statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
int.associated_iff_nat_abs {a b : ℤ} : associated a b ↔ a.nat_abs = b.nat_abs | begin
rw [←dvd_dvd_iff_associated, ←int.nat_abs_dvd_iff_dvd,
←int.nat_abs_dvd_iff_dvd, dvd_dvd_iff_associated],
exact associated_iff_eq,
end | theorem | int.associated_iff_nat_abs | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"associated",
"associated_iff_eq",
"dvd_dvd_iff_associated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.associated_iff {a b : ℤ} : associated a b ↔ (a = b ∨ a = -b) | begin
rw int.associated_iff_nat_abs,
exact int.nat_abs_eq_nat_abs_iff,
end | lemma | int.associated_iff | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"associated",
"int.associated_iff_nat_abs",
"int.nat_abs_eq_nat_abs_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zmultiples_nat_abs (a : ℤ) :
add_subgroup.zmultiples (a.nat_abs : ℤ) = add_subgroup.zmultiples a | le_antisymm
(add_subgroup.zmultiples_le_of_mem (mem_zmultiples_iff.mpr (dvd_nat_abs.mpr (dvd_refl a))))
(add_subgroup.zmultiples_le_of_mem (mem_zmultiples_iff.mpr (nat_abs_dvd.mpr (dvd_refl a)))) | lemma | int.zmultiples_nat_abs | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"add_subgroup.zmultiples",
"dvd_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_nat_abs (a : ℤ) : ideal.span ({a.nat_abs} : set ℤ) = ideal.span {a} | by { rw ideal.span_singleton_eq_span_singleton, exact (associated_nat_abs _).symm } | lemma | int.span_nat_abs | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"ideal.span",
"ideal.span_singleton_eq_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_of_mul_eq_pow_bit1_left {a b c : ℤ}
(hab : is_coprime a b) {k : ℕ} (h : a * b = c ^ (bit1 k)) : ∃ d, a = d ^ (bit1 k) | begin
obtain ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow' hab h,
replace hd := hd.symm,
rw [associated_iff_nat_abs, nat_abs_eq_nat_abs_iff, ←neg_pow_bit1] at hd,
obtain rfl|rfl := hd; exact ⟨_, rfl⟩,
end | theorem | int.eq_pow_of_mul_eq_pow_bit1_left | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"exists_associated_pow_of_mul_eq_pow'",
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_of_mul_eq_pow_bit1_right {a b c : ℤ}
(hab : is_coprime a b) {k : ℕ} (h : a * b = c ^ (bit1 k)) : ∃ d, b = d ^ (bit1 k) | eq_pow_of_mul_eq_pow_bit1_left hab.symm (by rwa mul_comm at h) | theorem | int.eq_pow_of_mul_eq_pow_bit1_right | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"is_coprime",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_pow_of_mul_eq_pow_bit1 {a b c : ℤ}
(hab : is_coprime a b) {k : ℕ} (h : a * b = c ^ (bit1 k)) :
(∃ d, a = d ^ (bit1 k)) ∧ (∃ e, b = e ^ (bit1 k)) | ⟨eq_pow_of_mul_eq_pow_bit1_left hab h, eq_pow_of_mul_eq_pow_bit1_right hab h⟩ | theorem | int.eq_pow_of_mul_eq_pow_bit1 | ring_theory.int | src/ring_theory/int/basic.lean | [
"algebra.euclidean_domain.basic",
"data.nat.factors",
"ring_theory.coprime.basic",
"ring_theory.principal_ideal_domain"
] | [
"is_coprime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_unit_of_le (hS : S ≤ A⁰) (s : S) : is_unit (algebra_map A K s) | by apply is_localization.map_units K (⟨s.1, hS s.2⟩ : A⁰) | lemma | localization.map_is_unit_of_le | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra_map",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_to_fraction_ring (B : Type*) [comm_ring B] [algebra A B]
[is_localization S B] (hS : S ≤ A⁰) :
B →ₐ[A] K | { commutes' := λ a, by simp,
..is_localization.lift (map_is_unit_of_le K S hS) } | def | localization.map_to_fraction_ring | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra",
"comm_ring",
"is_localization",
"is_localization.lift"
] | The canonical map from a localization of `A` at `S` to the fraction ring
of `A`, given that `S ≤ A⁰`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_to_fraction_ring_apply {B : Type*} [comm_ring B] [algebra A B]
[is_localization S B] (hS : S ≤ A⁰) (b : B) :
map_to_fraction_ring K S B hS b = is_localization.lift (map_is_unit_of_le K S hS) b | rfl | lemma | localization.map_to_fraction_ring_apply | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra",
"comm_ring",
"is_localization",
"is_localization.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_map_to_fraction_ring_iff (B : Type*) [comm_ring B] [algebra A B]
[is_localization S B] (hS : S ≤ A⁰) (x : K) :
x ∈ (map_to_fraction_ring K S B hS).range ↔
∃ (a s : A) (hs : s ∈ S), x = is_localization.mk' K a ⟨s, hS hs⟩ | ⟨ by { rintro ⟨x,rfl⟩, obtain ⟨a,s,rfl⟩ := is_localization.mk'_surjective S x,
use [a, s, s.2], apply is_localization.lift_mk' },
by { rintro ⟨a,s,hs,rfl⟩, use is_localization.mk' _ a ⟨s,hs⟩,
apply is_localization.lift_mk' } ⟩ | lemma | localization.mem_range_map_to_fraction_ring_iff | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra",
"comm_ring",
"is_localization",
"is_localization.lift_mk'",
"is_localization.mk'",
"is_localization.mk'_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_range_map_to_fraction_ring (B : Type*) [comm_ring B] [algebra A B]
[is_localization S B] (hS : S ≤ A⁰) :
is_localization S (map_to_fraction_ring K S B hS).range | is_localization.is_localization_of_alg_equiv S $ show B ≃ₐ[A] _, from alg_equiv.of_bijective
(map_to_fraction_ring K S B hS).range_restrict
begin
refine ⟨λ a b h, _, set.surjective_onto_range⟩,
refine (is_localization.lift_injective_iff _).2 (λ a b, _) (subtype.ext_iff.1 h),
exact ⟨λ h, congr_arg _ (is_localizati... | instance | localization.is_localization_range_map_to_fraction_ring | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"alg_equiv.of_bijective",
"algebra",
"comm_ring",
"is_fraction_ring.injective",
"is_localization",
"is_localization.injective",
"is_localization.is_localization_of_alg_equiv",
"is_localization.lift_injective_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_ring_range_map_to_fraction_ring
(B : Type*) [comm_ring B] [algebra A B]
[is_localization S B] (hS : S ≤ A⁰) :
is_fraction_ring (map_to_fraction_ring K S B hS).range K | is_fraction_ring.is_fraction_ring_of_is_localization S _ _ hS | instance | localization.is_fraction_ring_range_map_to_fraction_ring | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra",
"comm_ring",
"is_fraction_ring",
"is_fraction_ring.is_fraction_ring_of_is_localization",
"is_localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra (hS : S ≤ A⁰) : subalgebra A K | (map_to_fraction_ring K S (localization S) hS).range.copy
{ x | ∃ (a s : A) (hs : s ∈ S), x = is_localization.mk' K a ⟨s, hS hs⟩ } $
by { ext, symmetry, apply mem_range_map_to_fraction_ring_iff } | def | localization.subalgebra | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"is_localization.mk'",
"localization",
"subalgebra"
] | Given a commutative ring `A` with fraction ring `K`, and a submonoid `S` of `A` which
contains no zero divisor, this is the localization of `A` at `S`, considered as
a subalgebra of `K` over `A`.
The carrier of this subalgebra is defined as the set of all `x : K` of the form
`is_localization.mk' K a ⟨s, _⟩`, where `s ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_subalgebra :
is_localization S (subalgebra K S hS) | by { dunfold localization.subalgebra, rw subalgebra.copy_eq, apply_instance } | instance | localization.subalgebra.is_localization_subalgebra | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"is_localization",
"localization.subalgebra",
"subalgebra",
"subalgebra.copy_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_ring : is_fraction_ring (subalgebra K S hS) K | is_fraction_ring.is_fraction_ring_of_is_localization S _ _ hS | instance | localization.subalgebra.is_fraction_ring | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"is_fraction_ring",
"is_fraction_ring.is_fraction_ring_of_is_localization",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_range_map_to_fraction_ring_iff_of_field
(B : Type*) [comm_ring B] [algebra A B] [is_localization S B] (x : K) :
x ∈ (map_to_fraction_ring K S B hS).range ↔
∃ (a s : A) (hs : s ∈ S), x = algebra_map A K a * (algebra_map A K s)⁻¹ | begin
rw mem_range_map_to_fraction_ring_iff,
iterate 3 { congr' with }, convert iff.rfl, rw units.coe_inv, refl,
end | lemma | localization.subalgebra.mem_range_map_to_fraction_ring_iff_of_field | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra",
"algebra_map",
"comm_ring",
"is_localization",
"units.coe_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_field : _root_.subalgebra A K | (map_to_fraction_ring K S (localization S) hS).range.copy
{ x | ∃ (a s : A) (hs : s ∈ S), x = algebra_map A K a * (algebra_map A K s)⁻¹ } $
by { ext, symmetry, apply mem_range_map_to_fraction_ring_iff_of_field } | def | localization.subalgebra.of_field | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"algebra_map",
"localization"
] | Given a domain `A` with fraction field `K`, and a submonoid `S` of `A` which
contains no zero divisor, this is the localization of `A` at `S`, considered as
a subalgebra of `K` over `A`.
The carrier of this subalgebra is defined as the set of all `x : K` of the form
`algebra_map A K a * (algebra_map A K s)⁻¹` where `a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization_of_field :
is_localization S (subalgebra.of_field K S hS) | by { dunfold localization.subalgebra.of_field, rw subalgebra.copy_eq, apply_instance } | instance | localization.subalgebra.is_localization_of_field | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"is_localization",
"localization.subalgebra.of_field",
"subalgebra.copy_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fraction_ring_of_field : is_fraction_ring (subalgebra.of_field K S hS) K | is_fraction_ring.is_fraction_ring_of_is_localization S _ _ hS | instance | localization.subalgebra.is_fraction_ring_of_field | ring_theory.localization | src/ring_theory/localization/as_subring.lean | [
"ring_theory.localization.localization_localization"
] | [
"is_fraction_ring",
"is_fraction_ring.is_fraction_ring_of_is_localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_compl :
submonoid R | { carrier := (Iᶜ : set R),
one_mem' := by convert I.ne_top_iff_one.1 hp.1; refl,
mul_mem' := λ x y hnx hny hxy, or.cases_on (hp.mem_or_mem hxy) hnx hny } | def | ideal.prime_compl | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"submonoid"
] | The complement of a prime ideal `I ⊆ R` is a submonoid of `R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prime_compl_le_non_zero_divisors [no_zero_divisors R] : I.prime_compl ≤ non_zero_divisors R | le_non_zero_divisors_of_no_zero_divisors $ not_not_intro I.zero_mem | lemma | ideal.prime_compl_le_non_zero_divisors | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"le_non_zero_divisors_of_no_zero_divisors",
"no_zero_divisors",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.at_prime | is_localization I.prime_compl S | abbreviation | is_localization.at_prime | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"is_localization"
] | Given a prime ideal `P`, the typeclass `is_localization.at_prime S P` states that `S` is
isomorphic to the localization of `R` at the complement of `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization.at_prime | localization I.prime_compl | abbreviation | localization.at_prime | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"localization"
] | Given a prime ideal `P`, `localization.at_prime S P` is a localization of
`R` at the complement of `P`, as a quotient type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_prime.nontrivial [is_localization.at_prime S I] : nontrivial S | nontrivial_of_ne (0 : S) 1 $ λ hze,
begin
rw [←(algebra_map R S).map_one, ←(algebra_map R S).map_zero] at hze,
obtain ⟨t, ht⟩ := (eq_iff_exists I.prime_compl S).1 hze,
have htz : (t : R) = 0, by simpa using ht.symm,
exact t.2 (htz.symm ▸ I.zero_mem : ↑t ∈ I)
end | lemma | is_localization.at_prime.nontrivial | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"is_localization.at_prime",
"map_one",
"nontrivial",
"nontrivial_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_prime.local_ring [is_localization.at_prime S I] : local_ring S | local_ring.of_nonunits_add
begin
intros x y hx hy hu,
cases is_unit_iff_exists_inv.1 hu with z hxyz,
have : ∀ {r : R} {s : I.prime_compl}, mk' S r s ∈ nonunits S → r ∈ I, from
λ (r : R) (s : I.prime_compl), not_imp_comm.1
(λ nr, is_unit_iff_exists_inv.2 ⟨mk' S ↑s (⟨r, nr⟩ : I.prime_compl),
mk'_m... | theorem | is_localization.at_prime.local_ring | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"is_localization.at_prime",
"local_ring",
"local_ring.of_nonunits_add",
"mk'",
"mk'_surjective",
"mul_one",
"nonunits",
"one_mul",
"submonoid.coe_mul",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_prime.local_ring : local_ring (localization I.prime_compl) | is_localization.at_prime.local_ring (localization I.prime_compl) I | instance | localization.at_prime.local_ring | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"is_localization.at_prime.local_ring",
"local_ring",
"localization"
] | The localization of `R` at the complement of a prime ideal is a local ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain_of_local_at_prime {P : ideal A} (hp : P.is_prime) :
is_domain (localization.at_prime P) | is_domain_localization P.prime_compl_le_non_zero_divisors | instance | is_localization.is_domain_of_local_at_prime | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ideal",
"is_domain",
"localization.at_prime"
] | The localization of an integral domain at the complement of a prime ideal is an integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_to_map_iff (x : R) :
is_unit ((algebra_map R S) x) ↔ x ∈ I.prime_compl | ⟨λ h hx, (is_prime_of_is_prime_disjoint I.prime_compl S I hI disjoint_compl_left).ne_top $
(ideal.map (algebra_map R S) I).eq_top_of_is_unit_mem (ideal.mem_map_of_mem _ hx) h,
λ h, map_units S ⟨x, h⟩⟩ | lemma | is_localization.at_prime.is_unit_to_map_iff | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"disjoint_compl_left",
"ideal.map",
"ideal.mem_map_of_mem",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_map_mem_maximal_iff (x : R) (h : _root_.local_ring S := local_ring S I) :
algebra_map R S x ∈ local_ring.maximal_ideal S ↔ x ∈ I | not_iff_not.mp $ by
simpa only [local_ring.mem_maximal_ideal, mem_nonunits_iff, not_not]
using is_unit_to_map_iff S I x | lemma | is_localization.at_prime.to_map_mem_maximal_iff | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"local_ring",
"local_ring.maximal_ideal",
"local_ring.mem_maximal_ideal",
"mem_nonunits_iff",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_maximal_ideal (h : _root_.local_ring S := local_ring S I) :
(local_ring.maximal_ideal S).comap (algebra_map R S) = I | ideal.ext $ λ x, by simpa only [ideal.mem_comap] using to_map_mem_maximal_iff _ I x | lemma | is_localization.at_prime.comap_maximal_ideal | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"ideal.ext",
"ideal.mem_comap",
"local_ring",
"local_ring.maximal_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_mk'_iff (x : R) (y : I.prime_compl) :
is_unit (mk' S x y) ↔ x ∈ I.prime_compl | ⟨λ h hx, mk'_mem_iff.mpr ((to_map_mem_maximal_iff S I x).mpr hx) h,
λ h, is_unit_iff_exists_inv.mpr ⟨mk' S ↑y ⟨x, h⟩, mk'_mul_mk'_eq_one ⟨x, h⟩ y⟩⟩ | lemma | is_localization.at_prime.is_unit_mk'_iff | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"is_unit",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mem_maximal_iff (x : R) (y : I.prime_compl) (h : _root_.local_ring S := local_ring S I) :
mk' S x y ∈ local_ring.maximal_ideal S ↔ x ∈ I | not_iff_not.mp $ by
simpa only [local_ring.mem_maximal_ideal, mem_nonunits_iff, not_not]
using is_unit_mk'_iff S I x y | lemma | is_localization.at_prime.mk'_mem_maximal_iff | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"local_ring",
"local_ring.maximal_ideal",
"local_ring.mem_maximal_ideal",
"mem_nonunits_iff",
"mk'",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
at_prime.comap_maximal_ideal :
ideal.comap (algebra_map R (localization.at_prime I))
(local_ring.maximal_ideal (localization I.prime_compl)) = I | at_prime.comap_maximal_ideal _ _ | lemma | localization.at_prime.comap_maximal_ideal | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"ideal.comap",
"local_ring.maximal_ideal",
"localization",
"localization.at_prime"
] | The unique maximal ideal of the localization at `I.prime_compl` lies over the ideal `I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
at_prime.map_eq_maximal_ideal :
ideal.map (algebra_map R (localization.at_prime I)) I =
(local_ring.maximal_ideal (localization I.prime_compl)) | begin
convert congr_arg (ideal.map _) at_prime.comap_maximal_ideal.symm,
rw map_comap I.prime_compl
end | lemma | localization.at_prime.map_eq_maximal_ideal | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"ideal.map",
"local_ring.maximal_ideal",
"localization",
"localization.at_prime"
] | The image of `I` in the localization at `I.prime_compl` is a maximal ideal, and in particular
it is the unique maximal ideal given by the local ring structure `at_prime.local_ring` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_comap_prime_compl_iff {J : ideal P} [hJ : J.is_prime] {f : R →+* P} :
I.prime_compl ≤ J.prime_compl.comap f ↔ J.comap f ≤ I | ⟨λ h x hx, by { contrapose! hx, exact h hx },
λ h x hx hfxJ, hx (h hfxJ)⟩ | lemma | localization.le_comap_prime_compl_iff | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom (J : ideal P) [hJ : J.is_prime] (f : R →+* P)
(hIJ : I = J.comap f) :
localization.at_prime I →+* localization.at_prime J | is_localization.map (localization.at_prime J) f (le_comap_prime_compl_iff.mpr (ge_of_eq hIJ)) | def | localization.local_ring_hom | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ge_of_eq",
"ideal",
"is_localization.map",
"localization.at_prime"
] | For a ring hom `f : R →+* S` and a prime ideal `J` in `S`, the induced ring hom from the
localization of `R` at `J.comap f` to the localization of `S` at `J`.
To make this definition more flexible, we allow any ideal `I` of `R` as input, together with a proof
that `I = J.comap f`. This can be useful when `I` is not de... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_ring_hom_to_map (J : ideal P) [hJ : J.is_prime] (f : R →+* P)
(hIJ : I = J.comap f) (x : R) :
local_ring_hom I J f hIJ (algebra_map _ _ x) = algebra_map _ _ (f x) | map_eq _ _ | lemma | localization.local_ring_hom_to_map | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"ideal",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom_mk' (J : ideal P) [hJ : J.is_prime] (f : R →+* P)
(hIJ : I = J.comap f) (x : R) (y : I.prime_compl) :
local_ring_hom I J f hIJ (is_localization.mk' _ x y) =
is_localization.mk' (localization.at_prime J) (f x)
(⟨f y, le_comap_prime_compl_iff.mpr (ge_of_eq hIJ) y.2⟩ : J.prime_compl) | map_mk' _ _ _ | lemma | localization.local_ring_hom_mk' | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ge_of_eq",
"ideal",
"is_localization.mk'",
"localization.at_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_local_ring_hom_local_ring_hom (J : ideal P) [hJ : J.is_prime] (f : R →+* P)
(hIJ : I = J.comap f) :
is_local_ring_hom (local_ring_hom I J f hIJ) | is_local_ring_hom.mk $ λ x hx,
begin
rcases is_localization.mk'_surjective I.prime_compl x with ⟨r, s, rfl⟩,
rw local_ring_hom_mk' at hx,
rw at_prime.is_unit_mk'_iff at hx ⊢,
exact λ hr, hx ((set_like.ext_iff.mp hIJ r).mp hr),
end | instance | localization.is_local_ring_hom_local_ring_hom | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ideal",
"is_local_ring_hom",
"is_localization.mk'_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom_unique (J : ideal P) [hJ : J.is_prime] (f : R →+* P)
(hIJ : I = J.comap f) {j : localization.at_prime I →+* localization.at_prime J}
(hj : ∀ x : R, j (algebra_map _ _ x) = algebra_map _ _ (f x)) :
local_ring_hom I J f hIJ = j | map_unique _ _ hj | lemma | localization.local_ring_hom_unique | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"algebra_map",
"ideal",
"localization.at_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom_id :
local_ring_hom I I (ring_hom.id R) (ideal.comap_id I).symm = ring_hom.id _ | local_ring_hom_unique _ _ _ _ (λ x, rfl) | lemma | localization.local_ring_hom_id | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"ideal.comap_id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_ring_hom_comp {S : Type*} [comm_semiring S]
(J : ideal S) [hJ : J.is_prime] (K : ideal P) [hK : K.is_prime]
(f : R →+* S) (hIJ : I = J.comap f) (g : S →+* P) (hJK : J = K.comap g) :
local_ring_hom I K (g.comp f) (by rw [hIJ, hJK, ideal.comap_comap f g]) =
(local_ring_hom J K g hJK).comp (local_ring_hom I ... | local_ring_hom_unique _ _ _ _
(λ r, by simp only [function.comp_app, ring_hom.coe_comp, local_ring_hom_to_map]) | lemma | localization.local_ring_hom_comp | ring_theory.localization | src/ring_theory/localization/at_prime.lean | [
"ring_theory.ideal.local_ring",
"ring_theory.localization.ideal"
] | [
"comm_semiring",
"ideal",
"ideal.comap_comap",
"ring_hom.coe_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization : Prop | (map_units [] : ∀ y : M, is_unit (algebra_map R S y))
(surj [] : ∀ z : S, ∃ x : R × M, z * algebra_map R S x.2 = algebra_map R S x.1)
(eq_iff_exists [] : ∀ {x y}, algebra_map R S x = algebra_map R S y ↔ ∃ c : M, ↑c * x = ↑c * y) | class | is_localization | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_unit"
] | The typeclass `is_localization (M : submodule R) S` where `S` is an `R`-algebra
expresses that `S` is isomorphic to the localization of `R` at `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le (N : submonoid R) (h₁ : M ≤ N)
(h₂ : ∀ r ∈ N, is_unit (algebra_map R S r)) : is_localization N S | { map_units := λ r, h₂ r r.2,
surj := λ s, by { obtain ⟨⟨x, y, hy⟩, H⟩ := is_localization.surj M s, exact ⟨⟨x, y, h₁ hy⟩, H⟩ },
eq_iff_exists := λ x y, begin
split,
{ rw is_localization.eq_iff_exists M,
rintro ⟨c, hc⟩,
exact ⟨⟨c, h₁ c.2⟩, hc⟩ },
{ rintro ⟨c, h⟩,
simpa only [set_like.co... | lemma | is_localization.of_le | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_localization",
"is_unit",
"map_mul",
"mul_right_inj",
"set_like.coe_mk",
"submonoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_localization_with_zero_map : submonoid.localization_with_zero_map M S | { to_fun := algebra_map R S,
map_units' := is_localization.map_units _,
surj' := is_localization.surj _,
eq_iff_exists' := λ _ _, is_localization.eq_iff_exists _ _,
.. algebra_map R S } | def | is_localization.to_localization_with_zero_map | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"submonoid.localization_with_zero_map"
] | `is_localization.to_localization_with_zero_map M S` shows `S` is the monoid localization of
`R` at `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_localization_map : submonoid.localization_map M S | (to_localization_with_zero_map M S).to_localization_map | abbreviation | is_localization.to_localization_map | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"submonoid.localization_map"
] | `is_localization.to_localization_map M S` shows `S` is the monoid localization of `R` at `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_localization_map_to_map :
(to_localization_map M S).to_map = (algebra_map R S : R →*₀ S) | rfl | lemma | is_localization.to_localization_map_to_map | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_localization_map_to_map_apply (x) :
(to_localization_map M S).to_map x = algebra_map R S x | rfl | lemma | is_localization.to_localization_map_to_map_apply | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sec (z : S) : R × M | classical.some $ is_localization.surj _ z | def | is_localization.sec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | Given a localization map `f : M →* N`, a section function sending `z : N` to some
`(x, y) : M × S` such that `f x * (f y)⁻¹ = z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_localization_map_sec : (to_localization_map M S).sec = sec M | rfl | lemma | is_localization.to_localization_map_sec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sec_spec (z : S) :
z * algebra_map R S (is_localization.sec M z).2 =
algebra_map R S (is_localization.sec M z).1 | classical.some_spec $ is_localization.surj _ z | lemma | is_localization.sec_spec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_localization.sec"
] | Given `z : S`, `is_localization.sec M z` is defined to be a pair `(x, y) : R × M` such
that `z * f y = f x` (so this lemma is true by definition). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sec_spec' (z : S) :
algebra_map R S (is_localization.sec M z).1 =
algebra_map R S (is_localization.sec M z).2 * z | by rw [mul_comm, sec_spec] | lemma | is_localization.sec_spec' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_localization.sec",
"mul_comm"
] | Given `z : S`, `is_localization.sec M z` is defined to be a pair `(x, y) : R × M` such
that `z * f y = f x`, so this lemma is just an application of `S`'s commutativity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_right_cancel {x y} {c : M} (h : algebra_map R S (c * x) = algebra_map R S (c * y)) :
algebra_map R S x = algebra_map R S y | (to_localization_map M S).map_right_cancel h | lemma | is_localization.map_right_cancel | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_left_cancel {x y} {c : M} (h : algebra_map R S (x * c) = algebra_map R S (y * c)) :
algebra_map R S x = algebra_map R S y | (to_localization_map M S).map_left_cancel h | lemma | is_localization.map_left_cancel | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_fst_eq_zero {z x} {y : M}
(h : z * algebra_map R S y = algebra_map R S x) (hx : x = 0) : z = 0 | by { rw [hx, (algebra_map R S).map_zero] at h,
exact (is_unit.mul_left_eq_zero (is_localization.map_units S y)).1 h} | lemma | is_localization.eq_zero_of_fst_eq_zero | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_unit.mul_left_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff (r : R) :
algebra_map R S r = 0 ↔ ∃ m : M, ↑m * r = 0 | begin
split,
intro h,
{ obtain ⟨m, hm⟩ := (is_localization.eq_iff_exists M S).mp
((algebra_map R S).map_zero.trans h.symm),
exact ⟨m, by simpa using hm.symm⟩ },
{ rintro ⟨m, hm⟩,
rw [← (is_localization.map_units S m).mul_right_inj, mul_zero, ← ring_hom.map_mul, hm,
ring_hom.map_zero] }
end | lemma | is_localization.map_eq_zero_iff | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mul_right_inj",
"mul_zero",
"ring_hom.map_mul",
"ring_hom.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (x : R) (y : M) : S | (to_localization_map M S).mk' x y | def | is_localization.mk' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | `is_localization.mk' S` is the surjection sending `(x, y) : R × M` to
`f x * (f y)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_sec (z : S) :
mk' S (is_localization.sec M z).1 (is_localization.sec M z).2 = z | (to_localization_map M S).mk'_sec _ | lemma | is_localization.mk'_sec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_localization.sec",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul (x₁ x₂ : R) (y₁ y₂ : M) :
mk' S (x₁ * x₂) (y₁ * y₂) = mk' S x₁ y₁ * mk' S x₂ y₂ | (to_localization_map M S).mk'_mul _ _ _ _ | lemma | is_localization.mk'_mul | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_one (x) : mk' S x (1 : M) = algebra_map R S x | (to_localization_map M S).mk'_one _ | lemma | is_localization.mk'_one | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'",
"mk'_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_spec (x) (y : M) :
mk' S x y * algebra_map R S y = algebra_map R S x | (to_localization_map M S).mk'_spec _ _ | lemma | is_localization.mk'_spec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_spec' (x) (y : M) :
algebra_map R S y * mk' S x y = algebra_map R S x | (to_localization_map M S).mk'_spec' _ _ | lemma | is_localization.mk'_spec' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_spec_mk (x) (y : R) (hy : y ∈ M) :
mk' S x ⟨y, hy⟩ * algebra_map R S y = algebra_map R S x | mk'_spec S x ⟨y, hy⟩ | lemma | is_localization.mk'_spec_mk | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_spec'_mk (x) (y : R) (hy : y ∈ M) :
algebra_map R S y * mk' S x ⟨y, hy⟩ = algebra_map R S x | mk'_spec' S x ⟨y, hy⟩ | lemma | is_localization.mk'_spec'_mk | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mk'_iff_mul_eq {x} {y : M} {z} :
z = mk' S x y ↔ z * algebra_map R S y = algebra_map R S x | (to_localization_map M S).eq_mk'_iff_mul_eq | theorem | is_localization.eq_mk'_iff_mul_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_iff_eq_mul {x} {y : M} {z} :
mk' S x y = z ↔ algebra_map R S x = z * algebra_map R S y | (to_localization_map M S).mk'_eq_iff_eq_mul | theorem | is_localization.mk'_eq_iff_eq_mul | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_add_eq_iff_add_mul_eq_mul {x} {y : M} {z₁ z₂} :
mk' S x y + z₁ = z₂ ↔ algebra_map R S x + z₁ * algebra_map R S y = z₂ * algebra_map R S y | by rw [←mk'_spec S x y, ←is_unit.mul_left_inj (is_localization.map_units S y), right_distrib] | theorem | is_localization.mk'_add_eq_iff_add_mul_eq_mul | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'",
"right_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_surjective (z : S) : ∃ x (y : M), mk' S x y = z | let ⟨r, hr⟩ := is_localization.surj _ z in ⟨r.1, r.2, (eq_mk'_iff_mul_eq.2 hr).symm⟩ | lemma | is_localization.mk'_surjective | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'",
"mk'_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fintype' [fintype R] : fintype S | have _ := classical.prop_decidable, by exactI
fintype.of_surjective (function.uncurry $ is_localization.mk' S)
(λ a, prod.exists'.mpr $ is_localization.mk'_surjective M a) | def | is_localization.fintype' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"fintype",
"fintype.of_surjective",
"is_localization.mk'",
"is_localization.mk'_surjective"
] | The localization of a `fintype` is a `fintype`. Cannot be an instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_of_zero_mem (h : (0 : R) ∈ M) : unique S | unique_of_zero_eq_one $ by simpa using is_localization.map_units S ⟨0, h⟩ | def | is_localization.unique_of_zero_mem | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"unique",
"unique_of_zero_eq_one"
] | Localizing at a submonoid with 0 inside it leads to the trivial ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_eq_iff_eq {x₁ x₂} {y₁ y₂ : M} :
mk' S x₁ y₁ = mk' S x₂ y₂ ↔ algebra_map R S (y₂ * x₁) = algebra_map R S (y₁ * x₂) | (to_localization_map M S).mk'_eq_iff_eq | lemma | is_localization.mk'_eq_iff_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_iff_eq' {x₁ x₂} {y₁ y₂ : M} :
mk' S x₁ y₁ = mk' S x₂ y₂ ↔ algebra_map R S (x₁ * y₂) = algebra_map R S (x₂ * y₁) | (to_localization_map M S).mk'_eq_iff_eq' | lemma | is_localization.mk'_eq_iff_eq' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mem_iff {x} {y : M} {I : ideal S} : mk' S x y ∈ I ↔ algebra_map R S x ∈ I | begin
split;
intro h,
{ rw [← mk'_spec S x y, mul_comm],
exact I.mul_mem_left ((algebra_map R S) y) h },
{ rw ← mk'_spec S x y at h,
obtain ⟨b, hb⟩ := is_unit_iff_exists_inv.1 (map_units S y),
have := I.mul_mem_left b h,
rwa [mul_comm, mul_assoc, hb, mul_one] at this }
end | lemma | is_localization.mk'_mem_iff | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"ideal",
"mk'",
"mul_assoc",
"mul_comm",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq {a₁ b₁} {a₂ b₂ : M} :
mk' S a₁ a₂ = mk' S b₁ b₂ ↔ ∃ c : M, ↑c * (↑b₂ * a₁) = c * (a₂ * b₁) | (to_localization_map M S).eq | lemma | is_localization.eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_zero_iff (x : R) (s : M) :
mk' S x s = 0 ↔ ∃ (m : M), ↑m * x = 0 | by rw [← (map_units S s).mul_left_inj, mk'_spec, zero_mul, map_eq_zero_iff M] | lemma | is_localization.mk'_eq_zero_iff | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'",
"mul_left_inj",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_zero (s : M) : is_localization.mk' S 0 s = 0 | by rw [eq_comm, is_localization.eq_mk'_iff_mul_eq, zero_mul, map_zero] | lemma | is_localization.mk'_zero | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_localization.eq_mk'_iff_mul_eq",
"is_localization.mk'",
"mk'_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mk'_ne_zero {x : R} {y : M} (hxy : is_localization.mk' S x y ≠ 0) : x ≠ 0 | begin
rintro rfl,
exact hxy (is_localization.mk'_zero _)
end | lemma | is_localization.ne_zero_of_mk'_ne_zero | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_localization.mk'",
"is_localization.mk'_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_eq {x y} :
algebra_map R S x = algebra_map R S y ↔ algebra_map R P x = algebra_map R P y | (to_localization_map M S).eq_iff_eq (to_localization_map M P) | lemma | is_localization.eq_iff_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_iff_mk'_eq {x₁ x₂}
{y₁ y₂ : M} : mk' S x₁ y₁ = mk' S x₂ y₂ ↔ mk' P x₁ y₁ = mk' P x₂ y₂ | (to_localization_map M S).mk'_eq_iff_mk'_eq (to_localization_map M P) | lemma | is_localization.mk'_eq_iff_mk'_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_of_eq {a₁ b₁ : R} {a₂ b₂ : M} (H : ↑a₂ * b₁ = ↑b₂ * a₁) :
mk' S a₁ a₂ = mk' S b₁ b₂ | (to_localization_map M S).mk'_eq_of_eq H | lemma | is_localization.mk'_eq_of_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_of_eq' {a₁ b₁ : R} {a₂ b₂ : M} (H : b₁ * ↑a₂ = a₁ * ↑b₂) :
mk' S a₁ a₂ = mk' S b₁ b₂ | (to_localization_map M S).mk'_eq_of_eq' H | lemma | is_localization.mk'_eq_of_eq' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_self {x : R} (hx : x ∈ M) : mk' S x ⟨x, hx⟩ = 1 | (to_localization_map M S).mk'_self _ hx | lemma | is_localization.mk'_self | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_self' {x : M} : mk' S (x : R) x = 1 | (to_localization_map M S).mk'_self' _ | lemma | is_localization.mk'_self' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_self'' {x : M} : mk' S x.1 x = 1 | mk'_self' _ | lemma | is_localization.mk'_self'' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mk'_eq_mk'_of_mul (x y : R) (z : M) :
(algebra_map R S) x * mk' S y z = mk' S (x * y) z | (to_localization_map M S).mul_mk'_eq_mk'_of_mul _ _ _ | lemma | is_localization.mul_mk'_eq_mk'_of_mul | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_mul_mk'_one (x : R) (y : M) :
mk' S x y = (algebra_map R S) x * mk' S 1 y | ((to_localization_map M S).mul_mk'_one_eq_mk' _ _).symm | lemma | is_localization.mk'_eq_mul_mk'_one | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_cancel_left (x : R) (y : M) :
mk' S (y * x : R) y = (algebra_map R S) x | (to_localization_map M S).mk'_mul_cancel_left _ _ | lemma | is_localization.mk'_mul_cancel_left | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_cancel_right (x : R) (y : M) :
mk' S (x * y) y = (algebra_map R S) x | (to_localization_map M S).mk'_mul_cancel_right _ _ | lemma | is_localization.mk'_mul_cancel_right | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_mk'_eq_one (x y : M) :
mk' S (x : R) y * mk' S (y : R) x = 1 | by rw [←mk'_mul, mul_comm]; exact mk'_self _ _ | lemma | is_localization.mk'_mul_mk'_eq_one | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_mk'_eq_one' (x : R) (y : M) (h : x ∈ M) :
mk' S x y * mk' S (y : R) ⟨x, h⟩ = 1 | mk'_mul_mk'_eq_one ⟨x, h⟩ _ | lemma | is_localization.mk'_mul_mk'_eq_one' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_comp (j : S →+* P) (y : M) :
is_unit (j.comp (algebra_map R S) y) | (to_localization_map M S).is_unit_comp j.to_monoid_hom _ | lemma | is_localization.is_unit_comp | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq {g : R →+* P} (hg : ∀ y : M, is_unit (g y)) {x y}
(h : (algebra_map R S) x = (algebra_map R S) y) :
g x = g y | @submonoid.localization_map.eq_of_eq _ _ _ _ _ _ _
(to_localization_map M S) g.to_monoid_hom hg _ _ h | lemma | is_localization.eq_of_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"is_unit",
"submonoid.localization_map.eq_of_eq"
] | Given a localization map `f : R →+* S` for a submonoid `M ⊆ R` and a map of `comm_semiring`s
`g : R →+* P` such that `g(M) ⊆ units P`, `f x = f y → g x = g y` for all `x y : R`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_add (x₁ x₂ : R) (y₁ y₂ : M) :
mk' S (x₁ * y₂ + x₂ * y₁) (y₁ * y₂) = mk' S x₁ y₁ + mk' S x₂ y₂ | mk'_eq_iff_eq_mul.2 $ eq.symm
begin
rw [mul_comm (_ + _), mul_add, mul_mk'_eq_mk'_of_mul, mk'_add_eq_iff_add_mul_eq_mul,
mul_comm (_ * _), ←mul_assoc, add_comm, ←map_mul, mul_mk'_eq_mk'_of_mul,
mk'_add_eq_iff_add_mul_eq_mul],
simp only [map_add, submonoid.coe_mul, map_mul],
ring
end | lemma | is_localization.mk'_add | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"map_mul",
"mk'",
"mk'_add",
"mul_comm",
"ring",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_inv_left {g : R →+* P} (h : ∀ y : M, is_unit (g y)) (y : M) (w z₁ z₂ : P) :
w * ↑(is_unit.lift_right (g.to_monoid_hom.restrict M) h y)⁻¹ + z₁ = z₂
↔ w + g y * z₁ = g y * z₂ | begin
rw [mul_comm, ←one_mul z₁, ←units.inv_mul (is_unit.lift_right (g.to_monoid_hom.restrict M) h y),
mul_assoc, ←mul_add, units.inv_mul_eq_iff_eq_mul, units.inv_mul_cancel_left,
is_unit.coe_lift_right],
simp only [ring_hom.to_monoid_hom_eq_coe, monoid_hom.restrict_apply, ring_hom.coe_monoid_hom]
end | lemma | is_localization.mul_add_inv_left | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_unit",
"is_unit.coe_lift_right",
"is_unit.lift_right",
"monoid_hom.restrict_apply",
"mul_assoc",
"mul_comm",
"ring_hom.coe_monoid_hom",
"ring_hom.to_monoid_hom_eq_coe",
"units.inv_mul_cancel_left",
"units.inv_mul_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_spec_mul_add {g : R →+* P} (hg : ∀ y : M, is_unit (g y)) (z w w' v) :
((to_localization_with_zero_map M S).lift g.to_monoid_with_zero_hom hg) z * w + w' = v
↔ g ((to_localization_map M S).sec z).1 * w + g ((to_localization_map M S).sec z).2 * w'
= g ((to_localization_map M S).sec z).2 * v | begin
show (_ * _) * _ + _ = _ ↔ _ = _,
erw [mul_comm, ←mul_assoc, mul_add_inv_left hg, mul_comm],
refl
end | lemma | is_localization.lift_spec_mul_add | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_unit",
"lift",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift {g : R →+* P} (hg : ∀ y : M, is_unit (g y)) : S →+* P | { map_add' :=
begin
intros x y,
erw [(to_localization_map M S).lift_spec, mul_add, mul_comm, eq_comm, lift_spec_mul_add,
add_comm, mul_comm,mul_assoc,mul_comm,mul_assoc, lift_spec_mul_add],
simp_rw ←mul_assoc,
show g _ * g _ * g _ + g _ * g _ * g _ = g _ * g _ * g _,
simp_rw [←map_mul g, ←ma... | def | is_localization.lift | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_unit",
"lift",
"map_mul",
"mul_assoc",
"mul_comm",
"ring",
"submonoid.localization_with_zero_map.lift"
] | Given a localization map `f : R →+* S` for a submonoid `M ⊆ R` and a map of `comm_semiring`s
`g : R →+* P` such that `g y` is invertible for all `y : M`, the homomorphism induced from
`S` to `P` sending `z : S` to `g x * (g y)⁻¹`, where `(x, y) : R × M` are such that
`z = f x * (f y)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk' (x y) :
lift hg (mk' S x y) = g x * ↑(is_unit.lift_right (g.to_monoid_hom.restrict M) hg y)⁻¹ | (to_localization_map M S).lift_mk' _ _ _ | lemma | is_localization.lift_mk' | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"is_unit.lift_right",
"lift",
"mk'"
] | Given a localization map `f : R →+* S` for a submonoid `M ⊆ R` and a map of `comm_semiring`s
`g : R →* P` such that `g y` is invertible for all `y : M`, the homomorphism induced from
`S` to `P` maps `f x * (f y)⁻¹` to `g x * (g y)⁻¹` for all `x : R, y ∈ M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk'_spec (x v) (y : M) :
lift hg (mk' S x y) = v ↔ g x = g y * v | (to_localization_map M S).lift_mk'_spec _ _ _ _ | lemma | is_localization.lift_mk'_spec | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"lift",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_eq (x : R) :
lift hg ((algebra_map R S) x) = g x | (to_localization_map M S).lift_eq _ _ | lemma | is_localization.lift_eq | ring_theory.localization | src/ring_theory/localization/basic.lean | [
"algebra.algebra.tower",
"algebra.ring.equiv",
"group_theory.monoid_localization",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors",
"tactic.ring_exp"
] | [
"algebra_map",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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