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 |
|---|---|---|---|---|---|---|---|---|---|---|
lift_eq_iff {x y : R × M} :
lift hg (mk' S x.1 x.2) = lift hg (mk' S y.1 y.2) ↔ g (x.1 * y.2) = g (y.1 * x.2) | (to_localization_map M S).lift_eq_iff _ | lemma | is_localization.lift_eq_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"
] | [
"lift",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp : (lift hg).comp (algebra_map R S) = g | ring_hom.ext $ monoid_hom.ext_iff.1 $ (to_localization_map M S).lift_comp _ | lemma | is_localization.lift_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",
"lift",
"lift_comp",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_of_comp (j : S →+* P) :
lift (is_unit_comp M j) = j | ring_hom.ext $ monoid_hom.ext_iff.1 $ (to_localization_map M S).lift_of_comp j.to_monoid_hom | lemma | is_localization.lift_of_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"
] | [
"lift",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_ext ⦃j k : S →* P⦄
(h : j.comp (algebra_map R S : R →* S) = k.comp (algebra_map R S)) : j = k | submonoid.localization_map.epic_of_localization_map (to_localization_map M S) $
monoid_hom.congr_fun h | lemma | is_localization.monoid_hom_ext | 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",
"monoid_hom.congr_fun",
"submonoid.localization_map.epic_of_localization_map"
] | See note [partially-applied ext lemmas] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_hom_ext ⦃j k : S →+* P⦄
(h : j.comp (algebra_map R S) = k.comp (algebra_map R S)) : j = k | ring_hom.coe_monoid_hom_injective $ monoid_hom_ext M $ monoid_hom.ext $ ring_hom.congr_fun h | lemma | is_localization.ring_hom_ext | 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",
"monoid_hom.ext",
"ring_hom.coe_monoid_hom_injective",
"ring_hom.congr_fun",
"ring_hom_ext"
] | See note [partially-applied ext lemmas] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_hom_subsingleton [algebra R P] : subsingleton (S →ₐ[R] P) | ⟨λ f g, alg_hom.coe_ring_hom_injective $ is_localization.ring_hom_ext M $
by rw [f.comp_algebra_map, g.comp_algebra_map]⟩ | lemma | is_localization.alg_hom_subsingleton | 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"
] | [
"alg_hom.coe_ring_hom_injective",
"algebra",
"is_localization.ring_hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (j k : S → P) (hj1 : j 1 = 1) (hk1 : k 1 = 1)
(hjm : ∀ a b, j (a * b) = j a * j b) (hkm : ∀ a b, k (a * b) = k a * k b)
(h : ∀ a, j (algebra_map R S a) = k (algebra_map R S a)) : j = k | monoid_hom.mk.inj (monoid_hom_ext M $ monoid_hom.ext h : (⟨j, hj1, hjm⟩ : S →* P) = ⟨k, hk1, hkm⟩) | lemma | is_localization.ext | 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",
"monoid_hom.ext"
] | To show `j` and `k` agree on the whole localization, it suffices to show they agree
on the image of the base ring, if they preserve `1` and `*`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_unique {j : S →+* P}
(hj : ∀ x, j ((algebra_map R S) x) = g x) : lift hg = j | ring_hom.ext $ monoid_hom.ext_iff.1 $ @submonoid.localization_map.lift_unique
_ _ _ _ _ _ _ (to_localization_map M S) g.to_monoid_hom hg j.to_monoid_hom hj | lemma | is_localization.lift_unique | 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",
"lift_unique",
"ring_hom.ext",
"submonoid.localization_map.lift_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_id (x) : lift (map_units S : ∀ y : M, is_unit _) x = x | (to_localization_map M S).lift_id _ | lemma | is_localization.lift_id | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_surjective_iff :
surjective (lift hg : S → P) ↔ ∀ v : P, ∃ x : R × M, v * g x.2 = g x.1 | (to_localization_map M S).lift_surjective_iff hg | lemma | is_localization.lift_surjective_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"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_injective_iff :
injective (lift hg : S → P) ↔ ∀ x y, algebra_map R S x = algebra_map R S y ↔ g x = g y | (to_localization_map M S).lift_injective_iff hg | lemma | is_localization.lift_injective_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",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (g : R →+* P) (hy : M ≤ T.comap g) : S →+* Q | @lift R _ M _ _ _ _ _ _ ((algebra_map P Q).comp g) (λ y, map_units _ ⟨g y, hy y.2⟩) | def | is_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"
] | [
"algebra_map",
"lift"
] | Map a homomorphism `g : R →+* P` to `S →+* Q`, where `S` and `Q` are
localizations of `R` and `P` at `M` and `T` respectively,
such that `g(M) ⊆ T`.
We send `z : S` to `algebra_map P Q (g x) * (algebra_map P Q (g y))⁻¹`, where
`(x, y) : R × M` are such that `z = f x * (f y)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq (x) :
map Q g hy ((algebra_map R S) x) = algebra_map P Q (g x) | lift_eq (λ y, map_units _ ⟨g y, hy y.2⟩) x | lemma | is_localization.map_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",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp :
(map Q g hy).comp (algebra_map R S) = (algebra_map P Q).comp g | lift_comp $ λ y, map_units _ ⟨g y, hy y.2⟩ | lemma | is_localization.map_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",
"lift_comp",
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk' (x) (y : M) :
map Q g hy (mk' S x y) = mk' Q (g x) ⟨g y, hy y.2⟩ | @submonoid.localization_map.map_mk' _ _ _ _ _ _ _ (to_localization_map M S)
g.to_monoid_hom _ (λ y, hy y.2) _ _ (to_localization_map T Q) _ _ | lemma | is_localization.map_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'",
"submonoid.localization_map.map_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (z : S) (h : M ≤ M.comap (ring_hom.id R) := le_refl M) :
map S (ring_hom.id _) h z = z | lift_id _ | lemma | is_localization.map_id | 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_id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unique (j : S →+* Q)
(hj : ∀ x : R, j (algebra_map R S x) = algebra_map P Q (g x)) : map Q g hy = j | lift_unique (λ y, map_units _ ⟨g y, hy y.2⟩) hj | lemma | is_localization.map_unique | 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_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_map {A : Type*} [comm_semiring A] {U : submonoid A} {W} [comm_semiring W]
[algebra A W] [is_localization U W] {l : P →+* A} (hl : T ≤ U.comap l) :
(map W l hl).comp (map Q g hy : S →+* _) = map W (l.comp g) (λ x hx, hl (hy hx)) | ring_hom.ext $ λ x, @submonoid.localization_map.map_map _ _ _ _ _ P _ (to_localization_map M S) g _
_ _ _ _ _ _ _ _ _ (to_localization_map U W) l _ x | lemma | is_localization.map_comp_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",
"comm_semiring",
"is_localization",
"ring_hom.ext",
"submonoid",
"submonoid.localization_map.map_map"
] | If `comm_semiring` homs `g : R →+* P, l : P →+* A` induce maps of localizations, the composition
of the induced maps equals the map of localizations induced by `l ∘ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_map {A : Type*} [comm_semiring A] {U : submonoid A} {W} [comm_semiring W]
[algebra A W] [is_localization U W] {l : P →+* A} (hl : T ≤ U.comap l) (x : S) :
map W l hl (map Q g hy x) = map W (l.comp g) (λ x hx, hl (hy hx)) x | by rw ←map_comp_map hy hl; refl | lemma | is_localization.map_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",
"comm_semiring",
"is_localization",
"submonoid"
] | If `comm_semiring` homs `g : R →+* P, l : P →+* A` induce maps of localizations, the composition
of the induced maps equals the map of localizations induced by `l ∘ g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul (x : S) (z : R) :
map Q g hy (z • x : S) = g z • map Q g hy x | by rw [algebra.smul_def, algebra.smul_def, ring_hom.map_mul, map_eq] | lemma | is_localization.map_smul | 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.smul_def",
"map_eq",
"ring_hom.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv_of_ring_equiv (h : R ≃+* P) (H : M.map h.to_monoid_hom = T) :
S ≃+* Q | have H' : T.map h.symm.to_monoid_hom = M,
by { rw [← M.map_id, ← H, submonoid.map_map], congr, ext, apply h.symm_apply_apply },
{ to_fun := map Q (h : R →+* P) (M.le_comap_of_map_le (le_of_eq H)),
inv_fun := map S (h.symm : P →+* R) (T.le_comap_of_map_le (le_of_eq H')),
left_inv := λ x, by { rw [map_map, map_unique... | def | is_localization.ring_equiv_of_ring_equiv | 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",
"inv_fun",
"ring_hom.id",
"ring_hom.id_apply",
"submonoid.map_map"
] | If `S`, `Q` are localizations of `R` and `P` at submonoids `M, T` respectively, an
isomorphism `j : R ≃+* P` such that `j(M) = T` induces an isomorphism of localizations
`S ≃+* Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ring_equiv_of_ring_equiv_eq_map {j : R ≃+* P} (H : M.map j.to_monoid_hom = T) :
(ring_equiv_of_ring_equiv S Q j H : S →+* Q) =
map Q (j : R →+* P) (M.le_comap_of_map_le (le_of_eq H)) | rfl | lemma | is_localization.ring_equiv_of_ring_equiv_eq_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv_of_ring_equiv_eq {j : R ≃+* P} (H : M.map j.to_monoid_hom = T) (x) :
ring_equiv_of_ring_equiv S Q j H ((algebra_map R S) x) = algebra_map P Q (j x) | map_eq _ _ | lemma | is_localization.ring_equiv_of_ring_equiv_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",
"map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv_of_ring_equiv_mk' {j : R ≃+* P} (H : M.map j.to_monoid_hom = T) (x : R) (y : M) :
ring_equiv_of_ring_equiv S Q j H (mk' S x y) =
mk' Q (j x) ⟨j y, show j y ∈ T, from H ▸ set.mem_image_of_mem j y.2⟩ | map_mk' _ _ _ | lemma | is_localization.ring_equiv_of_ring_equiv_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'",
"set.mem_image_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv : S ≃ₐ[R] Q | { commutes' := ring_equiv_of_ring_equiv_eq _,
.. ring_equiv_of_ring_equiv S Q (ring_equiv.refl R) M.map_id } | def | is_localization.alg_equiv | 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"
] | [
"alg_equiv",
"ring_equiv.refl"
] | If `S`, `Q` are localizations of `R` at the submonoid `M` respectively,
there is an isomorphism of localizations `S ≃ₐ[R] Q`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_equiv_mk' (x : R) (y : M) : alg_equiv M S Q (mk' S x y) = mk' Q x y | map_mk' _ _ _ | lemma | is_localization.alg_equiv_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"
] | [
"alg_equiv",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv_symm_mk' (x : R) (y : M) : (alg_equiv M S Q).symm (mk' Q x y) = mk' S x y | map_mk' _ _ _ | lemma | is_localization.alg_equiv_symm_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"
] | [
"alg_equiv",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_of_alg_equiv [algebra R P] [is_localization M S] (h : S ≃ₐ[R] P) :
is_localization M P | begin
constructor,
{ intro y,
convert (is_localization.map_units S y).map h.to_alg_hom.to_ring_hom.to_monoid_hom,
exact (h.commutes y).symm },
{ intro y,
obtain ⟨⟨x, s⟩, e⟩ := is_localization.surj M (h.symm y),
apply_fun h at e,
simp only [h.map_mul, h.apply_symm_apply, h.commutes] at e,
e... | lemma | is_localization.is_localization_of_alg_equiv | 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",
"is_localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_iff_of_alg_equiv [algebra R P] (h : S ≃ₐ[R] P) :
is_localization M S ↔ is_localization M P | ⟨λ _, by exactI is_localization_of_alg_equiv M h,
λ _, by exactI is_localization_of_alg_equiv M h.symm⟩ | lemma | is_localization.is_localization_iff_of_alg_equiv | 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",
"is_localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_iff_of_ring_equiv (h : S ≃+* P) :
is_localization M S ↔
@@is_localization _ M P _ (h.to_ring_hom.comp $ algebra_map R S).to_algebra | begin
letI := (h.to_ring_hom.comp $ algebra_map R S).to_algebra,
exact is_localization_iff_of_alg_equiv M { commutes' := λ _, rfl, ..h },
end | lemma | is_localization.is_localization_iff_of_ring_equiv | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_of_base_ring_equiv [is_localization M S] (h : R ≃+* P) :
@@is_localization _ (M.map h.to_monoid_hom) S _
((algebra_map R S).comp h.symm.to_ring_hom).to_algebra | begin
constructor,
{ rintros ⟨_, ⟨y, hy, rfl⟩⟩,
convert is_localization.map_units S ⟨y, hy⟩,
dsimp only [ring_hom.algebra_map_to_algebra, ring_hom.comp_apply],
exact congr_arg _ (h.symm_apply_apply _) },
{ intro y,
obtain ⟨⟨x, s⟩, e⟩ := is_localization.surj M y,
refine ⟨⟨h x, _, _, s.prop, rfl... | lemma | is_localization.is_localization_of_base_ring_equiv | 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",
"ring_equiv.apply_symm_apply",
"ring_equiv.map_mul",
"ring_hom.algebra_map_to_algebra",
"ring_hom.comp_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization_iff_of_base_ring_equiv (h : R ≃+* P) :
is_localization M S ↔ @@is_localization _ (M.map h.to_monoid_hom) S _
((algebra_map R S).comp h.symm.to_ring_hom).to_algebra | begin
refine ⟨λ _, by exactI is_localization_of_base_ring_equiv _ _ h, _⟩,
letI := ((algebra_map R S).comp h.symm.to_ring_hom).to_algebra,
intro H,
convert @@is_localization_of_base_ring_equiv _ _ _ _ _ _ H h.symm,
{ erw [submonoid.map_equiv_eq_comap_symm, submonoid.comap_map_eq_of_injective],
exact h.to_... | lemma | is_localization.is_localization_iff_of_base_ring_equiv | 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.algebra_ext",
"algebra_map",
"is_localization",
"ring_equiv.symm_symm",
"ring_equiv.symm_to_ring_hom_comp_to_ring_hom",
"ring_hom.algebra_map_to_algebra",
"ring_hom.comp_assoc",
"ring_hom.comp_id",
"submonoid.comap_map_eq_of_injective",
"submonoid.map_equiv_eq_comap_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_zero_divisors_le_comap [is_localization M S] :
non_zero_divisors R ≤ (non_zero_divisors S).comap (algebra_map R S) | begin
rintros a ha b (e : b * algebra_map R S a = 0),
obtain ⟨x, s, rfl⟩ := mk'_surjective M b,
rw [← @mk'_one R _ M, ← mk'_mul, ← (algebra_map R S).map_zero, ← @mk'_one R _ M,
is_localization.eq] at e,
obtain ⟨c, e⟩ := e,
rw [mul_zero, mul_zero, submonoid.coe_one, one_mul, ←mul_assoc] at e,
rw mk'_eq_z... | lemma | is_localization.non_zero_divisors_le_comap | 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_localization.eq",
"mk'_one",
"mk'_surjective",
"mul_zero",
"non_zero_divisors",
"one_mul",
"submonoid.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_non_zero_divisors_le [is_localization M S] :
(non_zero_divisors R).map (algebra_map R S) ≤ non_zero_divisors S | submonoid.map_le_iff_le_comap.mpr (non_zero_divisors_le_comap M S) | lemma | is_localization.map_non_zero_divisors_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",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (z w : localization M) : localization M | localization.lift_on₂ z w
(λ a b c d, mk ((b : R) * c + d * a) (b * d)) $
λ a a' b b' c c' d d' h1 h2, mk_eq_mk_iff.2
begin
rw r_eq_r' at h1 h2 ⊢,
cases h1 with t₅ ht₅,
cases h2 with t₆ ht₆,
use t₅ * t₆,
dsimp only,
calc (↑t₅ * ↑t₆) * ((↑b' * ↑d') * ((b : R) * c + d * a)) =
(t₆ * (d' * c)) * (t₅ * ... | def | localization.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"
] | [
"localization",
"localization.lift_on₂",
"ring"
] | Addition in a ring localization is defined as `⟨a, b⟩ + ⟨c, d⟩ = ⟨b * c + d * a, b * d⟩`.
Should not be confused with `add_localization.add`, which is defined as
`⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_mk (a b c d) : (mk a b : localization M) + mk c d = mk (b * c + d * a) (b * d) | by { unfold has_add.add localization.add, apply lift_on₂_mk } | lemma | localization.add_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"
] | [
"localization",
"localization.add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mk_self (a b c) : (mk a b : localization M) + mk c b = mk (a + c) b | begin
rw [add_mk, mk_eq_mk_iff, r_eq_r'],
refine (r' M).symm ⟨1, _⟩,
simp only [submonoid.coe_one, submonoid.coe_mul],
ring
end | lemma | localization.add_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"
] | [
"localization",
"ring",
"submonoid.coe_mul",
"submonoid.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tac | `[
{ intros,
simp only [add_mk, localization.mk_mul, ← localization.mk_zero 1],
refine mk_eq_mk_iff.mpr (r_of_eq _),
simp only [submonoid.coe_mul],
ring }] | def | localization.tac | 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"
] | [
"localization.mk_mul",
"localization.mk_zero",
"ring",
"submonoid.coe_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_add_monoid_hom (b : M) : R →+ localization M | { to_fun := λ a, mk a b,
map_zero' := mk_zero _,
map_add' := λ x y, (add_mk_self _ _ _).symm } | def | localization.mk_add_monoid_hom | 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"
] | [
"localization"
] | For any given denominator `b : M`, the map `a ↦ a / b` is an `add_monoid_hom` from `R` to
`localization M` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_sum {ι : Type*} (f : ι → R) (s : finset ι) (b : M) :
mk (∑ i in s, f i) b = ∑ i in s, mk (f i) b | (mk_add_monoid_hom b).map_sum f s | lemma | localization.mk_sum | 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"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_list_sum (l : list R) (b : M) :
mk l.sum b = (l.map $ λ a, mk a b).sum | (mk_add_monoid_hom b).map_list_sum l | lemma | localization.mk_list_sum | 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 | |
mk_multiset_sum (l : multiset R) (b : M) :
mk l.sum b = (l.map $ λ a, mk a b).sum | (mk_add_monoid_hom b).map_multiset_sum l | lemma | localization.mk_multiset_sum | 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"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_localization_map_eq_monoid_of :
to_localization_map M (localization M) = monoid_of M | rfl | lemma | localization.to_localization_map_eq_monoid_of | 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"
] | [
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_of_eq_algebra_map (x) :
(monoid_of M).to_map x = algebra_map R (localization M) x | rfl | lemma | localization.monoid_of_eq_algebra_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",
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_one_eq_algebra_map (x) : mk x 1 = algebra_map R (localization M) x | rfl | lemma | localization.mk_one_eq_algebra_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",
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk'_apply (x y) : mk x y = is_localization.mk' (localization M) x y | by rw [mk_eq_monoid_of_mk'_apply, mk', to_localization_map_eq_monoid_of] | lemma | localization.mk_eq_mk'_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"
] | [
"is_localization.mk'",
"localization",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk' : (mk : R → M → localization M) = is_localization.mk' (localization M) | mk_eq_monoid_of_mk' | lemma | localization.mk_eq_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_localization.mk'",
"localization",
"mk_eq_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_algebra_map {A : Type*} [comm_semiring A] [algebra A R] (m : A) :
mk (algebra_map A R m) 1 = algebra_map A (localization M) m | by rw [mk_eq_mk', mk'_eq_iff_eq_mul, submonoid.coe_one, map_one, mul_one]; refl | lemma | localization.mk_algebra_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",
"algebra_map",
"comm_semiring",
"localization",
"map_one",
"mk_eq_mk'",
"mul_one",
"submonoid.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_nat_cast (m : ℕ) : (mk m 1 : localization M) = m | by simpa using @mk_algebra_map R _ M ℕ _ _ m | lemma | localization.mk_nat_cast | 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"
] | [
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv : localization M ≃ₐ[R] S | is_localization.alg_equiv M _ _ | def | localization.alg_equiv | 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"
] | [
"alg_equiv",
"is_localization.alg_equiv",
"localization"
] | The localization of `R` at `M` as a quotient type is isomorphic to any other localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.is_localization.unique (R Rₘ) [comm_semiring R] [comm_semiring Rₘ]
(M : submonoid R) [subsingleton R] [algebra R Rₘ] [is_localization M Rₘ] : unique Rₘ | have inhabited Rₘ := ⟨1⟩,
by exactI (alg_equiv M Rₘ).symm.injective.unique | def | is_localization.unique | 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"
] | [
"alg_equiv",
"algebra",
"comm_semiring",
"is_localization",
"submonoid",
"unique"
] | The localization of a singleton is a singleton. Cannot be an instance due to metavariables. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
alg_equiv_mk' (x : R) (y : M) :
alg_equiv M S (mk' (localization M) x y) = mk' S x y | alg_equiv_mk' _ _ | lemma | localization.alg_equiv_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"
] | [
"alg_equiv",
"localization",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv_symm_mk' (x : R) (y : M) :
(alg_equiv M S).symm (mk' S x y) = mk' (localization M) x y | alg_equiv_symm_mk' _ _ | lemma | localization.alg_equiv_symm_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"
] | [
"alg_equiv",
"localization",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv_mk (x y) :
alg_equiv M S (mk x y) = mk' S x y | by rw [mk_eq_mk', alg_equiv_mk'] | lemma | localization.alg_equiv_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"
] | [
"alg_equiv",
"mk'",
"mk_eq_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_equiv_symm_mk (x : R) (y : M) :
(alg_equiv M S).symm (mk' S x y) = mk x y | by rw [mk_eq_mk', alg_equiv_symm_mk'] | lemma | localization.alg_equiv_symm_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"
] | [
"alg_equiv",
"mk'",
"mk_eq_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (z : localization M) : localization M | localization.lift_on z (λ a b, mk (-a) b) $
λ a b c d h, mk_eq_mk_iff.2
begin
rw r_eq_r' at h ⊢,
cases h with t ht,
use t,
rw [mul_neg, mul_neg, ht],
ring_nf,
end | def | localization.neg | 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"
] | [
"localization",
"localization.lift_on",
"mul_neg"
] | Negation in a ring localization is defined as `-⟨a, b⟩ = ⟨-a, b⟩`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_mk (a b) : -(mk a b : localization M) = mk (-a) b | by { unfold has_neg.neg localization.neg, apply lift_on_mk } | lemma | localization.neg_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"
] | [
"localization",
"localization.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_mk (a c) (b d) : (mk a b : localization M) - mk c d = mk (d * a - b * c) (b * d) | calc mk a b - mk c d
= mk a b + (- mk c d) : sub_eq_add_neg _ _
... = mk a b + (mk (-c) d) : by rw neg_mk
... = mk (b * (-c) + d * a) (b * d) : add_mk _ _ _ _
... = mk (d * a - b * c) (b * d) : by congr'; ring | lemma | localization.sub_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"
] | [
"localization",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_int_cast (m : ℤ) : (mk m 1 : localization M) = m | by simpa using @mk_algebra_map R _ M ℤ _ _ m | lemma | localization.mk_int_cast | 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"
] | [
"localization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_map_eq_zero_iff {x : R} (hM : M ≤ non_zero_divisors R) :
algebra_map R S x = 0 ↔ x = 0 | begin
rw ← (algebra_map R S).map_zero,
split; intro h,
{ cases (eq_iff_exists M S).mp h with c hc,
rw [mul_zero, mul_comm] at hc,
exact hM c.2 x hc },
{ rw h },
end | lemma | is_localization.to_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_comm",
"mul_zero",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (hM : M ≤ non_zero_divisors R) :
injective (algebra_map R S) | begin
rw injective_iff_map_eq_zero (algebra_map R S),
intros a ha,
rwa to_map_eq_zero_iff S hM at ha
end | lemma | is_localization.injective | 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",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_map_ne_zero_of_mem_non_zero_divisors [nontrivial R]
(hM : M ≤ non_zero_divisors R) {x : R} (hx : x ∈ non_zero_divisors R) : algebra_map R S x ≠ 0 | show (algebra_map R S).to_monoid_with_zero_hom x ≠ 0,
from map_ne_zero_of_mem_non_zero_divisors (algebra_map R S) (is_localization.injective S hM) hx | lemma | is_localization.to_map_ne_zero_of_mem_non_zero_divisors | 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.injective",
"map_ne_zero_of_mem_non_zero_divisors",
"non_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sec_snd_ne_zero [nontrivial R] (hM : M ≤ non_zero_divisors R) (x : S) :
((sec M x).snd : R) ≠ 0 | non_zero_divisors.coe_ne_zero ⟨(sec M x).snd.val, hM (sec M x).snd.property⟩ | lemma | is_localization.sec_snd_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"
] | [
"non_zero_divisors",
"non_zero_divisors.coe_ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sec_fst_ne_zero [nontrivial R] [no_zero_divisors S] (hM : M ≤ non_zero_divisors R) {x : S}
(hx : x ≠ 0) : (sec M x).fst ≠ 0 | begin
have hsec := sec_spec M x,
intro hfst,
rw [hfst, map_zero, mul_eq_zero, _root_.map_eq_zero_iff] at hsec,
{ exact or.elim hsec hx (sec_snd_ne_zero hM x) },
{ exact is_localization.injective S hM }
end | lemma | is_localization.sec_fst_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.injective",
"mul_eq_zero",
"no_zero_divisors",
"non_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective_of_injective
(hg : function.injective g) [is_localization (M.map g : submonoid P) Q] :
function.injective (map Q g M.le_comap_map : S → Q) | begin
rw injective_iff_map_eq_zero,
intros z hz,
obtain ⟨a, b, rfl⟩ := mk'_surjective M z,
rw [map_mk', mk'_eq_zero_iff] at hz,
obtain ⟨⟨m', hm'⟩, hm⟩ := hz,
rw submonoid.mem_map at hm',
obtain ⟨n, hn, hnm⟩ := hm',
rw [subtype.coe_mk, ← hnm, ← map_mul, ← map_zero g] at hm,
rw [mk'_eq_zero_iff],
exa... | lemma | is_localization.map_injective_of_injective | 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",
"map_mul",
"mk'_surjective",
"submonoid",
"submonoid.mem_map",
"subtype.coe_mk"
] | Injectivity of a map descends to the map induced on localizations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
no_zero_divisors_of_le_non_zero_divisors
[algebra A S] {M : submonoid A} [is_localization M S]
(hM : M ≤ non_zero_divisors A) : no_zero_divisors S | { eq_zero_or_eq_zero_of_mul_eq_zero :=
begin
intros z w h,
cases surj M z with x hx,
cases surj M w with y hy,
have : z * w * algebra_map A S y.2 * algebra_map A S x.2 =
algebra_map A S x.1 * algebra_map A S y.1,
by rw [mul_assoc z, hy, ←hx]; ring,
rw [h, zero_mul, zero_m... | theorem | is_localization.no_zero_divisors_of_le_non_zero_divisors | 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",
"algebra_map",
"is_localization",
"map_mul",
"mul_assoc",
"no_zero_divisors",
"non_zero_divisors",
"ring",
"submonoid",
"zero_mul"
] | A `comm_ring` `S` which is the localization of a ring `R` without zero divisors at a subset of
non-zero elements does not have zero divisors.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain_of_le_non_zero_divisors
[algebra A S] {M : submonoid A} [is_localization M S]
(hM : M ≤ non_zero_divisors A) : is_domain S | begin
apply no_zero_divisors.to_is_domain _,
{ exact ⟨⟨(algebra_map A S) 0, (algebra_map A S) 1,
λ h, zero_ne_one (is_localization.injective S hM h)⟩⟩ },
{ exact no_zero_divisors_of_le_non_zero_divisors _ hM }
end | theorem | is_localization.is_domain_of_le_non_zero_divisors | 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",
"algebra_map",
"is_domain",
"is_localization",
"is_localization.injective",
"no_zero_divisors.to_is_domain",
"non_zero_divisors",
"submonoid",
"zero_ne_one"
] | A `comm_ring` `S` which is the localization of an integral domain `R` at a subset of
non-zero elements is an integral domain.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_domain_localization {M : submonoid A} (hM : M ≤ non_zero_divisors A) :
is_domain (localization M) | is_domain_of_le_non_zero_divisors _ hM | theorem | is_localization.is_domain_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"
] | [
"is_domain",
"localization",
"non_zero_divisors",
"submonoid"
] | The localization at of an integral domain to a set of non-zero elements is an integral domain.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_field.localization_map_bijective
{R Rₘ : Type*} [comm_ring R] [comm_ring Rₘ]
{M : submonoid R} (hM : (0 : R) ∉ M) (hR : is_field R)
[algebra R Rₘ] [is_localization M Rₘ] : function.bijective (algebra_map R Rₘ) | begin
letI := hR.to_field,
replace hM := le_non_zero_divisors_of_no_zero_divisors hM,
refine ⟨is_localization.injective _ hM, λ x, _⟩,
obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M x,
obtain ⟨n, hn⟩ := hR.mul_inv_cancel (non_zero_divisors.ne_zero $ hM hm),
exact ⟨r * n, by erw [eq_mk'_iff_mul_eq, ←map_mul, m... | lemma | is_field.localization_map_bijective | 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",
"algebra_map",
"comm_ring",
"is_field",
"is_localization",
"le_non_zero_divisors_of_no_zero_divisors",
"mk'_surjective",
"mul_assoc",
"mul_comm",
"mul_one",
"non_zero_divisors.ne_zero",
"submonoid"
] | If `R` is a field, then localizing at a submonoid not containing `0` adds no new elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field.localization_map_bijective
{K Kₘ : Type*} [field K] [comm_ring Kₘ] {M : submonoid K} (hM : (0 : K) ∉ M)
[algebra K Kₘ] [is_localization M Kₘ] : function.bijective (algebra_map K Kₘ) | (field.to_is_field K).localization_map_bijective hM | lemma | field.localization_map_bijective | 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",
"algebra_map",
"comm_ring",
"field",
"field.to_is_field",
"is_localization",
"submonoid"
] | If `R` is a field, then localizing at a submonoid not containing `0` adds no new elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
localization_algebra : algebra Rₘ Sₘ | (map Sₘ (algebra_map R S)
(show _ ≤ (algebra.algebra_map_submonoid S M).comap _, from M.le_comap_map)
: Rₘ →+* Sₘ).to_algebra | def | localization_algebra | 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",
"algebra.algebra_map_submonoid",
"algebra_map"
] | Definition of the natural algebra induced by the localization of an algebra.
Given an algebra `R → S`, a submonoid `R` of `M`, and a localization `Rₘ` for `M`,
let `Sₘ` be the localization of `S` to the image of `M` under `algebra_map R S`.
Then this is the natural algebra structure on `Rₘ → Sₘ`, such that the entire s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.map_units_map_submonoid (y : M) : is_unit (algebra_map R Sₘ y) | begin
rw is_scalar_tower.algebra_map_apply _ S,
exact is_localization.map_units Sₘ ⟨algebra_map R S y, algebra.mem_algebra_map_submonoid_of_mem y⟩
end | lemma | is_localization.map_units_map_submonoid | 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.mem_algebra_map_submonoid_of_mem",
"algebra_map",
"is_scalar_tower.algebra_map_apply",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.algebra_map_mk' (x : R) (y : M) :
algebra_map Rₘ Sₘ (is_localization.mk' Rₘ x y) =
is_localization.mk' Sₘ (algebra_map R S x) ⟨algebra_map R S y,
algebra.mem_algebra_map_submonoid_of_mem y⟩ | begin
rw [is_localization.eq_mk'_iff_mul_eq, subtype.coe_mk, ← is_scalar_tower.algebra_map_apply,
← is_scalar_tower.algebra_map_apply, is_scalar_tower.algebra_map_apply R Rₘ Sₘ,
is_scalar_tower.algebra_map_apply R Rₘ Sₘ, ← _root_.map_mul,
mul_comm, is_localization.mul_mk'_eq_mk'_of_mul],
exact con... | lemma | is_localization.algebra_map_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.mem_algebra_map_submonoid_of_mem",
"algebra_map",
"is_localization.eq_mk'_iff_mul_eq",
"is_localization.mk'",
"is_localization.mk'_mul_cancel_left",
"is_localization.mul_mk'_eq_mk'_of_mul",
"is_scalar_tower.algebra_map_apply",
"mul_comm",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_localization.algebra_map_eq_map_map_submonoid :
algebra_map Rₘ Sₘ = map Sₘ (algebra_map R S)
(show _ ≤ (algebra.algebra_map_submonoid S M).comap _, from M.le_comap_map) | eq.symm $ is_localization.map_unique _ (algebra_map Rₘ Sₘ) (λ x,
by rw [← is_scalar_tower.algebra_map_apply R S Sₘ, ← is_scalar_tower.algebra_map_apply R Rₘ Sₘ]) | lemma | is_localization.algebra_map_eq_map_map_submonoid | 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.algebra_map_submonoid",
"algebra_map",
"is_localization.map_unique",
"is_scalar_tower.algebra_map_apply"
] | If the square below commutes, the bottom map is uniquely specified:
```
R → S
↓ ↓
Rₘ → Sₘ
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.algebra_map_apply_eq_map_map_submonoid (x) :
algebra_map Rₘ Sₘ x = map Sₘ (algebra_map R S)
(show _ ≤ (algebra.algebra_map_submonoid S M).comap _, from M.le_comap_map)
x | fun_like.congr_fun (is_localization.algebra_map_eq_map_map_submonoid _ _ _ _) x | lemma | is_localization.algebra_map_apply_eq_map_map_submonoid | 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.algebra_map_submonoid",
"algebra_map",
"fun_like.congr_fun",
"is_localization.algebra_map_eq_map_map_submonoid"
] | If the square below commutes, the bottom map is uniquely specified:
```
R → S
↓ ↓
Rₘ → Sₘ
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_localization.lift_algebra_map_eq_algebra_map :
@is_localization.lift R _ M Rₘ _ _ Sₘ _ _ (algebra_map R Sₘ)
(is_localization.map_units_map_submonoid S Sₘ) =
algebra_map Rₘ Sₘ | is_localization.lift_unique _ (λ x, (is_scalar_tower.algebra_map_apply _ _ _ _).symm) | lemma | is_localization.lift_algebra_map_eq_algebra_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",
"is_localization.lift",
"is_localization.lift_unique",
"is_localization.map_units_map_submonoid",
"is_scalar_tower.algebra_map_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
localization_algebra_injective (hRS : function.injective (algebra_map R S)) :
function.injective (@algebra_map Rₘ Sₘ _ _ (localization_algebra M S)) | is_localization.map_injective_of_injective M Rₘ Sₘ hRS | lemma | localization_algebra_injective | 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.map_injective_of_injective",
"localization_algebra"
] | Injectivity of the underlying `algebra_map` descends to the algebra induced by localization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_le : #L ≤ #R | begin
classical,
casesI fintype_or_infinite R,
{ exact cardinal.mk_le_of_surjective (is_artinian_ring.localization_surjective S _) },
erw [←cardinal.mul_eq_self $ cardinal.aleph_0_le_mk R],
set f : R × R → L := λ aa, is_localization.mk' _ aa.1 (if h : aa.2 ∈ S then ⟨aa.2, h⟩ else 1),
refine @cardinal.mk_le_... | lemma | is_localization.card_le | ring_theory.localization | src/ring_theory/localization/cardinality.lean | [
"set_theory.cardinal.ordinal",
"ring_theory.artinian"
] | [
"cardinal.aleph_0_le_mk",
"cardinal.mk_le_of_surjective",
"fintype_or_infinite",
"is_artinian_ring.localization_surjective",
"is_localization.mk'",
"is_localization.mk'_surjective",
"set_like.eta"
] | A localization always has cardinality less than or equal to the base ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card (hS : S ≤ R⁰) : #R = #L | (cardinal.mk_le_of_injective (is_localization.injective L hS)).antisymm (card_le S) | lemma | is_localization.card | ring_theory.localization | src/ring_theory/localization/cardinality.lean | [
"set_theory.cardinal.ordinal",
"ring_theory.artinian"
] | [
"cardinal.mk_le_of_injective",
"is_localization.injective"
] | If you do not localize at any zero-divisors, localization preserves cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_fraction_ring [comm_ring K] [algebra R K] | is_localization (non_zero_divisors R) K | abbreviation | is_fraction_ring | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra",
"comm_ring",
"is_localization",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rat.is_fraction_ring : is_fraction_ring ℤ ℚ | { map_units :=
begin
rintro ⟨x, hx⟩,
rw mem_non_zero_divisors_iff_ne_zero at hx,
simpa only [eq_int_cast, is_unit_iff_ne_zero, int.cast_eq_zero,
ne.def, subtype.coe_mk] using hx,
end,
surj :=
begin
rintro ⟨n, d, hd, h⟩,
refine ⟨⟨n, ⟨d, _⟩⟩, rat.mul_denom_eq_num⟩,
rwa [m... | instance | rat.is_fraction_ring | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"eq_int_cast",
"int.cast_eq_zero",
"int.cast_inj",
"int.coe_nat_ne_zero_iff_pos",
"is_fraction_ring",
"is_unit_iff_ne_zero",
"mem_non_zero_divisors_iff_ne_zero",
"mul_left_cancel₀",
"subtype.coe_mk"
] | The cast from `int` to `rat` as a `fraction_ring`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_map_eq_zero_iff {x : R} :
algebra_map R K x = 0 ↔ x = 0 | to_map_eq_zero_iff _ (le_of_eq rfl) | lemma | is_fraction_ring.to_map_eq_zero_iff | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective : function.injective (algebra_map R K) | is_localization.injective _ (le_of_eq rfl) | theorem | is_fraction_ring.injective | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_localization.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {a b : R} : (↑a : K) = ↑b ↔ a = b | (is_fraction_ring.injective R K).eq_iff | lemma | is_fraction_ring.coe_inj | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"is_fraction_ring.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_map_ne_zero_of_mem_non_zero_divisors [nontrivial R]
{x : R} (hx : x ∈ non_zero_divisors R) : algebra_map R K x ≠ 0 | is_localization.to_map_ne_zero_of_mem_non_zero_divisors _ le_rfl hx | lemma | is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"is_localization.to_map_ne_zero_of_mem_non_zero_divisors",
"le_rfl",
"non_zero_divisors",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_domain : is_domain K | is_domain_of_le_non_zero_divisors _ (le_refl (non_zero_divisors A)) | theorem | is_fraction_ring.is_domain | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"is_domain",
"non_zero_divisors"
] | A `comm_ring` `K` which is the localization of an integral domain `R` at `R - {0}` is an
integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv (z : K) : K | if h : z = 0 then 0 else
mk' K ↑(sec (non_zero_divisors A) z).2
⟨(sec _ z).1,
mem_non_zero_divisors_iff_ne_zero.2 $ λ h0, h $
eq_zero_of_fst_eq_zero (sec_spec (non_zero_divisors A) z) h0⟩ | def | is_fraction_ring.inv | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"mk'",
"non_zero_divisors"
] | The inverse of an element in the field of fractions of an integral domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_cancel (x : K) (hx : x ≠ 0) :
x * is_fraction_ring.inv A x = 1 | begin
rw [is_fraction_ring.inv, dif_neg hx, ←is_unit.mul_left_inj
(map_units K ⟨(sec _ x).1, mem_non_zero_divisors_iff_ne_zero.2 $
λ h0, hx $ eq_zero_of_fst_eq_zero (sec_spec (non_zero_divisors A) x) h0⟩),
one_mul, mul_assoc],
rw [mk'_spec, ←eq_mk'_iff_mul_eq],
exact (mk'_sec _ x).symm
end | lemma | is_fraction_ring.mul_inv_cancel | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"is_fraction_ring.inv",
"mul_assoc",
"mul_inv_cancel",
"non_zero_divisors",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_field : field K | { inv := is_fraction_ring.inv A,
mul_inv_cancel := is_fraction_ring.mul_inv_cancel A,
inv_zero := begin
change is_fraction_ring.inv A (0 : K) = 0,
rw [is_fraction_ring.inv],
exact dif_pos rfl
end,
.. is_fraction_ring.is_domain A,
.. show comm_ring K, by apply_instance } | def | is_fraction_ring.to_field | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"comm_ring",
"field",
"inv_zero",
"is_fraction_ring.inv",
"is_fraction_ring.is_domain",
"is_fraction_ring.mul_inv_cancel",
"mul_inv_cancel"
] | A `comm_ring` `K` which is the localization of an integral domain `R` at `R - {0}` is a field.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk'_mk_eq_div {r s} (hs : s ∈ non_zero_divisors A) :
mk' K r ⟨s, hs⟩ = algebra_map A K r / algebra_map A K s | mk'_eq_iff_eq_mul.2 $ (div_mul_cancel (algebra_map A K r)
(is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors hs)).symm | lemma | is_fraction_ring.mk'_mk_eq_div | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"div_mul_cancel",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"mk'",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_div {r} (s : non_zero_divisors A) :
mk' K r s = algebra_map A K r / algebra_map A K s | mk'_mk_eq_div s.2 | lemma | is_fraction_ring.mk'_eq_div | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"mk'",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_surjective (z : K) : ∃ (x y : A) (hy : y ∈ non_zero_divisors A),
algebra_map _ _ x / algebra_map _ _ y = z | let ⟨x, ⟨y, hy⟩, h⟩ := mk'_surjective (non_zero_divisors A) z
in ⟨x, y, hy, by rwa mk'_eq_div at h⟩ | lemma | is_fraction_ring.div_surjective | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"mk'_surjective",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_map_of_injective (hg : function.injective g)
(y : non_zero_divisors A) : is_unit (g y) | is_unit.mk0 (g y) $ show g.to_monoid_with_zero_hom y ≠ 0,
from map_ne_zero_of_mem_non_zero_divisors g hg y.2 | lemma | is_fraction_ring.is_unit_map_of_injective | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"is_unit",
"is_unit.mk0",
"map_ne_zero_of_mem_non_zero_divisors",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_zero_iff_eq_zero [algebra R K] [is_fraction_ring R K] {x : R}
{y : non_zero_divisors R} : mk' K x y = 0 ↔ x = 0 | begin
refine ⟨λ hxy, _, λ h, by rw [h, mk'_zero]⟩,
{ simp_rw [mk'_eq_zero_iff, mul_left_coe_non_zero_divisors_eq_zero_iff] at hxy,
exact (exists_const _).mp hxy },
end | lemma | is_fraction_ring.mk'_eq_zero_iff_eq_zero | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra",
"exists_const",
"is_fraction_ring",
"mk'",
"mk'_zero",
"mul_left_coe_non_zero_divisors_eq_zero_iff",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_eq_one_iff_eq {x : A} {y : non_zero_divisors A} : mk' K x y = 1 ↔ x = y | begin
refine ⟨_, λ hxy, by rw [hxy, mk'_self']⟩,
{ intro hxy, have hy : (algebra_map A K) ↑y ≠ (0 : K) :=
is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors y.property,
rw [is_fraction_ring.mk'_eq_div, div_eq_one_iff_eq hy] at hxy,
exact is_fraction_ring.injective A K hxy }
end | lemma | is_fraction_ring.mk'_eq_one_iff_eq | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"div_eq_one_iff_eq",
"is_fraction_ring.injective",
"is_fraction_ring.mk'_eq_div",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"mk'",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (hg : injective g) : K →+* L | lift $ λ (y : non_zero_divisors A), is_unit_map_of_injective hg y | def | is_fraction_ring.lift | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"lift",
"non_zero_divisors"
] | Given an integral domain `A` with field of fractions `K`,
and an injective ring hom `g : A →+* L` where `L` is a field, we get a
field hom sending `z : K` to `g x * (g y)⁻¹`, where `(x, y) : A × (non_zero_divisors A)` are
such that `z = f x * (f y)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_algebra_map (hg : injective g) (x) :
lift hg (algebra_map A K x) = g x | lift_eq _ _ | lemma | is_fraction_ring.lift_algebra_map | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra_map",
"lift"
] | Given an integral domain `A` with field of fractions `K`,
and an injective ring hom `g : A →+* L` where `L` is a field,
the field hom induced from `K` to `L` maps `x` to `g x` for all
`x : A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk' (hg : injective g) (x) (y : non_zero_divisors A) :
lift hg (mk' K x y) = g x / g y | by simp only [mk'_eq_div, map_div₀, lift_algebra_map] | lemma | is_fraction_ring.lift_mk' | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"lift",
"map_div₀",
"mk'",
"non_zero_divisors"
] | Given an integral domain `A` with field of fractions `K`,
and an injective ring hom `g : A →+* L` where `L` is a field,
field hom induced from `K` to `L` maps `f x / f y` to `g x / g y` for all
`x : A, y ∈ non_zero_divisors A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map {A B K L : Type*} [comm_ring A] [comm_ring B] [is_domain B]
[comm_ring K] [algebra A K] [is_fraction_ring A K] [comm_ring L] [algebra B L]
[is_fraction_ring B L] {j : A →+* B} (hj : injective j) :
K →+* L | map L j (show non_zero_divisors A ≤ (non_zero_divisors B).comap j,
from non_zero_divisors_le_comap_non_zero_divisors_of_injective j hj) | def | is_fraction_ring.map | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra",
"comm_ring",
"is_domain",
"is_fraction_ring",
"non_zero_divisors",
"non_zero_divisors_le_comap_non_zero_divisors_of_injective"
] | Given integral domains `A, B` with fields of fractions `K`, `L`
and an injective ring hom `j : A →+* B`, we get a field hom
sending `z : K` to `g (j x) * (g (j y))⁻¹`, where `(x, y) : A × (non_zero_divisors A)` are
such that `z = f x * (f y)⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
field_equiv_of_ring_equiv [algebra B L] [is_fraction_ring B L] (h : A ≃+* B) :
K ≃+* L | ring_equiv_of_ring_equiv K L h
begin
ext b,
show b ∈ h.to_equiv '' _ ↔ _,
erw [h.to_equiv.image_eq_preimage, set.preimage, set.mem_set_of_eq,
mem_non_zero_divisors_iff_ne_zero, mem_non_zero_divisors_iff_ne_zero],
exact h.symm.map_ne_zero_iff
end | def | is_fraction_ring.field_equiv_of_ring_equiv | ring_theory.localization | src/ring_theory/localization/fraction_ring.lean | [
"algebra.algebra.tower",
"ring_theory.localization.basic"
] | [
"algebra",
"is_fraction_ring",
"mem_non_zero_divisors_iff_ne_zero",
"set.preimage"
] | Given integral domains `A, B` and localization maps to their fields of fractions
`f : A →+* K, g : B →+* L`, an isomorphism `j : A ≃+* B` induces an isomorphism of
fields of fractions `K ≃+* L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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