units.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk_eq_mk_of_coe_ideal
{I J : (fractional_ideal R⁰ $ fraction_ring R)ˣ} {I' J' : ideal R}
(hI : (I : fractional_ideal R⁰ $ fraction_ring R) = I')
(hJ : (J : fractional_ideal R⁰ $ fraction_ring R) = J') :
class_group.mk I = class_group.mk J
↔ ∃ x y : R, x ≠ 0 ∧ y ≠ 0 ∧ ideal.span {x} * I' = ideal.... | begin
rw [class_group.mk_eq_mk],
split,
{ rintro ⟨x, rfl⟩,
rw [units.coe_mul, hI, coe_to_principal_ideal, mul_comm,
span_singleton_mul_coe_ideal_eq_coe_ideal] at hJ,
exact ⟨_, _, sec_fst_ne_zero le_rfl x.ne_zero, sec_snd_ne_zero le_rfl ↑x, hJ⟩ },
{ rintro ⟨x, y, hx, hy, h⟩,
split, rw [mul_co... | lemma | class_group.mk_eq_mk_of_coe_ideal | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.mk",
"class_group.mk_eq_mk",
"coe_to_principal_ideal",
"fraction_ring",
"fractional_ideal",
"ideal",
"ideal.span",
"le_rfl",
"mem_non_zero_divisors_of_ne_zero",
"mul_comm",
"units.coe_mul",
"units.eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk_eq_one_of_coe_ideal {I : (fractional_ideal R⁰ $ fraction_ring R)ˣ}
{I' : ideal R} (hI : (I : fractional_ideal R⁰ $ fraction_ring R) = I') :
class_group.mk I = 1 ↔ ∃ x : R, x ≠ 0 ∧ I' = ideal.span {x} | begin
rw [← map_one class_group.mk, class_group.mk_eq_mk_of_coe_ideal hI (_ : _ = ↑⊤)],
any_goals { refl },
split,
{ rintro ⟨x, y, hx, hy, h⟩,
rw [ideal.mul_top] at h,
rcases ideal.mem_span_singleton_mul.mp ((ideal.span_singleton_le_iff_mem _).mp h.ge)
with ⟨i, hi, rfl⟩,
rw [← ideal.span_singl... | lemma | class_group.mk_eq_one_of_coe_ideal | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.mk",
"class_group.mk_eq_mk_of_coe_ideal",
"fraction_ring",
"fractional_ideal",
"ideal",
"ideal.mul_top",
"ideal.span",
"ideal.span_singleton_le_iff_mem",
"ideal.span_singleton_mul_right_inj",
"ideal.span_singleton_mul_span_singleton",
"ideal.span_singleton_one",
"ideal.top_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.induction {P : class_group R → Prop}
(h : ∀ (I : (fractional_ideal R⁰ K)ˣ), P (class_group.mk I)) (x : class_group R) : P x | quotient_group.induction_on x (λ I, begin
convert h (units.map_equiv ↑(canonical_equiv R⁰ (fraction_ring R) K) I),
ext : 1,
rw [units.coe_map, units.coe_map_equiv],
exact (canonical_equiv_flip R⁰ K (fraction_ring R) I).symm
end) | lemma | class_group.induction | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group",
"class_group.mk",
"fraction_ring",
"fractional_ideal",
"quotient_group.induction_on",
"units.coe_map",
"units.coe_map_equiv",
"units.map_equiv"
] | Induction principle for the class group: to show something holds for all `x : class_group R`,
we can choose a fraction field `K` and show it holds for the equivalence class of each
`I : fractional_ideal R⁰ K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
class_group.equiv :
class_group R ≃* (fractional_ideal R⁰ K)ˣ ⧸ (to_principal_ideal R K).range | quotient_group.congr _ _
(units.map_equiv (fractional_ideal.canonical_equiv R⁰ (fraction_ring R) K :
fractional_ideal R⁰ (fraction_ring R) ≃* fractional_ideal R⁰ K)) $
begin
ext I,
simp only [subgroup.mem_map, mem_principal_ideals_iff, monoid_hom.coe_coe],
split,
{ rintro ⟨I, ⟨x, hx⟩, rfl⟩,
refine ⟨fr... | def | class_group.equiv | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group",
"fraction_ring",
"fraction_ring.alg_equiv",
"fractional_ideal",
"fractional_ideal.canonical_equiv",
"mem_principal_ideals_iff",
"monoid_hom.coe_coe",
"quotient_group.congr",
"ring_equiv.coe_to_mul_equiv",
"subgroup.mem_map",
"to_principal_ideal",
"units.coe_map_equiv",
"units.... | The definition of the class group does not depend on the choice of field of fractions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
class_group.equiv_mk (K' : Type*) [field K'] [algebra R K'] [is_fraction_ring R K']
(I : (fractional_ideal R⁰ K)ˣ) :
class_group.equiv K' (class_group.mk I) =
quotient_group.mk' _ (units.map_equiv ↑(fractional_ideal.canonical_equiv R⁰ K K') I) | begin
rw [class_group.equiv, class_group.mk, monoid_hom.comp_apply, quotient_group.congr_mk'],
congr,
ext : 1,
rw [units.coe_map_equiv, units.coe_map_equiv, units.coe_map],
exact fractional_ideal.canonical_equiv_canonical_equiv _ _ _ _ _
end | lemma | class_group.equiv_mk | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"class_group.equiv",
"class_group.mk",
"field",
"fractional_ideal",
"fractional_ideal.canonical_equiv",
"fractional_ideal.canonical_equiv_canonical_equiv",
"is_fraction_ring",
"monoid_hom.comp_apply",
"quotient_group.congr_mk'",
"quotient_group.mk'",
"units.coe_map",
"units.coe_ma... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk_canonical_equiv (K' : Type*) [field K'] [algebra R K']
[is_fraction_ring R K'] (I : (fractional_ideal R⁰ K)ˣ) :
class_group.mk (units.map ↑(canonical_equiv R⁰ K K') I : (fractional_ideal R⁰ K')ˣ) =
class_group.mk I | by rw [class_group.mk, monoid_hom.comp_apply, ← monoid_hom.comp_apply (units.map _),
← units.map_comp, ← ring_equiv.coe_monoid_hom_trans,
fractional_ideal.canonical_equiv_trans_canonical_equiv]; refl | lemma | class_group.mk_canonical_equiv | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"class_group.mk",
"field",
"fractional_ideal",
"fractional_ideal.canonical_equiv_trans_canonical_equiv",
"is_fraction_ring",
"monoid_hom.comp_apply",
"ring_equiv.coe_monoid_hom_trans",
"units.map",
"units.map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.mk0 [is_dedekind_domain R] :
(ideal R)⁰ →* (fractional_ideal R⁰ K)ˣ | { to_fun := λ I, units.mk0 I (coe_ideal_ne_zero.mpr $ mem_non_zero_divisors_iff_ne_zero.mp I.2),
map_one' := by simp,
map_mul' := λ x y, by simp } | def | fractional_ideal.mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"ideal",
"is_dedekind_domain",
"units.mk0"
] | Send a nonzero integral ideal to an invertible fractional ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fractional_ideal.coe_mk0 [is_dedekind_domain R] (I : (ideal R)⁰) :
(fractional_ideal.mk0 K I : fractional_ideal R⁰ K) = I | rfl | lemma | fractional_ideal.coe_mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"fractional_ideal",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.canonical_equiv_mk0 [is_dedekind_domain R]
(K' : Type*) [field K'] [algebra R K'] [is_fraction_ring R K'] (I : (ideal R)⁰) :
fractional_ideal.canonical_equiv R⁰ K K' (fractional_ideal.mk0 K I) = fractional_ideal.mk0 K' I | by simp only [fractional_ideal.coe_mk0, coe_coe, fractional_ideal.canonical_equiv_coe_ideal] | lemma | fractional_ideal.canonical_equiv_mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"coe_coe",
"field",
"fractional_ideal.canonical_equiv",
"fractional_ideal.canonical_equiv_coe_ideal",
"fractional_ideal.coe_mk0",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain",
"is_fraction_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_ideal.map_canonical_equiv_mk0 [is_dedekind_domain R]
(K' : Type*) [field K'] [algebra R K'] [is_fraction_ring R K'] (I : (ideal R)⁰) :
units.map ↑(fractional_ideal.canonical_equiv R⁰ K K') (fractional_ideal.mk0 K I) =
fractional_ideal.mk0 K' I | units.ext (fractional_ideal.canonical_equiv_mk0 K K' I) | lemma | fractional_ideal.map_canonical_equiv_mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra",
"field",
"fractional_ideal.canonical_equiv",
"fractional_ideal.canonical_equiv_mk0",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain",
"is_fraction_ring",
"units.ext",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk0 [is_dedekind_domain R] : (ideal R)⁰ →* class_group R | class_group.mk.comp (fractional_ideal.mk0 (fraction_ring R)) | def | class_group.mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group",
"fraction_ring",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain"
] | Send a nonzero ideal to the corresponding class in the class group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
class_group.mk_mk0 [is_dedekind_domain R] (I : (ideal R)⁰):
class_group.mk (fractional_ideal.mk0 K I) = class_group.mk0 I | by rw [class_group.mk0, monoid_hom.comp_apply,
← class_group.mk_canonical_equiv K (fraction_ring R),
fractional_ideal.map_canonical_equiv_mk0] | lemma | class_group.mk_mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.mk",
"class_group.mk0",
"class_group.mk_canonical_equiv",
"fraction_ring",
"fractional_ideal.map_canonical_equiv_mk0",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain",
"monoid_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.equiv_mk0 [is_dedekind_domain R] (I : (ideal R)⁰):
class_group.equiv K (class_group.mk0 I) =
quotient_group.mk' (to_principal_ideal R K).range (fractional_ideal.mk0 K I) | begin
rw [class_group.mk0, monoid_hom.comp_apply, class_group.equiv_mk],
congr,
ext,
simp
end | lemma | class_group.equiv_mk0 | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.equiv",
"class_group.equiv_mk",
"class_group.mk0",
"fractional_ideal.mk0",
"ideal",
"is_dedekind_domain",
"monoid_hom.comp_apply",
"quotient_group.mk'",
"to_principal_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk0_eq_mk0_iff_exists_fraction_ring [is_dedekind_domain R] {I J : (ideal R)⁰} :
class_group.mk0 I = class_group.mk0 J ↔ ∃ (x ≠ (0 : K)), span_singleton R⁰ x * I = J | begin
refine (class_group.equiv K).injective.eq_iff.symm.trans _,
simp only [class_group.equiv_mk0, quotient_group.mk'_eq_mk', mem_principal_ideals_iff,
coe_coe, units.ext_iff, units.coe_mul, fractional_ideal.coe_mk0, exists_prop],
split,
{ rintros ⟨X, ⟨x, hX⟩, hx⟩,
refine ⟨x, _, _⟩,
{ rintro rfl, s... | lemma | class_group.mk0_eq_mk0_iff_exists_fraction_ring | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.equiv",
"class_group.equiv_mk0",
"class_group.mk0",
"coe_coe",
"exists_prop",
"fractional_ideal.coe_mk0",
"ideal",
"is_dedekind_domain",
"mem_principal_ideals_iff",
"mul_comm",
"quotient_group.mk'_eq_mk'",
"units.coe_mul",
"units.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk0_eq_mk0_iff [is_dedekind_domain R] {I J : (ideal R)⁰} :
class_group.mk0 I = class_group.mk0 J ↔
∃ (x y : R) (hx : x ≠ 0) (hy : y ≠ 0), ideal.span {x} * (I : ideal R) = ideal.span {y} * J | begin
refine (class_group.mk0_eq_mk0_iff_exists_fraction_ring (fraction_ring R)).trans ⟨_, _⟩,
{ rintros ⟨z, hz, h⟩,
obtain ⟨x, ⟨y, hy⟩, rfl⟩ := is_localization.mk'_surjective R⁰ z,
refine ⟨x, y, _, mem_non_zero_divisors_iff_ne_zero.mp hy, _⟩,
{ rintro hx, apply hz,
rw [hx, is_fraction_ring.mk'_eq... | lemma | class_group.mk0_eq_mk0_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra_map",
"class_group.mk0",
"class_group.mk0_eq_mk0_iff_exists_fraction_ring",
"fraction_ring",
"fractional_ideal.mk'_mul_coe_ideal_eq_coe_ideal",
"ideal",
"ideal.span",
"is_dedekind_domain",
"is_fraction_ring.injective",
"is_fraction_ring.mk'_eq_div",
"is_localization.mk'_surjective",
"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk0_surjective [is_dedekind_domain R] :
function.surjective (class_group.mk0 : (ideal R)⁰ → class_group R) | begin
rintros ⟨I⟩,
obtain ⟨a, a_ne_zero', ha⟩ := I.1.2,
have a_ne_zero := mem_non_zero_divisors_iff_ne_zero.mp a_ne_zero',
have fa_ne_zero : (algebra_map R (fraction_ring R)) a ≠ 0 :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors a_ne_zero',
refine ⟨⟨{ carrier := { x | (algebra_map R _ a)⁻¹ * a... | lemma | class_group.mk0_surjective | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"algebra.smul_def",
"algebra_map",
"class_group",
"class_group.mk0",
"coe_coe",
"coe_to_principal_ideal",
"fraction_ring",
"fractional_ideal",
"fractional_ideal.coe_mk0",
"fractional_ideal.eq_span_singleton_mul",
"fractional_ideal.map_canonical_equiv_mk0",
"ideal",
"inv_mul_cancel",
"is_de... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk_eq_one_iff {I : (fractional_ideal R⁰ K)ˣ} :
class_group.mk I = 1 ↔ (I : submodule R K).is_principal | begin
simp only [← (class_group.equiv K).injective.eq_iff, _root_.map_one, class_group.equiv_mk,
quotient_group.mk'_apply, quotient_group.eq_one_iff, monoid_hom.mem_range, units.ext_iff,
coe_to_principal_ideal, units.coe_map_equiv, fractional_ideal.canonical_equiv_self, coe_coe,
ring_equiv.coe_mul_e... | lemma | class_group.mk_eq_one_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.equiv",
"class_group.equiv_mk",
"class_group.mk",
"coe_coe",
"coe_to_principal_ideal",
"fractional_ideal",
"fractional_ideal.canonical_equiv_self",
"monoid_hom.mem_range",
"mul_equiv.refl_apply",
"quotient_group.eq_one_iff",
"quotient_group.mk'_apply",
"ring_equiv.coe_mul_equiv_re... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
class_group.mk0_eq_one_iff [is_dedekind_domain R] {I : ideal R} (hI : I ∈ (ideal R)⁰) :
class_group.mk0 ⟨I, hI⟩ = 1 ↔ I.is_principal | class_group.mk_eq_one_iff.trans (coe_submodule_is_principal R _) | lemma | class_group.mk0_eq_one_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group.mk0",
"ideal",
"is_dedekind_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_class_group_eq_one [is_principal_ideal_ring R] : fintype.card (class_group R) = 1 | begin
rw fintype.card_eq_one_iff,
use 1,
refine class_group.induction (fraction_ring R) (λ I, _),
exact class_group.mk_eq_one_iff.mpr (I : fractional_ideal R⁰ (fraction_ring R)).is_principal
end | lemma | card_class_group_eq_one | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"class_group",
"class_group.induction",
"fintype.card",
"fintype.card_eq_one_iff",
"fraction_ring",
"fractional_ideal",
"is_principal_ideal_ring"
] | The class number of a principal ideal domain is `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_class_group_eq_one_iff [is_dedekind_domain R] [fintype (class_group R)] :
fintype.card (class_group R) = 1 ↔ is_principal_ideal_ring R | begin
split, swap, { introsI, convert card_class_group_eq_one, assumption, },
rw fintype.card_eq_one_iff,
rintros ⟨I, hI⟩,
have eq_one : ∀ J : class_group R, J = 1 := λ J, trans (hI J) (hI 1).symm,
refine ⟨λ I, _⟩,
by_cases hI : I = ⊥,
{ rw hI, exact bot_is_principal },
exact (class_group.mk0_eq_one_iff... | lemma | card_class_group_eq_one_iff | ring_theory | src/ring_theory/class_group.lean | [
"group_theory.quotient_group",
"ring_theory.dedekind_domain.ideal"
] | [
"bot_is_principal",
"card_class_group_eq_one",
"class_group",
"class_group.mk0_eq_one_iff",
"fintype",
"fintype.card",
"fintype.card_eq_one_iff",
"is_dedekind_domain",
"is_principal_ideal_ring"
] | The class number is `1` iff the ring of integers is a principal ideal domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.left_mul_matrix_complex (z : ℂ) :
algebra.left_mul_matrix complex.basis_one_I z = !![z.re, -z.im; z.im, z.re] | begin
ext i j,
rw [algebra.left_mul_matrix_eq_repr_mul, complex.coe_basis_one_I_repr, complex.coe_basis_one_I,
mul_re, mul_im, matrix.of_apply],
fin_cases j,
{ simp_rw [matrix.cons_val_zero, one_re, one_im, mul_zero, mul_one, sub_zero, zero_add],
fin_cases i; refl },
{ simp_rw [matrix.cons_val_one, ma... | lemma | algebra.left_mul_matrix_complex | ring_theory | src/ring_theory/complex.lean | [
"data.complex.module",
"ring_theory.norm",
"ring_theory.trace"
] | [
"algebra.left_mul_matrix",
"algebra.left_mul_matrix_eq_repr_mul",
"complex.basis_one_I",
"complex.coe_basis_one_I",
"complex.coe_basis_one_I_repr",
"matrix.cons_val_one",
"matrix.cons_val_zero",
"matrix.head_cons",
"matrix.of_apply",
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.trace_complex_apply (z : ℂ) : algebra.trace ℝ ℂ z = 2*z.re | begin
rw [algebra.trace_eq_matrix_trace complex.basis_one_I,
algebra.left_mul_matrix_complex, matrix.trace_fin_two],
exact (two_mul _).symm
end | lemma | algebra.trace_complex_apply | ring_theory | src/ring_theory/complex.lean | [
"data.complex.module",
"ring_theory.norm",
"ring_theory.trace"
] | [
"algebra.left_mul_matrix_complex",
"algebra.trace",
"algebra.trace_eq_matrix_trace",
"complex.basis_one_I",
"matrix.trace_fin_two",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.norm_complex_apply (z : ℂ) : algebra.norm ℝ z = z.norm_sq | begin
rw [algebra.norm_eq_matrix_det complex.basis_one_I,
algebra.left_mul_matrix_complex, matrix.det_fin_two, norm_sq_apply],
simp,
end | lemma | algebra.norm_complex_apply | ring_theory | src/ring_theory/complex.lean | [
"data.complex.module",
"ring_theory.norm",
"ring_theory.trace"
] | [
"algebra.left_mul_matrix_complex",
"algebra.norm",
"algebra.norm_eq_matrix_det",
"complex.basis_one_I",
"matrix.det_fin_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra.norm_complex_eq : algebra.norm ℝ = norm_sq.to_monoid_hom | monoid_hom.ext algebra.norm_complex_apply | lemma | algebra.norm_complex_eq | ring_theory | src/ring_theory/complex.lean | [
"data.complex.module",
"ring_theory.norm",
"ring_theory.trace"
] | [
"algebra.norm",
"algebra.norm_complex_apply",
"monoid_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_con (R : Type*) [has_add R] [has_mul R] extends setoid R | (add' : ∀ {w x y z}, r w x → r y z → r (w + y) (x + z))
(mul' : ∀ {w x y z}, r w x → r y z → r (w * y) (x * z)) | structure | ring_con | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_con_gen.rel [has_add R] [has_mul R] (r : R → R → Prop) : R → R → Prop
| of : Π x y, r x y → ring_con_gen.rel x y
| refl : Π x, ring_con_gen.rel x x
| symm : Π {x y}, ring_con_gen.rel x y → ring_con_gen.rel y x
| trans : Π {x y z}, ring_con_gen.rel x y → ring_con_gen.rel y z → ring_con_gen.rel x z
| add : Π {w x y ... | inductive | ring_con_gen.rel | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | The inductively defined smallest ring congruence relation containing a given binary
relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_con_gen [has_add R] [has_mul R] (r : R → R → Prop) : ring_con R | { r := ring_con_gen.rel r,
iseqv := ⟨ring_con_gen.rel.refl, @ring_con_gen.rel.symm _ _ _ _, @ring_con_gen.rel.trans _ _ _ _⟩,
add' := λ _ _ _ _, ring_con_gen.rel.add,
mul' := λ _ _ _ _, ring_con_gen.rel.mul } | def | ring_con_gen | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"ring_con",
"ring_con_gen.rel"
] | The inductively defined smallest ring congruence relation containing a given binary
relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add_con : add_con R | { ..c } | def | ring_con.to_add_con | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"add_con"
] | Every `ring_con` is also an `add_con` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_con : con R | { ..c } | def | ring_con.to_con | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"con"
] | Every `ring_con` is also a `con` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_eq_coe : c.r = c | rfl | lemma | ring_con.rel_eq_coe | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (x) : c x x | c.refl' x | lemma | ring_con.refl | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm {x y} : c x y → c y x | c.symm' | lemma | ring_con.symm | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {x y z} : c x y → c y z → c x z | c.trans' | lemma | ring_con.trans | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add {w x y z} : c w x → c y z → c (w + y) (x + z) | c.add' | lemma | ring_con.add | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul {w x y z} : c w x → c y z → c (w * y) (x * z) | c.mul' | lemma | ring_con.mul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_mk {s : setoid R} {ha hm a b} : ring_con.mk s ha hm a b ↔ setoid.r a b | iff.rfl | lemma | ring_con.rel_mk | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient | quotient c.to_setoid | def | ring_con.quotient | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | Defining the quotient by a congruence relation of a type with addition and multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_mk_eq_coe (x : R) : quot.mk c x = (x : c.quotient) | rfl | lemma | ring_con.quot_mk_eq_coe | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq {a b : R} : (a : c.quotient) = b ↔ c a b | quotient.eq' | lemma | ring_con.eq | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"quotient.eq'"
] | Two elements are related by a congruence relation `c` iff they are represented by the same
element of the quotient by `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_add (x y : R) : (↑(x + y) : c.quotient) = ↑x + ↑y | rfl | lemma | ring_con.coe_add | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (x y : R) : (↑(x * y) : c.quotient) = ↑x * ↑y | rfl | lemma | ring_con.coe_mul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : (↑(0 : R) : c.quotient) = 0 | rfl | lemma | ring_con.coe_zero | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : (↑(1 : R) : c.quotient) = 1 | rfl | lemma | ring_con.coe_one | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (a : α) (x : R) : (↑(a • x) : c.quotient) = a • x | rfl | lemma | ring_con.coe_smul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : R) : (↑(-x) : c.quotient) = -x | rfl | lemma | ring_con.coe_neg | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (x y : R) : (↑(x - y) : c.quotient) = x - y | rfl | lemma | ring_con.coe_sub | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zsmul : has_smul ℤ c.quotient | c.to_add_con^.quotient.has_zsmul | instance | ring_con.has_zsmul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul (z : ℤ) (x : R) : (↑(z • x) : c.quotient) = z • x | rfl | lemma | ring_con.coe_zsmul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul : has_smul ℕ c.quotient | c.to_add_con^.quotient.has_nsmul | instance | ring_con.has_nsmul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (n : ℕ) (x : R) : (↑(n • x) : c.quotient) = n • x | rfl | lemma | ring_con.coe_nsmul | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (x : R) (n : ℕ) : (↑(x ^ n) : c.quotient) = x ^ n | rfl | lemma | ring_con.coe_pow | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast (n : ℕ) : (↑(n : R) : c.quotient) = n | rfl | lemma | ring_con.coe_nat_cast | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast (n : ℕ) : (↑(n : R) : c.quotient) = n | rfl | lemma | ring_con.coe_int_cast | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_scalar_tower_right [has_add R] [mul_one_class R] [has_smul α R] [is_scalar_tower α R R]
(c : ring_con R) :
is_scalar_tower α c.quotient c.quotient | { smul_assoc := λ a, quotient.ind₂' $ by exact λ m₁ m₂,
congr_arg quotient.mk' $ smul_mul_assoc _ _ _ } | instance | ring_con.is_scalar_tower_right | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"has_smul",
"is_scalar_tower",
"mul_one_class",
"quotient.ind₂'",
"quotient.mk'",
"ring_con",
"smul_assoc",
"smul_mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class [has_add R] [mul_one_class R] [has_smul α R]
[is_scalar_tower α R R] [smul_comm_class α R R] (c : ring_con R) :
smul_comm_class α c.quotient c.quotient | { smul_comm := λ a, quotient.ind₂' $ by exact λ m₁ m₂,
congr_arg quotient.mk' $ (mul_smul_comm _ _ _).symm } | instance | ring_con.smul_comm_class | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"has_smul",
"is_scalar_tower",
"mul_one_class",
"mul_smul_comm",
"quotient.ind₂'",
"quotient.mk'",
"ring_con",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comm_class' [has_add R] [mul_one_class R] [has_smul α R]
[is_scalar_tower α R R] [smul_comm_class R α R] (c : ring_con R) :
smul_comm_class c.quotient α c.quotient | by haveI := smul_comm_class.symm R α R; exact smul_comm_class.symm _ _ _ | instance | ring_con.smul_comm_class' | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"has_smul",
"is_scalar_tower",
"mul_one_class",
"ring_con",
"smul_comm_class",
"smul_comm_class.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' [non_assoc_semiring R] (c : ring_con R) : R →+* c.quotient | { to_fun := quotient.mk', map_zero' := rfl, map_one' := rfl,
map_add' := λ _ _, rfl, map_mul' := λ _ _, rfl } | def | ring_con.mk' | ring_theory | src/ring_theory/congruence.lean | [
"algebra.group_ring_action.basic",
"algebra.hom.ring",
"algebra.ring.inj_surj",
"group_theory.congruence"
] | [
"mk'",
"non_assoc_semiring",
"quotient.mk'",
"ring_con"
] | The natural homomorphism from a ring to its quotient by a congruence relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr (A : Type u) {B : Type v} [comm_ring A] [comm_ring B] [algebra A B] [fintype ι]
(b : ι → B) | by { classical, exact (trace_matrix A b).det } | def | algebra.discr | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra",
"comm_ring",
"fintype"
] | Given an `A`-algebra `B` and `b`, an `ι`-indexed family of elements of `B`, we define
`discr A ι b` as the determinant of `trace_matrix A ι b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_def [decidable_eq ι] [fintype ι] (b : ι → B) :
discr A b = (trace_matrix A b).det | by convert rfl | lemma | algebra.discr_def | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discr_reindex (b : basis ι A B) (f : ι ≃ ι') :
discr A (b ∘ ⇑(f.symm)) = discr A b | begin
classical,
rw [← basis.coe_reindex, discr_def, trace_matrix_reindex, det_reindex_self, ← discr_def]
end | lemma | algebra.discr_reindex | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"basis",
"basis.coe_reindex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discr_zero_of_not_linear_independent [is_domain A] {b : ι → B}
(hli : ¬linear_independent A b) : discr A b = 0 | begin
classical,
obtain ⟨g, hg, i, hi⟩ := fintype.not_linear_independent_iff.1 hli,
have : (trace_matrix A b).mul_vec g = 0,
{ ext i,
have : ∀ j, (trace A B) (b i * b j) * g j = (trace A B) (((g j) • (b j)) * b i),
{ intro j, simp [mul_comm], },
simp only [mul_vec, dot_product, trace_matrix_apply, p... | lemma | algebra.discr_zero_of_not_linear_independent | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"by_contra",
"is_domain",
"linear_independent",
"linear_map.map_sum",
"linear_map.map_zero",
"matrix.eq_zero_of_mul_vec_eq_zero",
"mul_comm",
"zero_mul"
] | If `b` is not linear independent, then `algebra.discr A b = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_of_matrix_vec_mul [decidable_eq ι] (b : ι → B) (P : matrix ι ι A) :
discr A ((P.map (algebra_map A B)).vec_mul b) = P.det ^ 2 * discr A b | by rw [discr_def, trace_matrix_of_matrix_vec_mul, det_mul, det_mul, det_transpose, mul_comm,
← mul_assoc, discr_def, pow_two] | lemma | algebra.discr_of_matrix_vec_mul | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"matrix",
"mul_assoc",
"mul_comm",
"pow_two"
] | Relation between `algebra.discr A ι b` and
`algebra.discr A ((P.map (algebra_map A B)).vec_mul b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_of_matrix_mul_vec [decidable_eq ι] (b : ι → B) (P : matrix ι ι A) :
discr A ((P.map (algebra_map A B)).mul_vec b) = P.det ^ 2 * discr A b | by rw [discr_def, trace_matrix_of_matrix_mul_vec, det_mul, det_mul, det_transpose,
mul_comm, ← mul_assoc, discr_def, pow_two] | lemma | algebra.discr_of_matrix_mul_vec | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"matrix",
"mul_assoc",
"mul_comm",
"pow_two"
] | Relation between `algebra.discr A ι b` and
`algebra.discr A ((P.map (algebra_map A B)).mul_vec b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_not_zero_of_basis [is_separable K L] (b : basis ι K L) : discr K b ≠ 0 | begin
casesI is_empty_or_nonempty ι,
{ simp [discr] },
{ have := span_eq_top_of_linear_independent_of_card_eq_finrank b.linear_independent
(finrank_eq_card_basis b).symm,
classical,
rw [discr_def, trace_matrix],
simp_rw [← basis.mk_apply b.linear_independent this.ge],
rw [← trace_matrix, tra... | lemma | algebra.discr_not_zero_of_basis | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"basis",
"basis.mk_apply",
"bilin_form.nondegenerate_iff_det_ne_zero",
"is_empty_or_nonempty",
"is_separable",
"span_eq_top_of_linear_independent_of_card_eq_finrank",
"trace_form_nondegenerate"
] | Over a field, if `b` is a basis, then `algebra.discr K b ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_is_unit_of_basis [is_separable K L] (b : basis ι K L) : is_unit (discr K b) | is_unit.mk0 _ (discr_not_zero_of_basis _ _) | lemma | algebra.discr_is_unit_of_basis | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"basis",
"is_separable",
"is_unit",
"is_unit.mk0"
] | Over a field, if `b` is a basis, then `algebra.discr K b` is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_eq_det_embeddings_matrix_reindex_pow_two [decidable_eq ι] [is_separable K L]
(e : ι ≃ (L →ₐ[K] E)) : algebra_map K E (discr K b) =
(embeddings_matrix_reindex K E b e).det ^ 2 | by rw [discr_def, ring_hom.map_det, ring_hom.map_matrix_apply,
trace_matrix_eq_embeddings_matrix_reindex_mul_trans, det_mul, det_transpose, pow_two] | lemma | algebra.discr_eq_det_embeddings_matrix_reindex_pow_two | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"is_separable",
"pow_two",
"ring_hom.map_det"
] | If `L/K` is a field extension and `b : ι → L`, then `discr K b` is the square of the
determinant of the matrix whose `(i, j)` coefficient is `σⱼ (b i)`, where `σⱼ : L →ₐ[K] E` is the
embedding in an algebraically closed field `E` corresponding to `j : ι` via a bijection
`e : ι ≃ (L →ₐ[K] E)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_power_basis_eq_prod (e : fin pb.dim ≃ (L →ₐ[K] E)) [is_separable K L] :
algebra_map K E (discr K pb.basis) =
∏ i : fin pb.dim, ∏ j in Ioi i, (e j pb.gen- (e i pb.gen)) ^ 2 | begin
rw [discr_eq_det_embeddings_matrix_reindex_pow_two K E pb.basis e,
embeddings_matrix_reindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow],
congr, ext i,
rw [← prod_pow]
end | lemma | algebra.discr_power_basis_eq_prod | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"is_separable"
] | The discriminant of a power basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_power_basis_eq_prod' [is_separable K L] (e : fin pb.dim ≃ (L →ₐ[K] E)) :
algebra_map K E (discr K pb.basis) =
∏ i : fin pb.dim, ∏ j in Ioi i, -((e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen)) | begin
rw [discr_power_basis_eq_prod _ _ _ e],
congr, ext i, congr, ext j,
ring
end | lemma | algebra.discr_power_basis_eq_prod' | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"is_separable",
"ring"
] | A variation of `of_power_basis_eq_prod`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_power_basis_eq_prod'' [is_separable K L] (e : fin pb.dim ≃ (L →ₐ[K] E)) :
algebra_map K E (discr K pb.basis) =
(-1) ^ (n * (n - 1) / 2) * ∏ i : fin pb.dim, ∏ j in Ioi i,
(e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen) | begin
rw [discr_power_basis_eq_prod' _ _ _ e],
simp_rw [λ i j, neg_eq_neg_one_mul ((e j pb.gen- (e i pb.gen)) * (e i pb.gen- (e j pb.gen))),
prod_mul_distrib],
congr,
simp only [prod_pow_eq_pow_sum, prod_const],
congr,
rw [← @nat.cast_inj ℚ, nat.cast_sum],
have : ∀ (x : fin pb.dim), (↑x + 1) ≤ pb.dim ... | lemma | algebra.discr_power_basis_eq_prod'' | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"alg_hom.card",
"algebra_map",
"equiv.symm",
"fin.card_Ioi",
"fin.is_lt",
"finite_dimensional.finrank_pos_iff",
"finset.card_fin",
"fintype.card_fin",
"is_separable",
"mul_one",
"nat.cast_add",
"nat.cast_div",
"nat.cast_inj",
"nat.cast_mul",
"nat.cast_one",
"nat.cast_sub",
"nat.cast_... | A variation of `of_power_basis_eq_prod`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_power_basis_eq_norm [is_separable K L] : discr K pb.basis =
(-1) ^ (n * (n - 1) / 2) * (norm K (aeval pb.gen (minpoly K pb.gen).derivative)) | begin
let E := algebraic_closure L,
letI := λ (a b : E), classical.prop_decidable (eq a b),
have e : fin pb.dim ≃ (L →ₐ[K] E),
{ refine equiv_of_card_eq _,
rw [fintype.card_fin, alg_hom.card],
exact (power_basis.finrank pb).symm },
have hnodup : (map (algebra_map K E) (minpoly K pb.gen)).roots.nodup ... | lemma | algebra.discr_power_basis_eq_norm | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"alg_hom.card",
"alg_hom.congr_fun",
"algebra_map",
"algebraic_closure",
"equiv.apply_eq_iff_eq",
"equiv.injective",
"exists_prop",
"finset.mem_mk",
"finset.prod_mk",
"fintype.card_fin",
"heq_iff_eq",
"is_alg_closed.splits_codomain",
"is_separable",
"is_separable.is_integral",
"is_separa... | Formula for the discriminant of a power basis using the norm of the field extension. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_is_integral {b : ι → L} (h : ∀ i, is_integral R (b i)) :
is_integral R (discr K b) | begin
classical,
rw [discr_def],
exact is_integral.det (λ i j, is_integral_trace (is_integral_mul (h i) (h j)))
end | lemma | algebra.discr_is_integral | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"is_integral",
"is_integral.det",
"is_integral_mul"
] | If `K` and `L` are fields and `is_scalar_tower R K L`, and `b : ι → L` satisfies
` ∀ i, is_integral R (b i)`, then `is_integral R (discr K b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_eq_discr_of_to_matrix_coeff_is_integral [number_field K] {b : basis ι ℚ K}
{b' : basis ι' ℚ K} (h : ∀ i j, is_integral ℤ (b.to_matrix b' i j))
(h' : ∀ i j, is_integral ℤ (b'.to_matrix b i j)) :
discr ℚ b = discr ℚ b' | begin
replace h' : ∀ i j, is_integral ℤ (b'.to_matrix ((b.reindex (b.index_equiv b'))) i j),
{ intros i j,
convert h' i ((b.index_equiv b').symm j),
simpa },
classical,
rw [← (b.reindex (b.index_equiv b')).to_matrix_map_vec_mul b', discr_of_matrix_vec_mul,
← one_mul (discr ℚ b), basis.coe_reindex, d... | lemma | algebra.discr_eq_discr_of_to_matrix_coeff_is_integral | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map",
"basis",
"basis.coe_reindex",
"basis.to_matrix_mul_to_matrix_flip",
"is_fraction_ring.injective",
"is_integral",
"is_integral.det",
"is_unit",
"number_field",
"one_mul",
"ring_hom.map_mul",
"ring_hom.map_one"
] | If `b` and `b'` are `ℚ`-bases of a number field `K` such that
`∀ i j, is_integral ℤ (b.to_matrix b' i j)` and `∀ i j, is_integral ℤ (b'.to_matrix b i j)` then
`discr ℚ b = discr ℚ b'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discr_mul_is_integral_mem_adjoin [is_domain R] [is_separable K L] [is_integrally_closed R]
[is_fraction_ring R K] {B : power_basis K L} (hint : is_integral R B.gen) {z : L}
(hz : is_integral R z) : (discr K B.basis) • z ∈ adjoin R ({B.gen} : set L) | begin
have hinv : is_unit (trace_matrix K B.basis).det :=
by simpa [← discr_def] using discr_is_unit_of_basis _ B.basis,
have H : (trace_matrix K B.basis).det • (trace_matrix K B.basis).mul_vec (B.basis.equiv_fun z) =
(trace_matrix K B.basis).det • (λ i, trace K L (z * B.basis i)),
{ congr, exact trace_m... | lemma | algebra.discr_mul_is_integral_mem_adjoin | ring_theory | src/ring_theory/discriminant.lean | [
"ring_theory.trace",
"ring_theory.norm",
"number_theory.number_field.basic"
] | [
"algebra_map_smul",
"basis.equiv_fun_apply",
"finset.smul_sum",
"is_domain",
"is_fraction_ring",
"is_integral",
"is_integral.pow",
"is_integral_mul",
"is_integrally_closed",
"is_separable",
"is_unit",
"pi.smul_apply",
"power_basis",
"power_basis.coe_basis",
"set.mem_singleton",
"smul_a... | Let `K` be the fraction field of an integrally closed domain `R` and let `L` be a finite
separable extension of `K`. Let `B : power_basis K L` be such that `is_integral R B.gen`.
Then for all, `z : L` that are integral over `R`, we have
`(discr K B.basis) • z ∈ adjoin R ({B.gen} : set L)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_C_mul_X_pow_of_forall_coeff_mem {f : R[X]} {P : ideal R}
(hfP : ∀ (n : ℕ), ↑n < f.degree → f.coeff n ∈ P) :
map (mk P) f = C ((mk P) f.leading_coeff) * X ^ f.nat_degree | polynomial.ext (λ n, begin
by_cases hf0 : f = 0, { simp [hf0], },
rcases lt_trichotomy ↑n (degree f) with h | h | h,
{ erw [coeff_map, eq_zero_iff_mem.2 (hfP n h), coeff_C_mul, coeff_X_pow, if_neg, mul_zero],
rintro rfl, exact not_lt_of_ge degree_le_nat_degree h },
{ have : nat_degree f = n, from nat_degree... | lemma | polynomial.eisenstein_criterion_aux.map_eq_C_mul_X_pow_of_forall_coeff_mem | ring_theory | src/ring_theory/eisenstein_criterion.lean | [
"data.nat.cast.with_top",
"ring_theory.prime",
"ring_theory.polynomial.content",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"mul_one",
"mul_zero",
"polynomial.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nat_degree_of_map_eq_mul_X_pow {n : ℕ}
{P : ideal R} (hP : P.is_prime) {q : R[X]} {c : polynomial (R ⧸ P)}
(hq : map (mk P) q = c * X ^ n) (hc0 : c.degree = 0) : n ≤ q.nat_degree | with_bot.coe_le_coe.1
(calc ↑n = degree (q.map (mk P)) :
by rw [hq, degree_mul, hc0, zero_add, degree_pow, degree_X, nsmul_one, nat.cast_with_bot]
... ≤ degree q : degree_map_le _ _
... ≤ nat_degree q : degree_le_nat_degree) | lemma | polynomial.eisenstein_criterion_aux.le_nat_degree_of_map_eq_mul_X_pow | ring_theory | src/ring_theory/eisenstein_criterion.lean | [
"data.nat.cast.with_top",
"ring_theory.prime",
"ring_theory.polynomial.content",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"nat.cast_with_bot",
"nsmul_one",
"polynomial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval_zero_mem_ideal_of_eq_mul_X_pow {n : ℕ} {P : ideal R}
{q : R[X]} {c : polynomial (R ⧸ P)}
(hq : map (mk P) q = c * X ^ n) (hn0 : 0 < n) : eval 0 q ∈ P | by rw [← coeff_zero_eq_eval_zero, ← eq_zero_iff_mem, ← coeff_map,
coeff_zero_eq_eval_zero, hq, eval_mul, eval_pow, eval_X, zero_pow hn0, mul_zero] | lemma | polynomial.eisenstein_criterion_aux.eval_zero_mem_ideal_of_eq_mul_X_pow | ring_theory | src/ring_theory/eisenstein_criterion.lean | [
"data.nat.cast.with_top",
"ring_theory.prime",
"ring_theory.polynomial.content",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"mul_zero",
"polynomial",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit {p q : R[X]}
(hu : ∀ (x : R), C x ∣ p * q → is_unit x) (hpm : p.nat_degree = 0) :
is_unit p | begin
rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpm), is_unit_C],
refine hu _ _,
rw [← eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpm)],
exact dvd_mul_right _ _
end | lemma | polynomial.eisenstein_criterion_aux.is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit | ring_theory | src/ring_theory/eisenstein_criterion.lean | [
"data.nat.cast.with_top",
"ring_theory.prime",
"ring_theory.polynomial.content",
"ring_theory.ideal.quotient_operations"
] | [
"dvd_mul_right",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_of_eisenstein_criterion {f : R[X]} {P : ideal R} (hP : P.is_prime)
(hfl : f.leading_coeff ∉ P)
(hfP : ∀ n : ℕ, ↑n < degree f → f.coeff n ∈ P)
(hfd0 : 0 < degree f) (h0 : f.coeff 0 ∉ P^2)
(hu : f.is_primitive) : irreducible f | have hf0 : f ≠ 0, from λ _, by simp only [*, not_true, submodule.zero_mem, coeff_zero] at *,
have hf : f.map (mk P) = C (mk P (leading_coeff f)) * X ^ nat_degree f,
from map_eq_C_mul_X_pow_of_forall_coeff_mem hfP,
have hfd0 : 0 < f.nat_degree, from with_bot.coe_lt_coe.1
(lt_of_lt_of_le hfd0 degree_le_nat_degree),
⟨... | theorem | polynomial.irreducible_of_eisenstein_criterion | ring_theory | src/ring_theory/eisenstein_criterion.lean | [
"data.nat.cast.with_top",
"ring_theory.prime",
"ring_theory.polynomial.content",
"ring_theory.ideal.quotient_operations"
] | [
"ideal",
"ideal.mul_mem_mul",
"imp_false",
"irreducible",
"mul_comm",
"mul_eq_mul_prime_pow",
"mul_zero",
"nat.with_bot.add_eq_zero_iff",
"not_and_distrib",
"not_not",
"or_iff_not_imp_left",
"polynomial",
"polynomial.map_mul",
"prime",
"submodule.zero_mem",
"zero_mul"
] | If `f` is a non constant polynomial with coefficients in `R`, and `P` is a prime ideal in `R`,
then if every coefficient in `R` except the leading coefficient is in `P`, and
the trailing coefficient is not in `P^2` and no non units in `R` divide `f`, then `f` is
irreducible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_unramified : Prop | (comp_injective :
∀ ⦃B : Type u⦄ [comm_ring B], by exactI ∀ [algebra R B] (I : ideal B) (hI : I ^ 2 = ⊥), by exactI
function.injective ((ideal.quotient.mkₐ R I).comp : (A →ₐ[R] B) → (A →ₐ[R] B ⧸ I))) | class | algebra.formally_unramified | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra",
"comm_ring",
"ideal",
"ideal.quotient.mkₐ"
] | An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_smooth : Prop | (comp_surjective :
∀ ⦃B : Type u⦄ [comm_ring B], by exactI ∀ [algebra R B] (I : ideal B) (hI : I ^ 2 = ⊥), by exactI
function.surjective ((ideal.quotient.mkₐ R I).comp : (A →ₐ[R] B) → (A →ₐ[R] B ⧸ I))) | class | algebra.formally_smooth | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra",
"comm_ring",
"ideal",
"ideal.quotient.mkₐ"
] | An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_etale : Prop | (comp_bijective :
∀ ⦃B : Type u⦄ [comm_ring B], by exactI ∀ [algebra R B] (I : ideal B) (hI : I ^ 2 = ⊥), by exactI
function.bijective ((ideal.quotient.mkₐ R I).comp : (A →ₐ[R] B) → (A →ₐ[R] B ⧸ I))) | class | algebra.formally_etale | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"algebra",
"comm_ring",
"ideal",
"ideal.quotient.mkₐ"
] | An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
formally_etale.iff_unramified_and_smooth :
formally_etale R A ↔ formally_unramified R A ∧ formally_smooth R A | begin
rw [formally_unramified_iff, formally_smooth_iff, formally_etale_iff],
simp_rw ← forall_and_distrib,
refl
end | lemma | algebra.formally_etale.iff_unramified_and_smooth | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"forall_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.to_unramified [h : formally_etale R A] : formally_unramified R A | (formally_etale.iff_unramified_and_smooth.mp h).1 | instance | algebra.formally_etale.to_unramified | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.to_smooth [h : formally_etale R A] : formally_smooth R A | (formally_etale.iff_unramified_and_smooth.mp h).2 | instance | algebra.formally_etale.to_smooth | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_etale.of_unramified_and_smooth [h₁ : formally_unramified R A]
[h₂ : formally_smooth R A] : formally_etale R A | formally_etale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩ | lemma | algebra.formally_etale.of_unramified_and_smooth | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.lift_unique {B : Type u} [comm_ring B] [_RB : algebra R B]
[formally_unramified R A] (I : ideal B)
(hI : is_nilpotent I) (g₁ g₂ : A →ₐ[R] B) (h : (ideal.quotient.mkₐ R I).comp g₁ =
(ideal.quotient.mkₐ R I).comp g₂) : g₁ = g₂ | begin
revert g₁ g₂,
change function.injective (ideal.quotient.mkₐ R I).comp,
unfreezingI { revert _RB },
apply ideal.is_nilpotent.induction_on I hI,
{ introsI B _ I hI _, exact formally_unramified.comp_injective I hI },
{ introsI B _ I J hIJ h₁ h₂ _ g₁ g₂ e,
apply h₁,
apply h₂,
ext x,
replac... | lemma | algebra.formally_unramified.lift_unique | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.comp_apply",
"alg_hom.congr_fun",
"algebra",
"comm_ring",
"ideal",
"ideal.is_nilpotent.induction_on",
"ideal.mem_quotient_iff_mem",
"ideal.quotient.eq",
"ideal.quotient.mkₐ",
"ideal.quotient.mkₐ_eq_mk",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.ext [formally_unramified R A] (hI : is_nilpotent I)
{g₁ g₂ : A →ₐ[R] B} (H : ∀ x, ideal.quotient.mk I (g₁ x) = ideal.quotient.mk I (g₂ x)) :
g₁ = g₂ | formally_unramified.lift_unique I hI g₁ g₂ (alg_hom.ext H) | lemma | algebra.formally_unramified.ext | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.ext",
"ideal.quotient.mk",
"is_nilpotent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
formally_unramified.lift_unique_of_ring_hom [formally_unramified R A]
{C : Type u} [comm_ring C] (f : B →+* C) (hf : is_nilpotent f.ker)
(g₁ g₂ : A →ₐ[R] B) (h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ | formally_unramified.lift_unique _ hf _ _
begin
ext x,
have := ring_hom.congr_fun h x,
simpa only [ideal.quotient.eq, function.comp_app, alg_hom.coe_comp, ideal.quotient.mkₐ_eq_mk,
ring_hom.mem_ker, map_sub, sub_eq_zero],
end | lemma | algebra.formally_unramified.lift_unique_of_ring_hom | ring_theory | src/ring_theory/etale.lean | [
"ring_theory.quotient_nilpotent",
"ring_theory.kaehler"
] | [
"alg_hom.coe_comp",
"comm_ring",
"ideal.quotient.eq",
"ideal.quotient.mkₐ_eq_mk",
"is_nilpotent",
"ring_hom.congr_fun",
"ring_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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