fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Quotient.smul_top {R : Type*} [CommRing R] (a : R) (I : Ideal R) :
(a • ⊤ : Submodule R (R ⧸ I)) = Submodule.span R {Submodule.Quotient.mk a} := by
simp [← Ideal.Quotient.span_singleton_one, Algebra.smul_def, Submodule.smul_span] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | Quotient.smul_top | null |
KerLift.map_smul (f : A →ₐ[R₁] B) (r : R₁) (x : A ⧸ (RingHom.ker f)) :
f.kerLift (r • x) = r • f.kerLift x := by
obtain ⟨a, rfl⟩ := Quotient.mkₐ_surjective R₁ _ x
exact _root_.map_smul f _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | KerLift.map_smul | null |
kerLiftAlg (f : A →ₐ[R₁] B) : A ⧸ (RingHom.ker f) →ₐ[R₁] B :=
AlgHom.mk' (RingHom.kerLift (f : A →+* B)) fun _ _ => KerLift.map_smul f _ _
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | kerLiftAlg | The induced algebras morphism from the quotient by the kernel to the codomain.
This is an isomorphism if `f` has a right inverse (`quotientKerAlgEquivOfRightInverse`) /
is surjective (`quotientKerAlgEquivOfSurjective`). |
kerLiftAlg_mk (f : A →ₐ[R₁] B) (a : A) :
kerLiftAlg f (Quotient.mk (RingHom.ker f) a) = f a := by
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | kerLiftAlg_mk | null |
kerLiftAlg_toRingHom (f : A →ₐ[R₁] B) :
(kerLiftAlg f : A ⧸ ker f →+* B) = RingHom.kerLift (f : A →+* B) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | kerLiftAlg_toRingHom | null |
kerLiftAlg_injective (f : A →ₐ[R₁] B) : Function.Injective (kerLiftAlg f) :=
RingHom.kerLift_injective (R := A) (S := B) f | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | kerLiftAlg_injective | The induced algebra morphism from the quotient by the kernel is injective. |
@[simps!]
quotientKerAlgEquivOfRightInverse {f : A →ₐ[R₁] B} {g : B → A}
(hf : Function.RightInverse g f) : (A ⧸ RingHom.ker f) ≃ₐ[R₁] B :=
{ RingHom.quotientKerEquivOfRightInverse hf,
kerLiftAlg f with } | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientKerAlgEquivOfRightInverse | The **first isomorphism** theorem for algebras, computable version. |
@[simps!]
noncomputable quotientKerAlgEquivOfSurjective {f : A →ₐ[R₁] B} (hf : Function.Surjective f) :
(A ⧸ (RingHom.ker f)) ≃ₐ[R₁] B :=
quotientKerAlgEquivOfRightInverse (Classical.choose_spec hf.hasRightInverse) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientKerAlgEquivOfSurjective | The **first isomorphism theorem** for algebras. |
quotientMap {I : Ideal R} (J : Ideal S) [I.IsTwoSided] [J.IsTwoSided] (f : R →+* S)
(hIJ : I ≤ J.comap f) : R ⧸ I →+* S ⧸ J :=
Quotient.lift I ((Quotient.mk J).comp f) fun _ ha => by
simpa [Function.comp_apply, RingHom.coe_comp, Quotient.eq_zero_iff_mem] using hIJ ha
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap | The ring hom `R/I →+* S/J` induced by a ring hom `f : R →+* S` with `I ≤ f⁻¹(J)` |
quotientMap_mk {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} {H : J ≤ I.comap f} {x : R} :
quotientMap I f H (Quotient.mk J x) = Quotient.mk I (f x) :=
Quotient.lift_mk J _ _
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_mk | null |
quotientMap_algebraMap {J : Ideal A} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : A →+* S} {H : J ≤ I.comap f}
{x : R₁} : quotientMap I f H (algebraMap R₁ (A ⧸ J) x) = Quotient.mk I (f (algebraMap _ _ x)) :=
Quotient.lift_mk J _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_algebraMap | null |
quotientMap_comp_mk {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} (H : J ≤ I.comap f) :
(quotientMap I f H).comp (Quotient.mk J) = (Quotient.mk I).comp f :=
RingHom.ext fun x => by simp only [Function.comp_apply, RingHom.coe_comp, Ideal.quotientMap_mk] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_comp_mk | null |
ker_quotientMap_mk {I J : Ideal R} [I.IsTwoSided] [J.IsTwoSided] :
RingHom.ker (quotientMap (J.map _) (Quotient.mk I) le_comap_map) = I.map (Quotient.mk J) := by
rw [Ideal.quotientMap, Ideal.ker_quotient_lift, ← RingHom.comap_ker, Ideal.mk_ker,
Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective,
← RingHom.ker_eq_comap_bot, Ideal.mk_ker, Ideal.map_sup, Ideal.map_quotient_self, bot_sup_eq] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | ker_quotientMap_mk | null |
@[simps]
quotientEquiv : R ⧸ I ≃+* S ⧸ J where
__ := quotientMap J f (hIJ ▸ le_comap_map)
invFun := quotientMap I f.symm (hIJ ▸ (map_comap_of_equiv f).le)
left_inv := by
rintro ⟨r⟩
simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, RingHom.toFun_eq_coe,
quotientMap_mk, RingEquiv.coe_toRingHom, RingEquiv.symm_apply_apply]
right_inv := by
rintro ⟨s⟩
simp only [Submodule.Quotient.quot_mk_eq_mk, Quotient.mk_eq_mk, RingHom.toFun_eq_coe,
quotientMap_mk, RingEquiv.coe_toRingHom, RingEquiv.apply_symm_apply] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquiv | The ring equiv `R/I ≃+* S/J` induced by a ring equiv `f : R ≃+* S`, where `J = f(I)`. |
quotientEquiv_mk (x : R) :
quotientEquiv I J f hIJ (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J (f x) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquiv_mk | null |
quotientEquiv_symm_mk (x : S) :
(quotientEquiv I J f hIJ).symm (Ideal.Quotient.mk J x) = Ideal.Quotient.mk I (f.symm x) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquiv_symm_mk | null |
quotientMap_injective' {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} {H : J ≤ I.comap f} (h : I.comap f ≤ J) :
Function.Injective (quotientMap I f H) := by
refine (injective_iff_map_eq_zero (quotientMap I f H)).2 fun a ha => ?_
obtain ⟨r, rfl⟩ := Quotient.mk_surjective a
rw [quotientMap_mk, Quotient.eq_zero_iff_mem] at ha
exact Quotient.eq_zero_iff_mem.mpr (h ha) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_injective' | `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. |
quotientMap_injective {I : Ideal S} {f : R →+* S} [I.IsTwoSided] :
Function.Injective (quotientMap I f le_rfl) :=
quotientMap_injective' le_rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_injective | If we take `J = I.comap f` then `quotientMap` is injective automatically. |
quotientMap_surjective {J : Ideal R} {I : Ideal S} [I.IsTwoSided] [J.IsTwoSided]
{f : R →+* S} {H : J ≤ I.comap f}
(hf : Function.Surjective f) : Function.Surjective (quotientMap I f H) := fun x =>
let ⟨x, hx⟩ := Quotient.mk_surjective x
let ⟨y, hy⟩ := hf x
⟨(Quotient.mk J) y, by simp [hx, hy]⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMap_surjective | null |
comp_quotientMap_eq_of_comp_eq {R' S' : Type*} [Ring R'] [Ring S'] {f : R →+* S}
{f' : R' →+* S'} {g : R →+* R'} {g' : S →+* S'} (hfg : f'.comp g = g'.comp f)
(I : Ideal S') [I.IsTwoSided] :
let leq := le_of_eq (_root_.trans (comap_comap f g') (hfg ▸ comap_comap g f'))
(quotientMap I g' le_rfl).comp (quotientMap (I.comap g') f le_rfl) =
(quotientMap I f' le_rfl).comp (quotientMap (I.comap f') g leq) := by
refine RingHom.ext fun a => ?_
obtain ⟨r, rfl⟩ := Quotient.mk_surjective a
simp only [RingHom.comp_apply, quotientMap_mk]
exact (Ideal.Quotient.mk I).congr_arg (_root_.trans (g'.comp_apply f r).symm
(hfg ▸ f'.comp_apply g r)) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | comp_quotientMap_eq_of_comp_eq | Commutativity of a square is preserved when taking quotients by an ideal. |
quotientMapₐ (f : A →ₐ[R₁] B) (hIJ : I ≤ J.comap f) :
A ⧸ I →ₐ[R₁] B ⧸ J :=
{ quotientMap J (f : A →+* B) hIJ with commutes' := fun r => by simp only [RingHom.toFun_eq_coe,
quotientMap_algebraMap, AlgHom.coe_toRingHom, AlgHom.commutes, Quotient.mk_algebraMap] }
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientMapₐ | The algebra hom `A/I →+* B/J` induced by an algebra hom `f : A →ₐ[R₁] B` with `I ≤ f⁻¹(J)`. |
quotient_map_mkₐ (f : A →ₐ[R₁] B) (H : I ≤ J.comap f) {x : A} :
quotientMapₐ J f H (Quotient.mk I x) = Quotient.mkₐ R₁ J (f x) :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotient_map_mkₐ | null |
quotient_map_comp_mkₐ (f : A →ₐ[R₁] B) (H : I ≤ J.comap f) :
(quotientMapₐ J f H).comp (Quotient.mkₐ R₁ I) = (Quotient.mkₐ R₁ J).comp f :=
AlgHom.ext fun x => by simp only [quotient_map_mkₐ, Quotient.mkₐ_eq_mk, AlgHom.comp_apply]
variable (I) in | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotient_map_comp_mkₐ | null |
quotientEquivAlg (f : A ≃ₐ[R₁] B) (hIJ : J = I.map (f : A →+* B)) :
(A ⧸ I) ≃ₐ[R₁] B ⧸ J :=
{ quotientEquiv I J (f : A ≃+* B) hIJ with
commutes' r := by simp } | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquivAlg | The algebra equiv `A/I ≃ₐ[R] B/J` induced by an algebra equiv `f : A ≃ₐ[R] B`,
where`J = f(I)`. |
Quotient.algebraQuotientOfLEComap {R} [CommRing R] [Algebra R A] {p : Ideal R}
{P : Ideal A} [P.IsTwoSided] (h : p ≤ comap (algebraMap R A) P) :
Algebra (R ⧸ p) (A ⧸ P) where
algebraMap := quotientMap P (algebraMap R A) h
smul := Quotient.lift₂ (⟦· • ·⟧) fun r₁ a₁ r₂ a₂ hr ha ↦ Quotient.sound <| by
have := h (p.quotientRel_def.mp hr)
rw [mem_comap, map_sub] at this
simpa only [Algebra.smul_def] using P.quotientRel_def.mpr
(P.mul_sub_mul_mem this <| P.quotientRel_def.mp ha)
smul_def' := by rintro ⟨_⟩ ⟨_⟩; exact congr_arg (⟦·⟧) (Algebra.smul_def _ _)
commutes' := by rintro ⟨_⟩ ⟨_⟩; exact congr_arg (⟦·⟧) (Algebra.commutes _ _) | abbrev | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | Quotient.algebraQuotientOfLEComap | If `P` lies over `p`, then `R / p` has a canonical map to `A / P`. |
algebraMap_quotient_injective {R} [CommRing R] {I : Ideal A} [I.IsTwoSided] [Algebra R A] :
Function.Injective (algebraMap (R ⧸ I.comap (algebraMap R A)) (A ⧸ I)) := by
rintro ⟨a⟩ ⟨b⟩ hab
replace hab := Quotient.eq.mp hab
rw [← RingHom.map_sub] at hab
exact Quotient.eq.mpr hab
variable (R₁) | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | algebraMap_quotient_injective | null |
quotientEquivAlgOfEq {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :
(A ⧸ I) ≃ₐ[R₁] A ⧸ J :=
quotientEquivAlg I J AlgEquiv.refl <| h ▸ (map_id I).symm
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquivAlgOfEq | Quotienting by equal ideals gives equivalent algebras. |
quotientEquivAlgOfEq_mk {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) (x : A) :
quotientEquivAlgOfEq R₁ h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquivAlgOfEq_mk | null |
quotientEquivAlgOfEq_symm {I J : Ideal A} [I.IsTwoSided] [J.IsTwoSided] (h : I = J) :
(quotientEquivAlgOfEq R₁ h).symm = quotientEquivAlgOfEq R₁ h.symm := by
ext
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientEquivAlgOfEq_symm | null |
comap_map_mk {I J : Ideal R} [I.IsTwoSided] (h : I ≤ J) :
Ideal.comap (Ideal.Quotient.mk I) (Ideal.map (Ideal.Quotient.mk I) J) = J := by
ext; rw [← Ideal.mem_quotient_iff_mem h, Ideal.mem_comap] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | comap_map_mk | null |
noncomputable quotientKerEquivRange
{R A B : Type*} [CommSemiring R] [Ring A] [Algebra R A] [Semiring B] [Algebra R B]
(f : A →ₐ[R] B) :
(A ⧸ RingHom.ker f) ≃ₐ[R] f.range :=
(Ideal.quotientEquivAlgOfEq R (AlgHom.ker_rangeRestrict f).symm).trans <|
Ideal.quotientKerAlgEquivOfSurjective f.rangeRestrict_surjective | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotientKerEquivRange | The **first isomorphism theorem** for commutative algebras (`AlgHom.range` version). |
RingEquiv.quotientBot [Ring R] : R ⧸ (⊥ : Ideal R) ≃+* R :=
(Ideal.quotEquivOfEq (RingHom.ker_coe_equiv <| .refl _).symm).trans <|
RingHom.quotientKerEquivOfRightInverse (f := .id R) (g := _root_.id) fun _ ↦ rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | RingEquiv.quotientBot | The quotient of a ring by he zero ideal is isomorphic to the ring itself. |
RingEquiv.quotientBot_mk [Ring R] (r : R) :
RingEquiv.quotientBot R (Ideal.Quotient.mk ⊥ r) = r :=
rfl
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | RingEquiv.quotientBot_mk | null |
RingEquiv.quotientBot_symm_mk [Ring R] (r : R) :
(RingEquiv.quotientBot R).symm r = r :=
rfl
variable (R S) in | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | RingEquiv.quotientBot_symm_mk | null |
AlgEquiv.quotientBot [CommSemiring R] [Ring S] [Algebra R S] :
(S ⧸ (⊥ : Ideal S)) ≃ₐ[R] S where
__ := RingEquiv.quotientBot S
commutes' x := by simp [← Ideal.Quotient.mk_algebraMap]
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | AlgEquiv.quotientBot | `RingEquiv.quotientBot` as an algebra isomorphism. |
AlgEquiv.quotientBot_mk [CommSemiring R] [CommRing S] [Algebra R S] (s : S) :
AlgEquiv.quotientBot R S (Ideal.Quotient.mk ⊥ s) = s :=
rfl
@[simp] | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | AlgEquiv.quotientBot_mk | null |
AlgEquiv.quotientBot_symm_mk [CommSemiring R] [CommRing S] [Algebra R S]
(s : S) : (AlgEquiv.quotientBot R S).symm s = s :=
rfl | lemma | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | AlgEquiv.quotientBot_symm_mk | null |
quotLeftToQuotSup : R ⧸ I →+* R ⧸ I ⊔ J :=
Ideal.Quotient.factor le_sup_left | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotLeftToQuotSup | The obvious ring hom `R/I → R/(I ⊔ J)` |
ker_quotLeftToQuotSup : RingHom.ker (quotLeftToQuotSup I J) =
J.map (Ideal.Quotient.mk I) := by
simp only [mk_ker, sup_idem, sup_comm, quotLeftToQuotSup, Quotient.factor, ker_quotient_lift,
map_eq_iff_sup_ker_eq_of_surjective (Ideal.Quotient.mk I) Quotient.mk_surjective, ← sup_assoc] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | ker_quotLeftToQuotSup | The kernel of `quotLeftToQuotSup` |
quotQuotToQuotSup : (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) →+* R ⧸ I ⊔ J :=
Ideal.Quotient.lift (J.map (Ideal.Quotient.mk I)) (quotLeftToQuotSup I J)
(ker_quotLeftToQuotSup I J).symm.le | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotToQuotSup | The ring homomorphism `(R/I)/J' -> R/(I ⊔ J)` induced by `quotLeftToQuotSup` where `J'`
is the image of `J` in `R/I` |
quotQuotMk : R →+* (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) :=
(Ideal.Quotient.mk (J.map (Ideal.Quotient.mk I))).comp (Ideal.Quotient.mk I) | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotMk | The composite of the maps `R → (R/I)` and `(R/I) → (R/I)/J'` |
ker_quotQuotMk : RingHom.ker (quotQuotMk I J) = I ⊔ J := by
rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,
comap_map_of_surjective (Ideal.Quotient.mk I) Ideal.Quotient.mk_surjective, ← RingHom.ker,
mk_ker, sup_comm] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | ker_quotQuotMk | The kernel of `quotQuotMk` |
liftSupQuotQuotMk (I J : Ideal R) : R ⧸ I ⊔ J →+* (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) :=
Ideal.Quotient.lift (I ⊔ J) (quotQuotMk I J) (ker_quotQuotMk I J).symm.le | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | liftSupQuotQuotMk | The ring homomorphism `R/(I ⊔ J) → (R/I)/J' `induced by `quotQuotMk` |
quotQuotEquivQuotSup : (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) ≃+* R ⧸ I ⊔ J :=
RingEquiv.ofHomInv (quotQuotToQuotSup I J) (liftSupQuotQuotMk I J)
(by
repeat apply Ideal.Quotient.ringHom_ext
rfl)
(by
repeat apply Ideal.Quotient.ringHom_ext
rfl)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSup | `quotQuotToQuotSup` and `liftSupQuotQuotMk` are inverse isomorphisms. In the case where
`I ≤ J`, this is the Third Isomorphism Theorem (see `quotQuotEquivQuotOfLe`). |
quotQuotEquivQuotSup_quotQuotMk (x : R) :
quotQuotEquivQuotSup I J (quotQuotMk I J x) = Ideal.Quotient.mk (I ⊔ J) x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSup_quotQuotMk | null |
quotQuotEquivQuotSup_symm_quotQuotMk (x : R) :
(quotQuotEquivQuotSup I J).symm (Ideal.Quotient.mk (I ⊔ J) x) = quotQuotMk I J x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSup_symm_quotQuotMk | null |
quotQuotEquivComm : (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) ≃+*
(R ⧸ J) ⧸ I.map (Ideal.Quotient.mk J) :=
((quotQuotEquivQuotSup I J).trans (quotEquivOfEq (sup_comm ..))).trans
(quotQuotEquivQuotSup J I).symm
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm | The obvious isomorphism `(R/I)/J' → (R/J)/I'` |
quotQuotEquivComm_quotQuotMk (x : R) :
quotQuotEquivComm I J (quotQuotMk I J x) = quotQuotMk J I x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_quotQuotMk | null |
quotQuotEquivComm_comp_quotQuotMk :
RingHom.comp (↑(quotQuotEquivComm I J)) (quotQuotMk I J) = quotQuotMk J I :=
RingHom.ext <| quotQuotEquivComm_quotQuotMk I J
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_comp_quotQuotMk | null |
quotQuotEquivComm_symm : (quotQuotEquivComm I J).symm = quotQuotEquivComm J I := by
rfl
variable {I J} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_symm | null |
quotQuotEquivQuotOfLE (h : I ≤ J) : (R ⧸ I) ⧸ J.map (Ideal.Quotient.mk I) ≃+* R ⧸ J :=
(quotQuotEquivQuotSup I J).trans (Ideal.quotEquivOfEq <| sup_eq_right.mpr h)
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE | **The Third Isomorphism theorem** for rings. See `quotQuotEquivQuotSup` for a version
that does not assume an inclusion of ideals. |
quotQuotEquivQuotOfLE_quotQuotMk (x : R) (h : I ≤ J) :
quotQuotEquivQuotOfLE h (quotQuotMk I J x) = (Ideal.Quotient.mk J) x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_quotQuotMk | null |
quotQuotEquivQuotOfLE_symm_mk (x : R) (h : I ≤ J) :
(quotQuotEquivQuotOfLE h).symm ((Ideal.Quotient.mk J) x) = quotQuotMk I J x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_symm_mk | null |
quotQuotEquivQuotOfLE_comp_quotQuotMk (h : I ≤ J) :
RingHom.comp (↑(quotQuotEquivQuotOfLE h)) (quotQuotMk I J) = (Ideal.Quotient.mk J) := by
ext
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_comp_quotQuotMk | null |
quotQuotEquivQuotOfLE_symm_comp_mk (h : I ≤ J) :
RingHom.comp (↑(quotQuotEquivQuotOfLE h).symm) (Ideal.Quotient.mk J) = quotQuotMk I J := by
ext
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_symm_comp_mk | null |
@[simp]
quotQuotEquivComm_mk_mk [CommRing R] (I J : Ideal R) (x : R) :
quotQuotEquivComm I J (Ideal.Quotient.mk _ (Ideal.Quotient.mk _ x)) = algebraMap R _ x :=
rfl
variable [CommSemiring R] {A : Type v} [CommRing A] [Algebra R A] (I J : Ideal A)
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_mk_mk | null |
quotQuotEquivQuotSup_quot_quot_algebraMap (x : R) :
DoubleQuot.quotQuotEquivQuotSup I J (algebraMap R _ x) = algebraMap _ _ x :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSup_quot_quot_algebraMap | null |
quotQuotEquivComm_algebraMap (x : R) :
quotQuotEquivComm I J (algebraMap R _ x) = algebraMap _ _ x :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_algebraMap | null |
quotLeftToQuotSupₐ : A ⧸ I →ₐ[R] A ⧸ I ⊔ J :=
AlgHom.mk (quotLeftToQuotSup I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotLeftToQuotSupₐ | The natural algebra homomorphism `A / I → A / (I ⊔ J)`. |
quotLeftToQuotSupₐ_toRingHom :
(quotLeftToQuotSupₐ R I J : _ →+* _) = quotLeftToQuotSup I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotLeftToQuotSupₐ_toRingHom | null |
coe_quotLeftToQuotSupₐ : ⇑(quotLeftToQuotSupₐ R I J) = quotLeftToQuotSup I J :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotLeftToQuotSupₐ | null |
quotQuotToQuotSupₐ : (A ⧸ I) ⧸ J.map (Quotient.mkₐ R I) →ₐ[R] A ⧸ I ⊔ J :=
AlgHom.mk (quotQuotToQuotSup I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotToQuotSupₐ | The algebra homomorphism `(A / I) / J' -> A / (I ⊔ J)` induced by `quotQuotToQuotSup`,
where `J'` is the projection of `J` in `A / I`. |
quotQuotToQuotSupₐ_toRingHom :
((quotQuotToQuotSupₐ R I J) : _ ⧸ map (Ideal.Quotient.mkₐ R I) J →+* _) =
quotQuotToQuotSup I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotToQuotSupₐ_toRingHom | null |
coe_quotQuotToQuotSupₐ : ⇑(quotQuotToQuotSupₐ R I J) = quotQuotToQuotSup I J :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotToQuotSupₐ | null |
quotQuotMkₐ : A →ₐ[R] (A ⧸ I) ⧸ J.map (Quotient.mkₐ R I) :=
AlgHom.mk (quotQuotMk I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotMkₐ | The composition of the algebra homomorphisms `A → (A / I)` and `(A / I) → (A / I) / J'`,
where `J'` is the projection `J` in `A / I`. |
quotQuotMkₐ_toRingHom :
(quotQuotMkₐ R I J : _ →+* _ ⧸ J.map (Quotient.mkₐ R I)) = quotQuotMk I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotMkₐ_toRingHom | null |
coe_quotQuotMkₐ : ⇑(quotQuotMkₐ R I J) = quotQuotMk I J :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotMkₐ | null |
liftSupQuotQuotMkₐ (I J : Ideal A) : A ⧸ I ⊔ J →ₐ[R] (A ⧸ I) ⧸ J.map (Quotient.mkₐ R I) :=
AlgHom.mk (liftSupQuotQuotMk I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | liftSupQuotQuotMkₐ | The injective algebra homomorphism `A / (I ⊔ J) → (A / I) / J'`induced by `quot_quot_mk`,
where `J'` is the projection `J` in `A / I`. |
liftSupQuotQuotMkₐ_toRingHom :
(liftSupQuotQuotMkₐ R I J : _ →+* _ ⧸ J.map (Quotient.mkₐ R I)) = liftSupQuotQuotMk I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | liftSupQuotQuotMkₐ_toRingHom | null |
coe_liftSupQuotQuotMkₐ : ⇑(liftSupQuotQuotMkₐ R I J) = liftSupQuotQuotMk I J :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_liftSupQuotQuotMkₐ | null |
quotQuotEquivQuotSupₐ : ((A ⧸ I) ⧸ J.map (Quotient.mkₐ R I)) ≃ₐ[R] A ⧸ I ⊔ J :=
AlgEquiv.ofRingEquiv (f := quotQuotEquivQuotSup I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSupₐ | `quotQuotToQuotSup` and `liftSupQuotQuotMk` are inverse isomorphisms. In the case where
`I ≤ J`, this is the Third Isomorphism Theorem (see `DoubleQuot.quotQuotEquivQuotOfLE`). |
quotQuotEquivQuotSupₐ_toRingEquiv :
(quotQuotEquivQuotSupₐ R I J : _ ⧸ J.map (Quotient.mkₐ R I) ≃+* _) = quotQuotEquivQuotSup I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSupₐ_toRingEquiv | null |
coe_quotQuotEquivQuotSupₐ : ⇑(quotQuotEquivQuotSupₐ R I J) = quotQuotEquivQuotSup I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotEquivQuotSupₐ | null |
quotQuotEquivQuotSupₐ_symm_toRingEquiv :
((quotQuotEquivQuotSupₐ R I J).symm : _ ≃+* _ ⧸ J.map (Quotient.mkₐ R I)) =
(quotQuotEquivQuotSup I J).symm :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotSupₐ_symm_toRingEquiv | null |
coe_quotQuotEquivQuotSupₐ_symm :
⇑(quotQuotEquivQuotSupₐ R I J).symm = (quotQuotEquivQuotSup I J).symm :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotEquivQuotSupₐ_symm | null |
quotQuotEquivCommₐ :
((A ⧸ I) ⧸ J.map (Quotient.mkₐ R I)) ≃ₐ[R] (A ⧸ J) ⧸ I.map (Quotient.mkₐ R J) :=
AlgEquiv.ofRingEquiv (f := quotQuotEquivComm I J) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivCommₐ | 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`). |
quotQuotEquivCommₐ_toRingEquiv :
(quotQuotEquivCommₐ R I J : _ ⧸ J.map (Quotient.mkₐ R I) ≃+* _ ⧸ I.map (Quotient.mkₐ R J)) =
quotQuotEquivComm I J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivCommₐ_toRingEquiv | null |
coe_quotQuotEquivCommₐ : ⇑(quotQuotEquivCommₐ R I J) = ⇑(quotQuotEquivComm I J) :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotEquivCommₐ | null |
quotQuotEquivComm_symmₐ : (quotQuotEquivCommₐ R I J).symm = quotQuotEquivCommₐ R J I := by
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_symmₐ | null |
quotQuotEquivComm_comp_quotQuotMkₐ :
AlgHom.comp (↑(quotQuotEquivCommₐ R I J)) (quotQuotMkₐ R I J) = quotQuotMkₐ R J I :=
AlgHom.ext <| quotQuotEquivComm_quotQuotMk I J
variable {I J} | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivComm_comp_quotQuotMkₐ | null |
quotQuotEquivQuotOfLEₐ (h : I ≤ J) : ((A ⧸ I) ⧸ J.map (Quotient.mkₐ R I)) ≃ₐ[R] A ⧸ J :=
AlgEquiv.ofRingEquiv (f := quotQuotEquivQuotOfLE h) fun _ => rfl
@[simp] | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLEₐ | The **third isomorphism theorem** for algebras. See `quotQuotEquivQuotSupₐ` for version
that does not assume an inclusion of ideals. |
quotQuotEquivQuotOfLEₐ_toRingEquiv (h : I ≤ J) :
(quotQuotEquivQuotOfLEₐ R h : _ ⧸ J.map (Quotient.mkₐ R I) ≃+* _) = quotQuotEquivQuotOfLE h :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLEₐ_toRingEquiv | null |
coe_quotQuotEquivQuotOfLEₐ (h : I ≤ J) :
⇑(quotQuotEquivQuotOfLEₐ R h) = quotQuotEquivQuotOfLE h :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotEquivQuotOfLEₐ | null |
quotQuotEquivQuotOfLEₐ_symm_toRingEquiv (h : I ≤ J) :
((quotQuotEquivQuotOfLEₐ R h).symm : _ ≃+* _ ⧸ J.map (Quotient.mkₐ R I)) =
(quotQuotEquivQuotOfLE h).symm :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLEₐ_symm_toRingEquiv | null |
coe_quotQuotEquivQuotOfLEₐ_symm (h : I ≤ J) :
⇑(quotQuotEquivQuotOfLEₐ R h).symm = (quotQuotEquivQuotOfLE h).symm :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | coe_quotQuotEquivQuotOfLEₐ_symm | null |
quotQuotEquivQuotOfLE_comp_quotQuotMkₐ (h : I ≤ J) :
AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h)) (quotQuotMkₐ R I J) = Quotient.mkₐ R J :=
rfl
@[simp] | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_comp_quotQuotMkₐ | null |
quotQuotEquivQuotOfLE_symm_comp_mkₐ (h : I ≤ J) :
AlgHom.comp (↑(quotQuotEquivQuotOfLEₐ R h).symm) (Quotient.mkₐ R J) = quotQuotMkₐ R I J :=
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | quotQuotEquivQuotOfLE_symm_comp_mkₐ | null |
noncomputable
powQuotPowSuccLinearEquivMapMkPowSuccPow :
((I ^ n : Ideal R) ⧸ (I • ⊤ : Submodule R (I ^ n : Ideal R))) ≃ₗ[R]
Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n) := by
refine { LinearMap.codRestrict
(Submodule.restrictScalars _ (Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n)))
(Submodule.mapQ (I • ⊤) (I ^ (n + 1)) (Submodule.subtype (I ^ n)) ?_) ?_,
Equiv.ofBijective _ ⟨?_, ?_⟩ with }
· intro
simp [Submodule.mem_smul_top_iff, pow_succ']
· intro x
obtain ⟨⟨y, hy⟩, rfl⟩ := Submodule.Quotient.mk_surjective _ x
simp [Ideal.mem_sup_left hy]
· intro a b
obtain ⟨⟨x, hx⟩, rfl⟩ := Submodule.Quotient.mk_surjective _ a
obtain ⟨⟨y, hy⟩, rfl⟩ := Submodule.Quotient.mk_surjective _ b
simp [Ideal.Quotient.eq, Submodule.Quotient.eq, Submodule.mem_smul_top_iff, pow_succ']
· intro ⟨x, hx⟩
rw [Ideal.mem_map_iff_of_surjective _ Ideal.Quotient.mk_surjective] at hx
obtain ⟨y, hy, rfl⟩ := hx
refine ⟨Submodule.Quotient.mk ⟨y, hy⟩, ?_⟩
simp | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | powQuotPowSuccLinearEquivMapMkPowSuccPow | `I ^ n ⧸ I ^ (n + 1)` can be viewed as a quotient module and as ideal of `R ⧸ I ^ (n + 1)`.
This definition gives the `R`-linear equivalence between the two. |
noncomputable
powQuotPowSuccEquivMapMkPowSuccPow :
((I ^ n : Ideal R) ⧸ (I • ⊤ : Submodule R (I ^ n : Ideal R))) ≃
Ideal.map (Ideal.Quotient.mk (I ^ (n + 1))) (I ^ n) :=
powQuotPowSuccLinearEquivMapMkPowSuccPow I n | def | RingTheory | [
"Mathlib.Algebra.Algebra.Subalgebra.Operations",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Basic"
] | Mathlib/RingTheory/Ideal/Quotient/Operations.lean | powQuotPowSuccEquivMapMkPowSuccPow | `I ^ n ⧸ I ^ (n + 1)` can be viewed as a quotient module and as ideal of `R ⧸ I ^ (n + 1)`.
This definition gives the equivalence between the two, instead of the `R`-linear equivalence,
to bypass typeclass synthesis issues on complex `Module` goals. |
Ideal.Quotient.factor_ker (H : I ≤ J) [I.IsTwoSided] [J.IsTwoSided] :
RingHom.ker (factor H) = J.map (Ideal.Quotient.mk I) := by
ext x
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rcases Ideal.Quotient.mk_surjective x with ⟨r, hr⟩
rw [← hr] at h ⊢
simp only [factor, RingHom.mem_ker, lift_mk, eq_zero_iff_mem] at h
exact Ideal.mem_map_of_mem _ h
· rcases mem_image_of_mem_map_of_surjective _ Ideal.Quotient.mk_surjective h with ⟨r, hr, eq⟩
simpa [← eq, Ideal.Quotient.eq_zero_iff_mem] using hr | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | Ideal.Quotient.factor_ker | null |
Ideal.map_mk_comap_factor [J.IsTwoSided] [K.IsTwoSided] (hIJ : J ≤ I) (hJK : K ≤ J) :
(I.map (mk J)).comap (factor hJK) = I.map (mk K) := by
ext x
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rcases mem_image_of_mem_map_of_surjective (mk J) Quotient.mk_surjective h with ⟨r, hr, eq⟩
have : x - ((mk K) r) ∈ J.map (mk K) := by
simp [← factor_ker hJK, ← eq]
rcases mem_image_of_mem_map_of_surjective (mk K) Quotient.mk_surjective this with ⟨s, hs, eq'⟩
rw [← add_sub_cancel ((mk K) r) x, ← eq', ← map_add]
exact mem_map_of_mem (mk K) (Submodule.add_mem _ hr (hIJ hs))
· rcases mem_image_of_mem_map_of_surjective (mk K) Quotient.mk_surjective h with ⟨r, hr, eq⟩
simpa only [← eq] using mem_map_of_mem (mk J) hr | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | Ideal.map_mk_comap_factor | null |
@[simp]
mapQ_eq_factor (h : I ≤ J) (x : R ⧸ I) :
mapQ I J LinearMap.id h x = factor h x := rfl | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | mapQ_eq_factor | null |
pow_smul_top_le {m n : ℕ} (h : m ≤ n) : (I ^ n • ⊤ : Submodule R M) ≤ I ^ m • ⊤ :=
smul_mono_left (Ideal.pow_le_pow_right h) | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | pow_smul_top_le | null |
factorPow {m n : ℕ} (le : m ≤ n) :
M ⧸ (I ^ n • ⊤ : Submodule R M) →ₗ[R] M ⧸ (I ^ m • ⊤ : Submodule R M) :=
factor (smul_mono_left (Ideal.pow_le_pow_right le)) | abbrev | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | factorPow | The linear map from `M ⧸ I ^ m • ⊤` to `M ⧸ I ^ n • ⊤` induced by
the natural inclusion `I ^ n • ⊤ → I ^ m • ⊤`.
To future contributors: Before adding lemmas related to `Submodule.factorPow`, please
check whether it can be generalized to `Submodule.factor` and whether the
corresponding (more general) lemma for `Submodule.factor` already exists. |
factorPowSucc (m : ℕ) : M ⧸ (I ^ (m + 1) • ⊤ : Submodule R M) →ₗ[R]
M ⧸ (I ^ m • ⊤ : Submodule R M) := factorPow I M (Nat.le_succ m) | abbrev | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | factorPowSucc | `factorPow` for `n = m + 1` |
factorPow {m n : ℕ} (le : n ≤ m) : R ⧸ I ^ m →+* R ⧸ I ^ n :=
factor (pow_le_pow_right le) | abbrev | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | factorPow | The ring homomorphism from `R ⧸ I ^ m`
to `R ⧸ I ^ n` induced by the natural inclusion `I ^ n → I ^ m`.
To future contributors: Before adding lemmas related to `Ideal.factorPow`, please
check whether it can be generalized to `Ideal.factor` and whether the corresponding
(more general) lemma for `Ideal.factor` already exists. |
factorPowSucc (n : ℕ) : R ⧸ I ^ (n + 1) →+* R ⧸ I ^ n :=
factorPow I (Nat.le_succ n) | abbrev | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | factorPowSucc | `factorPow` for `m = n + 1` |
Ideal.map_mk_comap_factorPow {a b : ℕ} (apos : 0 < a) (le : a ≤ b) :
(I.map (mk (I ^ a))).comap (factorPow I le) = I.map (mk (I ^ b)) := by
apply Ideal.map_mk_comap_factor
exact pow_le_self (Nat.ne_zero_of_lt apos)
variable {I} in | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | Ideal.map_mk_comap_factorPow | null |
factorPowSucc.isUnit_of_isUnit_image {n : ℕ} (npos : n > 0) {a : R ⧸ I ^ (n + 1)}
(h : IsUnit (factorPow I n.le_succ a)) : IsUnit a := by
rcases isUnit_iff_exists.mp h with ⟨b, hb, _⟩
rcases factor_surjective (pow_le_pow_right n.le_succ) b with ⟨b', hb'⟩
rw [← hb', ← map_one (factorPow I n.le_succ), ← map_mul] at hb
apply (RingHom.sub_mem_ker_iff (factorPow I n.le_succ)).mpr at hb
rw [factor_ker (pow_le_pow_right n.le_succ)] at hb
rcases Ideal.mem_image_of_mem_map_of_surjective (Ideal.Quotient.mk (I ^ (n + 1)))
Ideal.Quotient.mk_surjective hb with ⟨c, hc, eq⟩
apply isUnit_of_mul_eq_one _ (b' * (1 - ((Ideal.Quotient.mk (I ^ (n + 1))) c)))
calc
_ = (a * b' - 1) * (1 - ((Ideal.Quotient.mk (I ^ (n + 1))) c)) +
(1 - ((Ideal.Quotient.mk (I ^ (n + 1))) c)) := by ring
_ = 1 := by
rw [← eq, mul_sub, mul_one, sub_add_sub_cancel', sub_eq_self, ← map_mul,
Ideal.Quotient.eq_zero_iff_mem, pow_add]
apply Ideal.mul_mem_mul hc (Ideal.mul_le_left (I := I ^ (n - 1)) _)
simpa only [← pow_add, Nat.sub_add_cancel npos] using hc | lemma | RingTheory | [
"Mathlib.LinearAlgebra.Quotient.Basic",
"Mathlib.RingTheory.Ideal.Quotient.Defs",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.RingTheory.Ideal.Maps"
] | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | factorPowSucc.isUnit_of_isUnit_image | null |
Subalgebra.isIntegral_iff (S : Subalgebra R A) :
Algebra.IsIntegral R S ↔ ∀ x ∈ S, IsIntegral R x :=
Algebra.isIntegral_def.trans <| .trans
(forall_congr' fun _ ↦ (isIntegral_algHom_iff S.val Subtype.val_injective).symm) Subtype.forall | theorem | RingTheory | [
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | Subalgebra.isIntegral_iff | null |
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