statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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quotient_map_surjective {J : ideal R} {I : ideal S} {f : R →+* S} {H : J ≤ I.comap f}
(hf : function.surjective f) : function.surjective (quotient_map I f H) | λ x, let ⟨x, hx⟩ := quotient.mk_surjective x in
let ⟨y, hy⟩ := hf x in ⟨(quotient.mk J) y, by simp [hx, hy]⟩ | lemma | ideal.quotient_map_surjective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_quotient_map_eq_of_comp_eq {R' S' : Type*} [comm_ring R'] [comm_ring S']
{f : R →+* S} {f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f)
(I : ideal S') : (quotient_map I g' le_rfl).comp (quotient_map (I.comap g') f le_rfl) =
(quotient_map I f' le_rfl).comp (quotient_map (I.comap... | begin
refine ring_hom.ext (λ a, _),
obtain ⟨r, rfl⟩ := quotient.mk_surjective a,
simp only [ring_hom.comp_apply, quotient_map_mk],
exact congr_arg (quotient.mk I) (trans (g'.comp_apply f r).symm (hfg ▸ (f'.comp_apply g r))),
end | lemma | ideal.comp_quotient_map_eq_of_comp_eq | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"comm_ring",
"ideal",
"le_rfl",
"quotient_map",
"ring_hom.comp_apply",
"ring_hom.ext"
] | Commutativity of a square is preserved when taking quotients by an ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_mapₐ {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (hIJ : I ≤ J.comap f) :
A ⧸ I →ₐ[R₁] B ⧸ J | { commutes' := λ r, by simp,
..quotient_map J (f : A →+* B) hIJ } | def | ideal.quotient_mapₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"quotient_map"
] | The algebra hom `A/I →+* B/J` induced by an algebra hom `f : A →ₐ[R₁] B` with `I ≤ f⁻¹(J)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_mkₐ {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (H : I ≤ J.comap f)
{x : A} : quotient_mapₐ J f H (quotient.mk I x) = quotient.mkₐ R₁ J (f x) | rfl | lemma | ideal.quotient_map_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_comp_mkₐ {I : ideal A} (J : ideal B) (f : A →ₐ[R₁] B) (H : I ≤ J.comap f) :
(quotient_mapₐ J f H).comp (quotient.mkₐ R₁ I) = (quotient.mkₐ R₁ J).comp f | alg_hom.ext (λ x, by simp only [quotient_map_mkₐ, quotient.mkₐ_eq_mk, alg_hom.comp_apply]) | lemma | ideal.quotient_map_comp_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.comp_apply",
"alg_hom.ext",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_alg (I : ideal A) (J : ideal B) (f : A ≃ₐ[R₁] B)
(hIJ : J = I.map (f : A →+* B)) :
(A ⧸ I) ≃ₐ[R₁] B ⧸ J | { commutes' := λ r, by simp,
..quotient_equiv I J (f : A ≃+* B) hIJ } | def | ideal.quotient_equiv_alg | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | The algebra equiv `A/I ≃ₐ[R] B/J` induced by an algebra equiv `f : A ≃ₐ[R] B`,
where`J = f(I)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_algebra {I : ideal A} [algebra R A] :
algebra (R ⧸ I.comap (algebra_map R A)) (A ⧸ I) | (quotient_map I (algebra_map R A) (le_of_eq rfl)).to_algebra | instance | ideal.quotient_algebra | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra",
"algebra_map",
"ideal",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_quotient_injective {I : ideal A} [algebra R A]:
function.injective (algebra_map (R ⧸ I.comap (algebra_map R A)) (A ⧸ I)) | begin
rintros ⟨a⟩ ⟨b⟩ hab,
replace hab := quotient.eq.mp hab,
rw ← ring_hom.map_sub at hab,
exact quotient.eq.mpr hab
end | lemma | ideal.algebra_map_quotient_injective | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra",
"algebra_map",
"ideal",
"ring_hom.map_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_alg_of_eq {I J : ideal A} (h : I = J) : (A ⧸ I) ≃ₐ[R₁] A ⧸ J | quotient_equiv_alg I J alg_equiv.refl $ h ▸ (map_id I).symm | def | ideal.quotient_equiv_alg_of_eq | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_equiv.refl",
"ideal",
"map_id"
] | Quotienting by equal ideals gives equivalent algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_equiv_alg_of_eq_mk {I J : ideal A} (h : I = J) (x : A) :
quotient_equiv_alg_of_eq R₁ h (ideal.quotient.mk I x) = ideal.quotient.mk J x | rfl | lemma | ideal.quotient_equiv_alg_of_eq_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_equiv_alg_of_eq_symm {I J : ideal A} (h : I = J) :
(quotient_equiv_alg_of_eq R₁ h).symm = quotient_equiv_alg_of_eq R₁ h.symm | by ext; refl | lemma | ideal.quotient_equiv_alg_of_eq_symm | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_left_to_quot_sup : R ⧸ I →+* R ⧸ (I ⊔ J) | ideal.quotient.factor I (I ⊔ J) le_sup_left | def | double_quot.quot_left_to_quot_sup | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.factor",
"le_sup_left"
] | The obvious ring hom `R/I → R/(I ⊔ J)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_quot_left_to_quot_sup :
(quot_left_to_quot_sup I J).ker = J.map (ideal.quotient.mk I) | by simp only [mk_ker, sup_idem, sup_comm, quot_left_to_quot_sup, quotient.factor, ker_quotient_lift,
map_eq_iff_sup_ker_eq_of_surjective I^.quotient.mk quotient.mk_surjective, ← sup_assoc] | lemma | double_quot.ker_quot_left_to_quot_sup | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"sup_assoc",
"sup_comm",
"sup_idem"
] | The kernel of `quot_left_to_quot_sup` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_to_quot_sup : (R ⧸ I) ⧸ J.map (ideal.quotient.mk I) →+* R ⧸ I ⊔ J | by exact ideal.quotient.lift (J.map (ideal.quotient.mk I)) (quot_left_to_quot_sup I J)
(ker_quot_left_to_quot_sup I J).symm.le | def | double_quot.quot_quot_to_quot_sup | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.lift",
"ideal.quotient.mk"
] | The ring homomorphism `(R/I)/J' -> R/(I ⊔ J)` induced by `quot_left_to_quot_sup` where `J'`
is the image of `J` in `R/I` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_mk : R →+* ((R ⧸ I) ⧸ J.map I^.quotient.mk) | by exact ((J.map I^.quotient.mk)^.quotient.mk).comp I^.quotient.mk | def | double_quot.quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | The composite of the maps `R → (R/I)` and `(R/I) → (R/I)/J'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ker_quot_quot_mk : (quot_quot_mk I J).ker = I ⊔ J | by rw [ring_hom.ker_eq_comap_bot, quot_quot_mk, ← comap_comap, ← ring_hom.ker, mk_ker,
comap_map_of_surjective (ideal.quotient.mk I) (quotient.mk_surjective), ← ring_hom.ker, mk_ker,
sup_comm] | lemma | double_quot.ker_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"ring_hom.ker",
"ring_hom.ker_eq_comap_bot",
"sup_comm"
] | The kernel of `quot_quot_mk` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_sup_quot_quot_mk (I J : ideal R) :
R ⧸ (I ⊔ J) →+* (R ⧸ I) ⧸ J.map (ideal.quotient.mk I) | ideal.quotient.lift (I ⊔ J) (quot_quot_mk I J) (ker_quot_quot_mk I J).symm.le | def | double_quot.lift_sup_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"ideal.quotient.lift",
"ideal.quotient.mk"
] | The ring homomorphism `R/(I ⊔ J) → (R/I)/J' `induced by `quot_quot_mk` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_quot_sup : (R ⧸ I) ⧸ J.map (ideal.quotient.mk I) ≃+* R ⧸ I ⊔ J | ring_equiv.of_hom_inv (quot_quot_to_quot_sup I J) (lift_sup_quot_quot_mk I J)
(by { ext z, refl }) (by { ext z, refl }) | def | double_quot.quot_quot_equiv_quot_sup | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk",
"ring_equiv.of_hom_inv"
] | `quot_quot_to_quot_add` and `lift_sup_double_qot_mk` are inverse isomorphisms. In the case where
`I ≤ J`, this is the Third Isomorphism Theorem (see `quot_quot_equiv_quot_of_le`) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_quot_sup_quot_quot_mk (x : R) :
quot_quot_equiv_quot_sup I J (quot_quot_mk I J x) = ideal.quotient.mk (I ⊔ J) x | rfl | lemma | double_quot.quot_quot_equiv_quot_sup_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_sup_symm_quot_quot_mk (x : R) :
(quot_quot_equiv_quot_sup I J).symm (ideal.quotient.mk (I ⊔ J) x) = quot_quot_mk I J x | rfl | lemma | double_quot.quot_quot_equiv_quot_sup_symm_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm :
(R ⧸ I) ⧸ J.map I^.quotient.mk ≃+* (R ⧸ J) ⧸ I.map J^.quotient.mk | ((quot_quot_equiv_quot_sup I J).trans (quot_equiv_of_eq sup_comm)).trans
(quot_quot_equiv_quot_sup J I).symm | def | double_quot.quot_quot_equiv_comm | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"sup_comm"
] | The obvious isomorphism `(R/I)/J' → (R/J)/I' ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_comm_quot_quot_mk (x : R) :
quot_quot_equiv_comm I J (quot_quot_mk I J x) = quot_quot_mk J I x | rfl | lemma | double_quot.quot_quot_equiv_comm_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_comp_quot_quot_mk :
ring_hom.comp ↑(quot_quot_equiv_comm I J) (quot_quot_mk I J) = quot_quot_mk J I | ring_hom.ext $ quot_quot_equiv_comm_quot_quot_mk I J | lemma | double_quot.quot_quot_equiv_comm_comp_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.comp",
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_symm :
(quot_quot_equiv_comm I J).symm = quot_quot_equiv_comm J I | rfl | lemma | double_quot.quot_quot_equiv_comm_symm | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le (h : I ≤ J) : (R ⧸ I) ⧸ (J.map I^.quotient.mk) ≃+* R ⧸ J | (quot_quot_equiv_quot_sup I J).trans (ideal.quot_equiv_of_eq $ sup_eq_right.mpr h) | def | double_quot.quot_quot_equiv_quot_of_le | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal.quot_equiv_of_eq"
] | **The Third Isomorphism theorem** for rings. See `quot_quot_equiv_quot_sup` for a version
that does not assume an inclusion of ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_quot_of_le_quot_quot_mk (x : R) (h : I ≤ J) :
quot_quot_equiv_quot_of_le h (quot_quot_mk I J x) = J^.quotient.mk x | rfl | lemma | double_quot.quot_quot_equiv_quot_of_le_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le_symm_mk (x : R) (h : I ≤ J) :
(quot_quot_equiv_quot_of_le h).symm (J^.quotient.mk x) = (quot_quot_mk I J x) | rfl | lemma | double_quot.quot_quot_equiv_quot_of_le_symm_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le_comp_quot_quot_mk (h : I ≤ J) :
ring_hom.comp ↑(quot_quot_equiv_quot_of_le h) (quot_quot_mk I J) = J^.quotient.mk | by ext ; refl | lemma | double_quot.quot_quot_equiv_quot_of_le_comp_quot_quot_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le_symm_comp_mk (h : I ≤ J) :
ring_hom.comp ↑(quot_quot_equiv_quot_of_le h).symm J^.quotient.mk = quot_quot_mk I J | by ext ; refl | lemma | double_quot.quot_quot_equiv_quot_of_le_symm_comp_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ring_hom.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_mk_mk [comm_ring R] (I J : ideal R) (x : R) :
quot_quot_equiv_comm I J (ideal.quotient.mk _ (ideal.quotient.mk _ x)) =
algebra_map R _ x | rfl | lemma | double_quot.quot_quot_equiv_comm_mk_mk | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"comm_ring",
"ideal",
"ideal.quotient.mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_sup_quot_quot_algebra_map (x : R) :
double_quot.quot_quot_equiv_quot_sup I J (algebra_map R _ x) = algebra_map _ _ x | rfl | lemma | double_quot.quot_quot_equiv_quot_sup_quot_quot_algebra_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map",
"double_quot.quot_quot_equiv_quot_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_algebra_map (x : R) :
quot_quot_equiv_comm I J (algebra_map R _ x) = algebra_map _ _ x | rfl | lemma | double_quot.quot_quot_equiv_comm_algebra_map | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_left_to_quot_supₐ : A ⧸ I →ₐ[R] A ⧸ (I ⊔ J) | alg_hom.mk (quot_left_to_quot_sup I J) rfl (map_mul _) rfl (map_add _) (λ _, rfl) | def | double_quot.quot_left_to_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"map_mul"
] | The natural algebra homomorphism `A / I → A / (I ⊔ J)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_left_to_quot_supₐ_to_ring_hom :
(quot_left_to_quot_supₐ R I J).to_ring_hom = quot_left_to_quot_sup I J | rfl | lemma | double_quot.quot_left_to_quot_supₐ_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_left_to_quot_supₐ :
⇑(quot_left_to_quot_supₐ R I J) = quot_left_to_quot_sup I J | rfl | lemma | double_quot.coe_quot_left_to_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_to_quot_supₐ : (A ⧸ I) ⧸ J.map (I^.quotient.mkₐ R) →ₐ[R] A ⧸ I ⊔ J | alg_hom.mk (quot_quot_to_quot_sup I J) rfl (map_mul _) rfl (map_add _) (λ _, rfl) | def | double_quot.quot_quot_to_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"map_mul"
] | The algebra homomorphism `(A / I) / J' -> A / (I ⊔ J) induced by `quot_left_to_quot_sup`,
where `J'` is the projection of `J` in `A / I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_to_quot_supₐ_to_ring_hom :
(quot_quot_to_quot_supₐ R I J).to_ring_hom = quot_quot_to_quot_sup I J | rfl | lemma | double_quot.quot_quot_to_quot_supₐ_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_to_quot_supₐ :
⇑(quot_quot_to_quot_supₐ R I J) = quot_quot_to_quot_sup I J | rfl | lemma | double_quot.coe_quot_quot_to_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_mkₐ : A →ₐ[R] ((A ⧸ I) ⧸ J.map (I^.quotient.mkₐ R)) | alg_hom.mk (quot_quot_mk I J) rfl (map_mul _) rfl (map_add _) (λ _, rfl) | def | double_quot.quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"map_mul"
] | The composition of the algebra homomorphisms `A → (A / I)` and `(A / I) → (A / I) / J'`,
where `J'` is the projection `J` in `A / I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_mkₐ_to_ring_hom :
(quot_quot_mkₐ R I J).to_ring_hom = quot_quot_mk I J | rfl | lemma | double_quot.quot_quot_mkₐ_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_mkₐ :
⇑(quot_quot_mkₐ R I J) = quot_quot_mk I J | rfl | lemma | double_quot.coe_quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_sup_quot_quot_mkₐ (I J : ideal A) :
A ⧸ (I ⊔ J) →ₐ[R] (A ⧸ I) ⧸ J.map (I^.quotient.mkₐ R) | alg_hom.mk (lift_sup_quot_quot_mk I J) rfl (map_mul _) rfl (map_add _) (λ _, rfl) | def | double_quot.lift_sup_quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"ideal",
"map_mul"
] | The injective algebra homomorphism `A / (I ⊔ J) → (A / I) / J'`induced by `quot_quot_mk`,
where `J'` is the projection `J` in `A / I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_sup_quot_quot_mkₐ_to_ring_hom :
(lift_sup_quot_quot_mkₐ R I J).to_ring_hom = lift_sup_quot_quot_mk I J | rfl | lemma | double_quot.lift_sup_quot_quot_mkₐ_to_ring_hom | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lift_sup_quot_quot_mkₐ :
⇑(lift_sup_quot_quot_mkₐ R I J) = lift_sup_quot_quot_mk I J | rfl | lemma | double_quot.coe_lift_sup_quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_supₐ : ((A ⧸ I) ⧸ J.map (I^.quotient.mkₐ R)) ≃ₐ[R] A ⧸ I ⊔ J | @alg_equiv.of_ring_equiv R _ _ _ _ _ _ _ (quot_quot_equiv_quot_sup I J) (λ _, rfl) | def | double_quot.quot_quot_equiv_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_equiv.of_ring_equiv"
] | `quot_quot_to_quot_add` and `lift_sup_quot_quot_mk` are inverse isomorphisms. In the case where
`I ≤ J`, this is the Third Isomorphism Theorem (see `quot_quot_equiv_quot_of_le`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_quot_supₐ_to_ring_equiv :
(quot_quot_equiv_quot_supₐ R I J).to_ring_equiv = quot_quot_equiv_quot_sup I J | rfl | lemma | double_quot.quot_quot_equiv_quot_supₐ_to_ring_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_equiv_quot_supₐ :
⇑(quot_quot_equiv_quot_supₐ R I J) = quot_quot_equiv_quot_sup I J | rfl | lemma | double_quot.coe_quot_quot_equiv_quot_supₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_supₐ_symm_to_ring_equiv:
(quot_quot_equiv_quot_supₐ R I J).symm.to_ring_equiv = (quot_quot_equiv_quot_sup I J).symm | rfl | lemma | double_quot.quot_quot_equiv_quot_supₐ_symm_to_ring_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_equiv_quot_supₐ_symm:
⇑(quot_quot_equiv_quot_supₐ R I J).symm = (quot_quot_equiv_quot_sup I J).symm | rfl | lemma | double_quot.coe_quot_quot_equiv_quot_supₐ_symm | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_commₐ :
((A ⧸ I) ⧸ J.map (I^.quotient.mkₐ R)) ≃ₐ[R]
((A ⧸ J) ⧸ I.map (J^.quotient.mkₐ R)) | @alg_equiv.of_ring_equiv R _ _ _ _ _ _ _ (quot_quot_equiv_comm I J) (λ _, rfl) | def | double_quot.quot_quot_equiv_commₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_equiv.of_ring_equiv"
] | The natural algebra isomorphism `(A / I) / J' → (A / J) / I'`,
where `J'` (resp. `I'`) is the projection of `J` in `A / I` (resp. `I` in `A / J`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_commₐ_to_ring_equiv :
(quot_quot_equiv_commₐ R I J).to_ring_equiv = quot_quot_equiv_comm I J | rfl | lemma | double_quot.quot_quot_equiv_commₐ_to_ring_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_equiv_commₐ :
⇑(quot_quot_equiv_commₐ R I J) = quot_quot_equiv_comm I J | rfl | lemma | double_quot.coe_quot_quot_equiv_commₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_symmₐ :
(quot_quot_equiv_commₐ R I J).symm = quot_quot_equiv_commₐ R J I | -- TODO: should be `rfl` but times out
alg_equiv.ext $ λ x, (fun_like.congr_fun (quot_quot_equiv_comm_symm I J) x : _) | lemma | double_quot.quot_quot_equiv_comm_symmₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_equiv.ext",
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_comm_comp_quot_quot_mkₐ :
alg_hom.comp ↑(quot_quot_equiv_commₐ R I J) (quot_quot_mkₐ R I J) = quot_quot_mkₐ R J I | alg_hom.ext $ quot_quot_equiv_comm_quot_quot_mk I J | lemma | double_quot.quot_quot_equiv_comm_comp_quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.comp",
"alg_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_leₐ (h : I ≤ J) :
((A ⧸ I) ⧸ (J.map (I^.quotient.mkₐ R))) ≃ₐ[R] A ⧸ J | @alg_equiv.of_ring_equiv R _ _ _ _ _ _ _ (quot_quot_equiv_quot_of_le h) (λ _, rfl) | def | double_quot.quot_quot_equiv_quot_of_leₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_equiv.of_ring_equiv"
] | The **third isomoprhism theorem** for rings. See `quot_quot_equiv_quot_sup` for version
that does not assume an inclusion of ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot_quot_equiv_quot_of_leₐ_to_ring_equiv (h : I ≤ J) :
(quot_quot_equiv_quot_of_leₐ R h).to_ring_equiv = quot_quot_equiv_quot_of_le h | rfl | lemma | double_quot.quot_quot_equiv_quot_of_leₐ_to_ring_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_equiv_quot_of_leₐ (h : I ≤ J) :
⇑(quot_quot_equiv_quot_of_leₐ R h) = quot_quot_equiv_quot_of_le h | rfl | lemma | double_quot.coe_quot_quot_equiv_quot_of_leₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_leₐ_symm_to_ring_equiv (h : I ≤ J) :
(quot_quot_equiv_quot_of_leₐ R h).symm.to_ring_equiv = (quot_quot_equiv_quot_of_le h).symm | rfl | lemma | double_quot.quot_quot_equiv_quot_of_leₐ_symm_to_ring_equiv | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_quot_quot_equiv_quot_of_leₐ_symm (h : I ≤ J) :
⇑(quot_quot_equiv_quot_of_leₐ R h).symm = (quot_quot_equiv_quot_of_le h).symm | rfl | lemma | double_quot.coe_quot_quot_equiv_quot_of_leₐ_symm | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le_comp_quot_quot_mkₐ (h : I ≤ J) :
alg_hom.comp ↑(quot_quot_equiv_quot_of_leₐ R h) (quot_quot_mkₐ R I J) =
J^.quotient.mkₐ R | rfl | lemma | double_quot.quot_quot_equiv_quot_of_le_comp_quot_quot_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quot_quot_equiv_quot_of_le_symm_comp_mkₐ (h : I ≤ J) :
alg_hom.comp ↑(quot_quot_equiv_quot_of_leₐ R h).symm (J^.quotient.mkₐ R) =
quot_quot_mkₐ R I J | rfl | lemma | double_quot.quot_quot_equiv_quot_of_le_symm_comp_mkₐ | ring_theory.ideal | src/ring_theory/ideal/quotient_operations.lean | [
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient"
] | [
"alg_hom.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_nat_gcd (m n : ℕ) : gcd m n = nat.gcd m n | rfl | lemma | gcd_eq_nat_gcd | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lcm_eq_nat_lcm (m n : ℕ) : lcm m n = nat.lcm m n | rfl | lemma | lcm_eq_nat_lcm | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_of_nonneg {z : ℤ} (h : 0 ≤ z) : normalize z = z | show z * ↑(ite _ _ _) = z, by rw [if_pos h, units.coe_one, mul_one] | lemma | int.normalize_of_nonneg | 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"
] | [
"mul_one",
"normalize",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_of_nonpos {z : ℤ} (h : z ≤ 0) : normalize z = -z | begin
obtain rfl | h := h.eq_or_lt,
{ simp },
{ change z * ↑(ite _ _ _) = -z,
rw [if_neg (not_le_of_gt h), units.coe_neg, units.coe_one, mul_neg_one] }
end | lemma | int.normalize_of_nonpos | 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"
] | [
"mul_neg_one",
"normalize",
"units.coe_neg",
"units.coe_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normalize_coe_nat (n : ℕ) : normalize (n : ℤ) = n | normalize_of_nonneg (coe_nat_le_coe_nat_of_le $ nat.zero_le n) | lemma | int.normalize_coe_nat | 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"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_normalize (z : ℤ) : |z| = normalize z | by cases le_total 0 z; simp [normalize_of_nonneg, normalize_of_nonpos, *] | lemma | int.abs_eq_normalize | 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"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_of_normalize_eq_self {z : ℤ} (hz : normalize z = z) : 0 ≤ z | abs_eq_self.1 $ by rw [abs_eq_normalize, hz] | lemma | int.nonneg_of_normalize_eq_self | 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"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonneg_iff_normalize_eq_self (z : ℤ) : normalize z = z ↔ 0 ≤ z | ⟨nonneg_of_normalize_eq_self, normalize_of_nonneg⟩ | lemma | int.nonneg_iff_normalize_eq_self | 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"
] | [
"normalize"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_associated_of_nonneg {a b : ℤ} (h : associated a b) (ha : 0 ≤ a) (hb : 0 ≤ b) : a = b | dvd_antisymm_of_normalize_eq (normalize_of_nonneg ha) (normalize_of_nonneg hb) h.dvd h.symm.dvd | lemma | int.eq_of_associated_of_nonneg | 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",
"dvd_antisymm_of_normalize_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_gcd (i j : ℤ) : ↑(int.gcd i j) = gcd_monoid.gcd i j | rfl | lemma | int.coe_gcd | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_lcm (i j : ℤ) : ↑(int.lcm i j) = gcd_monoid.lcm i j | rfl | lemma | int.coe_lcm | 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"
] | [
"int.lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_abs_gcd (i j : ℤ) : nat_abs (gcd_monoid.gcd i j) = int.gcd i j | rfl | lemma | int.nat_abs_gcd | 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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_abs_lcm (i j : ℤ) : nat_abs (gcd_monoid.lcm i j) = int.lcm i j | rfl | lemma | int.nat_abs_lcm | 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"
] | [
"int.lcm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_unit_of_abs (a : ℤ) : ∃ (u : ℤ) (h : is_unit u), (int.nat_abs a : ℤ) = u * a | begin
cases (nat_abs_eq a) with h,
{ use [1, is_unit_one], rw [← h, one_mul], },
{ use [-1, is_unit_one.neg], rw [← neg_eq_iff_eq_neg.mpr h],
simp only [neg_mul, one_mul] }
end | lemma | int.exists_unit_of_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"
] | [
"is_unit",
"is_unit_one",
"neg_mul",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_nat_abs {a b : ℤ} : int.gcd a b = nat.gcd a.nat_abs b.nat_abs | rfl | lemma | int.gcd_eq_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_eq_one_iff_coprime {a b : ℤ} : int.gcd a b = 1 ↔ is_coprime a b | begin
split,
{ intro hg,
obtain ⟨ua, hua, ha⟩ := exists_unit_of_abs a,
obtain ⟨ub, hub, hb⟩ := exists_unit_of_abs b,
use [(nat.gcd_a (int.nat_abs a) (int.nat_abs b)) * ua,
(nat.gcd_b (int.nat_abs a) (int.nat_abs b)) * ub],
rw [mul_assoc, ← ha, mul_assoc, ← hb, mul_comm, mul_comm _ (int.nat_a... | lemma | int.gcd_eq_one_iff_coprime | 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"
] | [
"by_contradiction",
"dvd_add",
"is_coprime",
"mul_assoc",
"mul_comm",
"nat.gcd_a",
"nat.gcd_b",
"nat.gcd_eq_gcd_ab",
"nat.prime.not_dvd_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprime_iff_nat_coprime {a b : ℤ} : is_coprime a b ↔ nat.coprime a.nat_abs b.nat_abs | by rw [←gcd_eq_one_iff_coprime, nat.coprime_iff_gcd_eq_one, gcd_eq_nat_abs] | lemma | int.coprime_iff_nat_coprime | 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",
"nat.coprime_iff_gcd_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gcd_ne_one_iff_gcd_mul_right_ne_one {a : ℤ} {m n : ℕ} :
a.gcd (m * n) ≠ 1 ↔ a.gcd m ≠ 1 ∨ a.gcd n ≠ 1 | by simp only [gcd_eq_one_iff_coprime, ← not_and_distrib, not_iff_not, is_coprime.mul_right_iff] | lemma | int.gcd_ne_one_iff_gcd_mul_right_ne_one | 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_right_iff",
"not_and_distrib",
"not_iff_not"
] | If `gcd a (m * n) ≠ 1`, then `gcd a m ≠ 1` or `gcd a n ≠ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_eq_one_of_gcd_mul_right_eq_one_left {a : ℤ} {m n : ℕ} (h : a.gcd (m * n) = 1) :
a.gcd m = 1 | nat.dvd_one.mp $ trans_rel_left _ (gcd_dvd_gcd_mul_right_right a m n) h | lemma | int.gcd_eq_one_of_gcd_mul_right_eq_one_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"
] | [
"gcd_dvd_gcd_mul_right_right"
] | If `gcd a (m * n) = 1`, then `gcd a m = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gcd_eq_one_of_gcd_mul_right_eq_one_right {a : ℤ} {m n : ℕ} (h : a.gcd (m * n) = 1) :
a.gcd n = 1 | nat.dvd_one.mp $ trans_rel_left _ (gcd_dvd_gcd_mul_left_right a n m) h | lemma | int.gcd_eq_one_of_gcd_mul_right_eq_one_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"
] | [
"gcd_dvd_gcd_mul_left_right"
] | If `gcd a (m * n) = 1`, then `gcd a n = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_of_gcd_eq_one {a b c : ℤ} (h : int.gcd a b = 1) (heq : a * b = c ^ 2) :
∃ (a0 : ℤ), a = a0 ^ 2 ∨ a = - (a0 ^ 2) | begin
have h' : is_unit (gcd_monoid.gcd a b), { rw [← coe_gcd, h, int.coe_nat_one], exact is_unit_one },
obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq,
use d,
rw ← hu,
cases int.units_eq_one_or u with hu' hu'; { rw hu', simp }
end | lemma | int.sq_of_gcd_eq_one | 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",
"int.units_eq_one_or",
"is_unit",
"is_unit_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_of_coprime {a b c : ℤ} (h : is_coprime a b) (heq : a * b = c ^ 2) :
∃ (a0 : ℤ), a = a0 ^ 2 ∨ a = - (a0 ^ 2) | sq_of_gcd_eq_one (gcd_eq_one_iff_coprime.mpr h) heq | lemma | int.sq_of_coprime | 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 | |
nat_abs_euclidean_domain_gcd (a b : ℤ) :
int.nat_abs (euclidean_domain.gcd a b) = int.gcd a b | begin
apply nat.dvd_antisymm; rw ← int.coe_nat_dvd,
{ rw int.nat_abs_dvd,
exact int.dvd_gcd (euclidean_domain.gcd_dvd_left _ _) (euclidean_domain.gcd_dvd_right _ _) },
{ rw int.dvd_nat_abs,
exact euclidean_domain.dvd_gcd (int.gcd_dvd_left _ _) (int.gcd_dvd_right _ _) }
end | lemma | int.nat_abs_euclidean_domain_gcd | 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"
] | [
"euclidean_domain.dvd_gcd",
"euclidean_domain.gcd",
"euclidean_domain.gcd_dvd_left",
"euclidean_domain.gcd_dvd_right",
"int.coe_nat_dvd",
"int.dvd_gcd",
"int.dvd_nat_abs",
"int.gcd_dvd_left",
"int.gcd_dvd_right",
"int.nat_abs_dvd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associates_int_equiv_nat : associates ℤ ≃ ℕ | begin
refine ⟨λz, z.out.nat_abs, λn, associates.mk n, _, _⟩,
{ refine (assume a, quotient.induction_on' a $ assume a,
associates.mk_eq_mk_iff_associated.2 $ associated.symm $ ⟨norm_unit a, _⟩),
show normalize a = int.nat_abs (normalize a),
rw [int.coe_nat_abs, int.abs_eq_normalize, normalize_idem] },
... | def | associates_int_equiv_nat | 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.symm",
"associates",
"associates.mk",
"int.abs_eq_normalize",
"int.coe_nat_abs",
"int.nat_abs_abs",
"normalize",
"normalize_idem",
"quotient.induction_on'"
] | Maps an associate class of integers consisting of `-n, n` to `n : ℕ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
int.prime.dvd_mul {m n : ℤ} {p : ℕ}
(hp : nat.prime p) (h : (p : ℤ) ∣ m * n) : p ∣ m.nat_abs ∨ p ∣ n.nat_abs | begin
apply (nat.prime.dvd_mul hp).mp,
rw ← int.nat_abs_mul,
exact int.coe_nat_dvd_left.mp h
end | lemma | int.prime.dvd_mul | 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"
] | [
"int.nat_abs_mul",
"nat.prime",
"nat.prime.dvd_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.prime.dvd_mul' {m n : ℤ} {p : ℕ}
(hp : nat.prime p) (h : (p : ℤ) ∣ m * n) : (p : ℤ) ∣ m ∨ (p : ℤ) ∣ n | begin
rw [int.coe_nat_dvd_left, int.coe_nat_dvd_left],
exact int.prime.dvd_mul hp h
end | lemma | int.prime.dvd_mul' | 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"
] | [
"int.coe_nat_dvd_left",
"int.prime.dvd_mul",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.prime.dvd_pow {n : ℤ} {k p : ℕ}
(hp : nat.prime p) (h : (p : ℤ) ∣ n ^ k) : p ∣ n.nat_abs | begin
apply @nat.prime.dvd_of_dvd_pow _ _ k hp,
rw ← int.nat_abs_pow,
exact int.coe_nat_dvd_left.mp h
end | lemma | int.prime.dvd_pow | 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"
] | [
"int.nat_abs_pow",
"nat.prime",
"nat.prime.dvd_of_dvd_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.prime.dvd_pow' {n : ℤ} {k p : ℕ}
(hp : nat.prime p) (h : (p : ℤ) ∣ n ^ k) : (p : ℤ) ∣ n | begin
rw int.coe_nat_dvd_left,
exact int.prime.dvd_pow hp h
end | lemma | int.prime.dvd_pow' | 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"
] | [
"int.coe_nat_dvd_left",
"int.prime.dvd_pow",
"nat.prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_two_or_dvd_of_dvd_two_mul_pow_self_two {m : ℤ} {p : ℕ}
(hp : nat.prime p) (h : (p : ℤ) ∣ 2 * m ^ 2) : p = 2 ∨ p ∣ int.nat_abs m | begin
cases int.prime.dvd_mul hp h with hp2 hpp,
{ apply or.intro_left,
exact le_antisymm (nat.le_of_dvd zero_lt_two hp2) (nat.prime.two_le hp) },
{ apply or.intro_right,
rw [sq, int.nat_abs_mul] at hpp,
exact (or_self _).mp ((nat.prime.dvd_mul hp).mp hpp)}
end | lemma | prime_two_or_dvd_of_dvd_two_mul_pow_self_two | 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"
] | [
"int.nat_abs_mul",
"int.prime.dvd_mul",
"nat.prime",
"nat.prime.dvd_mul",
"nat.prime.two_le",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.exists_prime_and_dvd {n : ℤ} (hn : n.nat_abs ≠ 1) : ∃ p, prime p ∧ p ∣ n | begin
obtain ⟨p, pp, pd⟩ := nat.exists_prime_and_dvd hn,
exact ⟨p, nat.prime_iff_prime_int.mp pp, int.coe_nat_dvd_left.mpr pd⟩,
end | lemma | int.exists_prime_and_dvd | 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"
] | [
"nat.exists_prime_and_dvd",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.factors_eq {n : ℕ} : normalized_factors n = n.factors | begin
cases n, { simp },
rw [← multiset.rel_eq, ← associated_eq_eq],
apply factors_unique (irreducible_of_normalized_factor) _,
{ rw [multiset.coe_prod, nat.prod_factors n.succ_ne_zero],
apply normalized_factors_prod (nat.succ_ne_zero _) },
{ apply_instance },
{ intros x hx,
rw [nat.irreducible_iff_... | theorem | nat.factors_eq | 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_eq_eq",
"multiset.coe_prod",
"multiset.rel_eq",
"nat.irreducible_iff_prime",
"nat.prime_iff",
"nat.prime_of_mem_factors",
"nat.prod_factors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.factors_multiset_prod_of_irreducible
{s : multiset ℕ} (h : ∀ (x : ℕ), x ∈ s → irreducible x) :
normalized_factors (s.prod) = s | begin
rw [← multiset.rel_eq, ← associated_eq_eq],
apply unique_factorization_monoid.factors_unique irreducible_of_normalized_factor h
(normalized_factors_prod _),
rw [ne.def, multiset.prod_eq_zero_iff],
intro con,
exact not_irreducible_zero (h 0 con),
end | lemma | nat.factors_multiset_prod_of_irreducible | 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_eq_eq",
"con",
"irreducible",
"multiset",
"multiset.prod_eq_zero_iff",
"multiset.rel_eq",
"not_irreducible_zero",
"unique_factorization_monoid.factors_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_int_iff_nat_abs_finite {a b : ℤ} : finite a b ↔ finite a.nat_abs b.nat_abs | by simp only [finite_def, ← int.nat_abs_dvd_iff_dvd, int.nat_abs_pow] | lemma | multiplicity.finite_int_iff_nat_abs_finite | 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"
] | [
"finite",
"int.nat_abs_dvd_iff_dvd",
"int.nat_abs_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_int_iff {a b : ℤ} : finite a b ↔ (a.nat_abs ≠ 1 ∧ b ≠ 0) | by rw [finite_int_iff_nat_abs_finite, finite_nat_iff, pos_iff_ne_zero, int.nat_abs_ne_zero] | lemma | multiplicity.finite_int_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"
] | [
"finite",
"int.nat_abs_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_nat : decidable_rel (λ a b : ℕ, (multiplicity a b).dom) | λ a b, decidable_of_iff _ finite_nat_iff.symm | instance | multiplicity.decidable_nat | 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"
] | [
"decidable_of_iff",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
decidable_int : decidable_rel (λ a b : ℤ, (multiplicity a b).dom) | λ a b, decidable_of_iff _ finite_int_iff.symm | instance | multiplicity.decidable_int | 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"
] | [
"decidable_of_iff",
"multiplicity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on_primes {P : ℕ → Prop} (h₀ : P 0) (h₁ : P 1)
(h : ∀ p a : ℕ, p.prime → P a → P (p * a)) (n : ℕ) : P n | begin
apply unique_factorization_monoid.induction_on_prime,
exact h₀,
{ intros n h,
rw nat.is_unit_iff.1 h,
exact h₁, },
{ intros a p _ hp ha,
exact h p a hp.nat_prime ha, },
end | lemma | induction_on_primes | 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"
] | [
"unique_factorization_monoid.induction_on_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.associated_nat_abs (k : ℤ) : associated k k.nat_abs | associated_of_dvd_dvd (int.coe_nat_dvd_right.mpr dvd_rfl) (int.nat_abs_dvd.mpr dvd_rfl) | lemma | int.associated_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_of_dvd_dvd",
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int.prime_iff_nat_abs_prime {k : ℤ} : prime k ↔ nat.prime k.nat_abs | (int.associated_nat_abs k).prime_iff.trans nat.prime_iff_prime_int.symm | lemma | int.prime_iff_nat_abs_prime | 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"
] | [
"int.associated_nat_abs",
"nat.prime",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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