Loogle!

Result

Found 116 declarations mentioning RingHom.range.

• RingHom.range 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) : Subring S
• RingHom.range_eq_map 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) : f.range = Subring.map f ⊤
• RingHom.fintypeRange 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] [Fintype R] [DecidableEq S] (f : R →+* S) : Fintype ↥f.range
• Subring.map_comap_eq 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) (t : Subring S) : Subring.map f (Subring.comap f t) = t ⊓ f.range
• Subring.range_subtype 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} [NonAssocRing R] (s : Subring R) : s.subtype.range = s
• RingHom.coe_range 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) : ↑f.range = Set.range ⇑f
• RingHom.range_eq_top_of_surjective 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) (hf : Function.Surjective ⇑f) : f.range = ⊤
• Subring.map_comap_eq_self 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {f : R →+* S} {t : Subring S} (h : t ≤ f.range) : Subring.map f (Subring.comap f t) = t
• RingHom.mem_range_self 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) (x : R) : f x ∈ f.range
• RingHom.range_eq_top 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {f : R →+* S} : f.range = ⊤ ↔ Function.Surjective ⇑f
• RingHom.map_range 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (g : S →+* T) (f : R →+* S) : Subring.map g f.range = (g.comp f).range
• RingHom.mem_range 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {f : R →+* S} {y : S} : y ∈ f.range ↔ ∃ x, f x = y
• RingHom.rangeRestrict 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) : R →+* ↥f.range
• RingHom.range_prodMap 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {R' : Type u_1} {S' : Type u_2} [Ring R'] [Ring S'] (f : R →+* S) (g : R' →+* S') : (f.prodMap g).range = f.range.prod g.range
• RingHom.domRestrict_comp_rangeRestrict 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocRing R] [NonAssocRing S] [NonAssocRing T] (g : S →+* T) (f : R →+* S) : (g.domRestrict f.range).comp f.rangeRestrict = g.comp f
• RingHom.rangeRestrict_surjective 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) : Function.Surjective ⇑f.rangeRestrict
• RingHom.coe_rangeRestrict 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] (f : R →+* S) (x : R) : ↑(f.rangeRestrict x) = f x
• RingEquiv.ofLeftInverse 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) : R ≃+* ↥f.range
• RingEquiv.ofLeftInverse_apply 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : R) : ↑((RingEquiv.ofLeftInverse h) x) = f x
• RingEquiv.ofLeftInverse_symm_apply 📋 Mathlib.Algebra.Ring.Subring.Basic
  {R : Type u} {S : Type v} [NonAssocRing R] [NonAssocRing S] {g : S → R} {f : R →+* S} (h : Function.LeftInverse g ⇑f) (x : ↥f.range) : (RingEquiv.ofLeftInverse h).symm x = g ↑x
• RingHom.ker_rangeRestrict 📋 Mathlib.RingTheory.Ideal.Maps
  {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) : RingHom.ker f.rangeRestrict = RingHom.ker f
• isNoetherianRing_range 📋 Mathlib.RingTheory.Noetherian.Basic
  {R : Type u_1} [Ring R] {S : Type u_2} [Ring S] (f : R →+* S) [IsNoetherianRing R] : IsNoetherianRing ↥f.range
• Subalgebra.range_algebraMap 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
  {R : Type u_3} {A : Type u_4} [CommRing R] [CommRing A] [Algebra R A] (S : Subalgebra R A) : (algebraMap (↥S) A).range = S.toSubring
• IsFractionRing.lift_fieldRange 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] {L : Type u_7} [Field L] [Algebra A K] [IsFractionRing A K] {g : A →+* L} (hg : Function.Injective ⇑g) : (IsFractionRing.lift hg).fieldRange = Subfield.closure ↑g.range
• IsFractionRing.lift_fieldRange_eq_of_range_eq 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] {L : Type u_7} [Field L] [Algebra A K] [IsFractionRing A K] {g : A →+* L} (hg : Function.Injective ⇑g) {s : Set L} (hs : g.range = Subring.closure s) : (IsFractionRing.lift hg).fieldRange = Subfield.closure s
• IsFractionRing.ringHom_fieldRange_eq_of_comp_eq 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_8} [Field L] {g : A →+* L} {f : K →+* L} (h : f.comp (algebraMap A K) = g) : f.fieldRange = Subfield.closure ↑g.range
• IsFractionRing.ringHom_fieldRange_eq_of_comp_eq_of_range_eq 📋 Mathlib.RingTheory.Localization.FractionRing
  {A : Type u_4} [CommRing A] {K : Type u_5} [Field K] [Algebra A K] [IsFractionRing A K] {L : Type u_8} [Field L] {g : A →+* L} {f : K →+* L} (h : f.comp (algebraMap A K) = g) {s : Set L} (hs : g.range = Subring.closure s) : f.fieldRange = Subfield.closure s
• RingHom.quotientKerEquivRange 📋 Mathlib.RingTheory.Ideal.Quotient.Operations
  {R : Type u} [Ring R] {S : Type v} [Ring S] (f : R →+* S) : R ⧸ RingHom.ker f ≃+* ↥f.range
• Irreducible.aeval_ne_zero_of_natDegree_ne_one 📋 Mathlib.Algebra.Polynomial.RingDivision
  {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [Ring S] [Algebra R S] [FaithfulSMul R S] {p : Polynomial R} (hp : Irreducible p) (hdeg : p.natDegree ≠ 1) {x : S} (hx : x ∈ (algebraMap R S).range) : (Polynomial.aeval x) p ≠ 0
• Polynomial.filter_roots_map_range_eq_map_roots 📋 Mathlib.Algebra.Polynomial.Roots
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [IsDomain A] [IsDomain B] {f : A →+* B} [DecidablePred fun x => x ∈ f.range] (hf : Function.Injective ⇑f) (p : Polynomial A) : Multiset.filter (fun x => x ∈ f.range) (Polynomial.map f p).roots = Multiset.map (⇑f) p.roots
• RingCon.kerLift_range_eq 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {P : Type u_3} [Ring M] [Ring P] {f : M →+* P} : (RingCon.kerLift f).range = f.range
• RingCon.range_lift 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {P : Type u_3} [Ring M] [Ring P] {c : RingCon M} {f : M →+* P} (H : c ≤ RingCon.ker f) : (c.lift f H).range = f.range
• RingCon.range_mk' 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} [Ring M] {c : RingCon M} : c.mk'.range = ⊤
• RingCon.quotientKerEquivRange 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {P : Type u_3} [Ring M] [Ring P] (f : M →+* P) : (RingCon.ker f).Quotient ≃+* ↥f.range
• RingCon.comapQuotientEquivRange 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {N : Type u_2} [Ring M] [Ring N] (c : RingCon M) (f : N →+* M) {d : RingCon N} (hcd : d = c.comap f) : d.Quotient ≃+* ↥(c.mk'.comp f).range
• RingCon.coe_comapQuotientEquivRange_mk 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {N : Type u_2} [Ring M] [Ring N] (c : RingCon M) (f : N →+* M) {d : RingCon N} (hcd : d = c.comap f) (x : N) : ↑((c.comapQuotientEquivRange f hcd) ↑x) = ↑(f x)
• RingCon.comapQuotientEquivRange_mk 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {N : Type u_2} [Ring M] [Ring N] (c : RingCon M) (f : N →+* M) {d : RingCon N} (hcd : d = c.comap f) (x : N) : (c.comapQuotientEquivRange f hcd) ↑x = ⟨↑(f x), ⋯⟩
• RingCon.comapQuotientEquivRange_symm_mk 📋 Mathlib.RingTheory.Congruence.Hom
  {M : Type u_1} {N : Type u_2} [Ring M] [Ring N] (c : RingCon M) (f : N →+* M) {d : RingCon N} (hcd : d = c.comap f) (x : N) : (c.comapQuotientEquivRange f hcd).symm ⟨↑(f x), ⋯⟩ = ↑x
• Polynomial.mem_map_range 📋 Mathlib.Algebra.Polynomial.Eval.Subring
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {p : Polynomial S} : p ∈ (Polynomial.mapRingHom f).range ↔ ∀ (n : ℕ), p.coeff n ∈ f.range
• Polynomial.notMem_map_range 📋 Mathlib.Algebra.Polynomial.Eval.Subring
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {p : Polynomial S} : p ∉ (Polynomial.mapRingHom f).range ↔ ∃ n, p.coeff n ∉ f.range
• Polynomial.Splits.of_splits_map 📋 Mathlib.Algebra.Polynomial.Splits
  {R : Type u_1} {S : Type u_2} [Field R] [CommRing S] [IsDomain S] {f : Polynomial R} (i : R →+* S) (hf : (Polynomial.map i f).Splits) (hi : ∀ a ∈ (Polynomial.map i f).roots, a ∈ i.range) : f.Splits
• Polynomial.Splits.mem_range_of_isRoot 📋 Mathlib.Algebra.Polynomial.Splits
  {R : Type u_1} {S : Type u_2} [Field R] [CommRing S] [IsDomain S] {f : Polynomial R} (hf : f.Splits) (hf0 : f ≠ 0) {i : R →+* S} {x : S} (hx : (Polynomial.map i f).IsRoot x) : x ∈ i.range
• Polynomial.Splits.of_splits_map_of_injective 📋 Mathlib.Algebra.Polynomial.Splits
  {R : Type u_1} [CommRing R] {f : Polynomial R} {S : Type u_4} [CommRing S] [IsDomain S] {i : R →+* S} (hi : Function.Injective ⇑i) (hf : (Polynomial.map i f).Splits) : (∀ a ∈ (Polynomial.map i f).roots, a ∈ i.range) → f.Splits
• Polynomial.Splits.mem_lift_of_roots_mem_range 📋 Mathlib.Algebra.Polynomial.Splits
  {R : Type u_1} [CommRing R] {f : Polynomial R} [IsDomain R] (hf : f.Splits) (hm : f.Monic) {S : Type u_4} [Ring S] (i : S →+* R) (hr : ∀ a ∈ f.roots, a ∈ i.range) : f ∈ Polynomial.lifts i
• IsLocalization.scaleRoots_commonDenom_mem_lifts 📋 Mathlib.RingTheory.Localization.Integral
  {R : Type u_1} [CommRing R] (M : Submonoid R) {Rₘ : Type u_3} [CommRing Rₘ] [Algebra R Rₘ] [IsLocalization M Rₘ] (p : Polynomial Rₘ) (hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range) : p.scaleRoots ((algebraMap R Rₘ) ↑(IsLocalization.commonDenom M p.support p.coeff)) ∈ Polynomial.lifts (algebraMap R Rₘ)
• exists_isIdempotentElem_eq_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e : S) (he : e ∈ f.range) (he' : IsIdempotentElem e) : ∃ e', IsIdempotentElem e' ∧ f e' = e
• OrthogonalIdempotents.lift_of_isNilpotent_ker 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {I : Type u_3} [Finite I] (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {e : I → S} (he : OrthogonalIdempotents e) (he' : ∀ (i : I), e i ∈ f.range) : ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) {I : Type u_3} [Fintype I] (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {e : I → S} (he : CompleteOrthogonalIdempotents e) (he' : ∀ (i : I), e i ∈ f.range) : ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• OrthogonalIdempotents.lift_of_isNilpotent_ker_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {n : ℕ} {e : Fin n → S} (he : OrthogonalIdempotents e) (he' : ∀ (i : Fin n), e i ∈ f.range) : ∃ e', OrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.lift_of_isNilpotent_ker_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) {n : ℕ} {e : Fin n → S} (he : CompleteOrthogonalIdempotents e) (he' : ∀ (i : Fin n), e i ∈ f.range) : ∃ e', CompleteOrthogonalIdempotents e' ∧ ⇑f ∘ e' = e
• CompleteOrthogonalIdempotents.exists_eq_comp_of_ker_eq_span 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] {I : Type u_3} [Fintype I] (f : R →+* S) (e₀ : R) (he₀ : IsIdempotentElem e₀) (hfe₀ : RingHom.ker f = Ideal.span {e₀}) (e : I → S) (he : CompleteOrthogonalIdempotents e) (hef : ∀ (i : I), e i ∈ f.range) : ∃ e', (CompleteOrthogonalIdempotents fun t => Option.rec e₀ e' t) ∧ e = ⇑f ∘ e'
• existsUnique_isIdempotentElem_eq_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e : S) (he : e ∈ f.range) (he' : IsIdempotentElem e) : ∃! e', IsIdempotentElem e' ∧ f e' = e
• exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotent_aux 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e₁ : S) (he : e₁ ∈ f.range) (he₁ : IsIdempotentElem e₁) (e₂ : R) (he₂ : IsIdempotentElem e₂) (he₁e₂ : e₁ * f e₂ = 0) : ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0
• exists_isIdempotentElem_mul_eq_zero_of_ker_isNilpotent 📋 Mathlib.RingTheory.Idempotents
  {R : Type u_1} {S : Type u_2} [Ring R] [Ring S] (f : R →+* S) (h : ∀ x ∈ RingHom.ker f, IsNilpotent x) (e₁ : S) (he : e₁ ∈ f.range) (he₁ : IsIdempotentElem e₁) (e₂ : R) (he₂ : IsIdempotentElem e₂) (he₁e₂ : e₁ * f e₂ = 0) (he₂e₁ : f e₂ * e₁ = 0) : ∃ e', IsIdempotentElem e' ∧ f e' = e₁ ∧ e' * e₂ = 0 ∧ e₂ * e' = 0
• isArtinianRing_range 📋 Mathlib.RingTheory.Artinian.Module
  {R : Type u_1} [Ring R] {S : Type u_2} [Ring S] (f : R →+* S) [IsArtinianRing R] : IsArtinianRing ↥f.range
• minpoly.natDegree_eq_one_iff 📋 Mathlib.FieldTheory.Minpoly.Basic
  {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] : (minpoly A x).natDegree = 1 ↔ x ∈ (algebraMap A B).range
• minpoly.mem_range_of_degree_eq_one 📋 Mathlib.FieldTheory.Minpoly.Basic
  (A : Type u_1) {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] (x : B) (hx : (minpoly A x).degree = 1) : x ∈ (algebraMap A B).range
• minpoly.degree_eq_one_iff 📋 Mathlib.FieldTheory.Minpoly.Basic
  {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] : (minpoly A x).degree = 1 ↔ x ∈ (algebraMap A B).range
• minpoly.two_le_natDegree_iff 📋 Mathlib.FieldTheory.Minpoly.Basic
  {A : Type u_1} {B : Type u_2} [CommRing A] [Ring B] [Algebra A B] {x : B} [Nontrivial B] (int : IsIntegral A x) : 2 ≤ (minpoly A x).natDegree ↔ x ∉ (algebraMap A B).range
• Ideal.eq_zero_of_polynomial_mem_map_range 📋 Mathlib.RingTheory.Polynomial.Quotient
  {R : Type u_1} [CommRing R] (I : Ideal (Polynomial R)) (x : ↥((Ideal.Quotient.mk I).comp Polynomial.C).range) (hx : Polynomial.C x ∈ Ideal.map (Polynomial.mapRingHom ((Ideal.Quotient.mk I).comp Polynomial.C).rangeRestrict) I) : x = 0
• IsIntegral.mem_range_algebraMap_of_minpoly_splits 📋 Mathlib.RingTheory.Adjoin.Field
  {R : Type u_1} {K : Type u_2} {L : Type u_3} [CommRing R] [Field K] [Field L] [Algebra R K] {x : L} [Algebra R L] [Algebra K L] [IsScalarTower R K L] (int : IsIntegral R x) (h : (Polynomial.map (algebraMap R K) (minpoly R x)).Splits) : x ∈ (algebraMap K L).range
• LinearMap.finrank_eq_of_isPerfPair 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [Field K] [Field L] [Algebra K L] [AddCommGroup M] [AddCommGroup N] [Module L M] [Module L N] [Module K M] [Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) [p.IsPerfPair] (M' : Submodule K M) (N' : Submodule K N) (hM : Submodule.span L ↑M' = ⊤) (hN : Submodule.span L ↑N' = ⊤) (hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range) : Module.finrank K ↥M' = Module.finrank L M
• LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMap 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [Field K] [Field L] [Algebra K L] [AddCommGroup M] [AddCommGroup N] [Module L M] [Module L N] [Module K M] [Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) [p.IsPerfPair] (M' : Submodule K M) (N' : Submodule K N) (hM : Submodule.span L ↑M' = ⊤) (hN : Submodule.span L ↑N' = ⊤) (hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range) : ∃ n b b', ∀ (i : Fin n), b i = ↑(b' i)
• LinearMap.IsPerfPair.restrictScalars_of_field 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [Field K] [Field L] [Algebra K L] [AddCommGroup M] [AddCommGroup N] [Module L M] [Module L N] [Module K M] [Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) [p.IsPerfPair] {M' : Type u_5} {N' : Type u_6} [AddCommGroup M'] [AddCommGroup N'] [Module K M'] [Module K N'] [IsScalarTower K L N] (i : M' →ₗ[K] M) (j : N' →ₗ[K] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hij : p.IsPerfectCompl (Submodule.span L ↑i.range) (Submodule.span L ↑j.range)) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap K L).range) : (i.restrictScalarsRange₂ j (Algebra.linearMap K L) ⋯ p hp).IsPerfPair
• LinearMap.restrictScalarsField_apply_apply 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {K : Type u_1} {L : Type u_2} {M : Type u_3} {N : Type u_4} [Field K] [Field L] [Algebra K L] [AddCommGroup M] [AddCommGroup N] [Module L M] [Module L N] [Module K M] [Module K N] [IsScalarTower K L M] (p : M →ₗ[L] N →ₗ[L] L) {M' : Type u_5} {N' : Type u_6} [AddCommGroup M'] [AddCommGroup N'] [Module K M'] [Module K N'] [IsScalarTower K L N] (i : M' →ₗ[K] M) (j : N' →ₗ[K] N) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap K L).range) (x : M') (y : N') : (algebraMap K L) (((i.restrictScalarsRange₂ j (Algebra.linearMap K L) ⋯ p hp) x) y) = (p (i x)) (j y)
• LinearMap.IsPerfPair.restrictScalars 📋 Mathlib.LinearAlgebra.PerfectPairing.Restrict
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (p : M →ₗ[R] N →ₗ[R] R) [p.IsPerfPair] {S : Type u_4} {M' : Type u_5} {N' : Type u_6} [CommRing S] [IsDomain S] [Algebra S R] [Module S M] [Module S N] [IsScalarTower S R M] [IsScalarTower S R N] [Module.IsTorsionFree S R] [Nontrivial R] [AddCommGroup M'] [Module S M'] [AddCommGroup N'] [Module S N'] (i : M' →ₗ[S] M) (j : N' →ₗ[S] N) (hi : Function.Injective ⇑i) (hj : Function.Injective ⇑j) (hM : Submodule.span R ↑i.range = ⊤) (hN : Submodule.span R ↑j.range = ⊤) (h₁ : ∀ (g : Module.Dual S N'), ∃ m, ↑S (p.toPerfPair (i m)) ∘ₗ j = Algebra.linearMap S R ∘ₗ g) (h₂ : ∀ (g : Module.Dual S M'), ∃ n, ↑S (p.flip.toPerfPair (j n)) ∘ₗ i = Algebra.linearMap S R ∘ₗ g) (hp : ∀ (m : M') (n : N'), (p (i m)) (j n) ∈ (algebraMap S R).range) : (i.restrictScalarsRange₂ j (Algebra.linearMap S R) ⋯ p hp).IsPerfPair
• ValuationRing.range_algebraMap_eq 📋 Mathlib.RingTheory.Valuation.ValuationRing
  (A : Type u) [CommRing A] (K : Type v) [Field K] [Algebra A K] [IsDomain A] [ValuationRing A] [IsFractionRing A K] : (ValuationRing.valuation A K).integer = (algebraMap A K).range
• RootPairing.toLinearMap_apply_apply_mem_range_algebraMap 📋 Mathlib.LinearAlgebra.RootSystem.IsValuedIn
  {ι : Type u_1} {R : Type u_2} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (P : RootPairing ι R M N) (S : Type u_6) [CommRing S] [Algebra S R] [P.IsValuedIn S] [Module S M] [Module S N] [IsScalarTower S R M] [IsScalarTower S R N] (x : M) (hx : x ∈ P.rootSpan S) (y : N) (hy : y ∈ P.corootSpan S) : (P.toLinearMap x) y ∈ (algebraMap S R).range
• minpoly.natSepDegree_eq_one_iff_pow_mem 📋 Mathlib.FieldTheory.SeparableDegree
  {F : Type u} {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E] (q : ℕ) [hF : ExpChar F q] {x : E} : (minpoly F x).natSepDegree = 1 ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• Polynomial.Monic.eq_X_pow_char_pow_sub_C_of_natSepDegree_eq_one_of_irreducible 📋 Mathlib.FieldTheory.SeparableDegree
  {F : Type u} [Field F] {f : Polynomial F} (q : ℕ) [ExpChar F q] (hm : f.Monic) (hi : Irreducible f) (h : f.natSepDegree = 1) : ∃ n y, (n = 0 ∨ y ∉ (frobenius F q).range) ∧ f = Polynomial.X ^ q ^ n - Polynomial.C y
• Polynomial.Monic.eq_X_pow_char_pow_sub_C_pow_of_natSepDegree_eq_one 📋 Mathlib.FieldTheory.SeparableDegree
  {F : Type u} [Field F] {f : Polynomial F} (q : ℕ) [ExpChar F q] (hm : f.Monic) (h : f.natSepDegree = 1) : ∃ m n y, m ≠ 0 ∧ (n = 0 ∨ y ∉ (frobenius F q).range) ∧ f = (Polynomial.X ^ q ^ n - Polynomial.C y) ^ m
• IsPurelyInseparable.inseparable 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  (F : Type u_1) {E : Type u_2} [CommRing F] [Ring E] [Algebra F E] [IsPurelyInseparable F E] (x : E) : IsSeparable F x → x ∈ (algebraMap F E).range
• IsPurelyInseparable.inseparable' 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  {F : Type u_1} {E : Type u_2} {inst✝ : CommRing F} {inst✝¹ : Ring E} {inst✝² : Algebra F E} [self : IsPurelyInseparable F E] (x : E) : IsSeparable F x → x ∈ (algebraMap F E).range
• IsPurelyInseparable.mk 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  {F : Type u_1} {E : Type u_2} [CommRing F] [Ring E] [Algebra F E] (isIntegral : Algebra.IsIntegral F E) (inseparable' : ∀ (x : E), IsSeparable F x → x ∈ (algebraMap F E).range) : IsPurelyInseparable F E
• isPurelyInseparable_iff 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  {F : Type u_1} {E : Type u_2} [CommRing F] [Ring E] [Algebra F E] : IsPurelyInseparable F E ↔ ∀ (x : E), IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range)
• IsPurelyInseparable.pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  (F : Type u) {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E] (q : ℕ) [ExpChar F q] (x : E) [IsPurelyInseparable F E] : ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• isPurelyInseparable_iff_pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  (F : Type u) {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E] (q : ℕ) [ExpChar F q] : IsPurelyInseparable F E ↔ ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• IsPurelyInseparable.exists_pow_mem_range_tensorProduct 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  {k : Type u_1} {K : Type u_2} {R : Type u_3} [Field k] [Field K] [Algebra k K] [CommRing R] [Algebra k R] [IsPurelyInseparable k K] (x : TensorProduct k R K) : ∃ n > 0, x ^ n ∈ (algebraMap R (TensorProduct k R K)).range
• IsPurelyInseparable.exists_pow_pow_mem_range_tensorProduct_of_expChar 📋 Mathlib.FieldTheory.PurelyInseparable.Basic
  {k : Type u_1} {K : Type u_2} {R : Type u_3} [Field k] [Field K] [Algebra k K] [CommRing R] [Algebra k R] [IsPurelyInseparable k K] (q : ℕ) [ExpChar k q] (x : TensorProduct k R K) : ∃ n, x ^ q ^ n ∈ (algebraMap R (TensorProduct k R K)).range
• Algebra.FormallyUnramified.range_eq_top_of_isPurelyInseparable 📋 Mathlib.RingTheory.Unramified.Field
  (K : Type u_1) (L : Type u_3) [Field K] [Field L] [Algebra K L] [Algebra.FormallyUnramified K L] [Algebra.EssFiniteType K L] [IsPurelyInseparable K L] : (algebraMap K L).range = ⊤
• mem_range_of_deriv_eq_zero 📋 Mathlib.RingTheory.Derivation.DifferentialRing
  (A : Type u_1) {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [Differential B] [Differential.ContainConstants A B] {x : B} (h : x′ = 0) : x ∈ (algebraMap A B).range
• Differential.ContainConstants.mem_range_of_deriv_eq_zero 📋 Mathlib.RingTheory.Derivation.DifferentialRing
  {A : Type u_1} {B : Type u_2} {inst✝ : CommRing A} {inst✝¹ : CommRing B} {inst✝² : Algebra A B} {inst✝³ : Differential B} [self : Differential.ContainConstants A B] {x : B} (h : x′ = 0) : x ∈ (algebraMap A B).range
• Differential.ContainConstants.mk 📋 Mathlib.RingTheory.Derivation.DifferentialRing
  {A : Type u_1} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] [Differential B] (mem_range_of_deriv_eq_zero : ∀ {x : B}, x′ = 0 → x ∈ (algebraMap A B).range) : Differential.ContainConstants A B
• AlgebraicGeometry.exists_lift_of_germInjective_aux 📋 Mathlib.AlgebraicGeometry.SpreadingOut
  {X : AlgebraicGeometry.Scheme} {R A : CommRingCat} {U : X.Opens} {x : ↥X} (hxU : x ∈ U) (φ : A ⟶ X.presheaf.stalk x) (φRA : R ⟶ A) (φRX : R ⟶ X.presheaf.obj (Opposite.op U)) (hφRA : (CommRingCat.Hom.hom φRA).FiniteType) (e : CategoryTheory.CategoryStruct.comp φRA φ = CategoryTheory.CategoryStruct.comp φRX (X.presheaf.germ U x hxU)) : ∃ V, ∃ (hxV : x ∈ V), V ≤ U ∧ (CommRingCat.Hom.hom φ).range ≤ (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).range
• NumberField.mem_span_integralBasis 📋 Mathlib.NumberTheory.NumberField.Basic
  (K : Type u_1) [Field K] [NumberField K] {x : K} : x ∈ Submodule.span ℤ (Set.range ⇑(NumberField.integralBasis K)) ↔ x ∈ (algebraMap (NumberField.RingOfIntegers K) K).range
• IsDedekindDomain.HeightOneSpectrum.mem_integers_of_valuation_le_one 📋 Mathlib.RingTheory.DedekindDomain.AdicValuation
  {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (x : K) (h : ∀ (v : IsDedekindDomain.HeightOneSpectrum R), (IsDedekindDomain.HeightOneSpectrum.valuation K v) x ≤ 1) : x ∈ (algebraMap R K).range
• Algebra.SubmersivePresentation.HasCoeffs.coeffs_subset_range 📋 Mathlib.RingTheory.Extension.Presentation.Core
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} {inst✝ : CommRing R} {inst✝¹ : CommRing S} {inst✝² : Algebra R S} {inst✝³ : Finite σ} {P : Algebra.SubmersivePresentation R S ι σ} {R₀ : Type u_5} {inst✝⁴ : CommRing R₀} {inst✝⁵ : Algebra R₀ R} {inst✝⁶ : Algebra R₀ S} {inst✝⁷ : IsScalarTower R₀ R S} [self : P.HasCoeffs R₀] : P.coeffs ⊆ ↑(algebraMap R₀ R).range
• Algebra.SubmersivePresentation.HasCoeffs.mk 📋 Mathlib.RingTheory.Extension.Presentation.Core
  {R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] [Finite σ] {P : Algebra.SubmersivePresentation R S ι σ} {R₀ : Type u_5} [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] (coeffs_subset_range : P.coeffs ⊆ ↑(algebraMap R₀ R).range) : P.HasCoeffs R₀
• IsIntegral.isAlmostIntegral_of_exists_smul_mem_range 📋 Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral
  {R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] {s : S} (H : IsIntegral R s) (h : ∃ t ∈ nonZeroDivisors R, t • s ∈ (algebraMap R S).range) : IsAlmostIntegral R s
• bijective_rangeRestrict_comp_of_valuationRing 📋 Mathlib.RingTheory.Valuation.LocalSubring
  {R : Type u_1} {S : Type u_2} {K : Type u_3} [CommRing R] [CommRing S] [Field K] [IsDomain R] [ValuationRing R] [IsLocalRing S] [Algebra R K] [IsFractionRing R K] (f : R →+* S) (g : S →+* K) (h : g.comp f = algebraMap R K) [IsLocalHom f] : Function.Bijective ⇑(g.rangeRestrict.comp f)
• AlgebraicGeometry.Proj.valuativeCriterion_existence_aux 📋 Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper
  {σ : Type u_1} {A : Type u_2} [CommRing A] [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] {O : Type u_3} [CommRing O] [IsDomain O] [ValuationRing O] {K : Type u_4} [Field K] [Algebra O K] [IsFractionRing O K] (φ₀ : ↥(𝒜 0) →+* O) (ι : Type u_5) [Finite ι] (x : ι → A) (h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤) (j : ι) (φ : HomogeneousLocalization.Away 𝒜 (x j) →+* K) (hcomm : (algebraMap O K).comp φ₀ = φ.comp (HomogeneousLocalization.fromZeroRingHom 𝒜 (Submonoid.powers (x j)))) (d : ι → ℕ) (hdi : ∀ (i : ι), 0 < d i) (hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)) : ∃ j₀ φ', φ'.comp (HomogeneousLocalization.awayMap 𝒜 ⋯ ⋯) = φ ∧ (φ'.comp (HomogeneousLocalization.awayMap 𝒜 ⋯ ⋯)).range ≤ (algebraMap O K).range
• instIsGaloisGroupSubtypeMemSubringRangeAlgebraMapOfIsScalarTower 📋 Mathlib.RingTheory.IsGaloisGroup.Basic
  (G : Type u_1) (A : Type u_2) (B : Type u_3) [Group G] [CommRing A] [CommRing B] [Algebra A B] (C : Type u_6) [CommRing C] [Algebra A C] [Algebra B C] [IsScalarTower A B C] [MulSemiringAction G C] [IsGaloisGroup G A C] : IsGaloisGroup G (↥(algebraMap A B).range) C
• IsPurelyInseparable.HasExponent.has_exponent 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} {inst✝ : CommRing K} {inst✝¹ : Ring L} {inst✝² : Algebra K L} [self : IsPurelyInseparable.HasExponent K L] : ∃ e, ∀ (a : L), a ^ ringExpChar K ^ e ∈ (algebraMap K L).range
• IsPurelyInseparable.HasExponent.mk 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [CommRing K] [Ring L] [Algebra K L] (has_exponent : ∃ e, ∀ (a : L), a ^ ringExpChar K ^ e ∈ (algebraMap K L).range) : IsPurelyInseparable.HasExponent K L
• IsPurelyInseparable.hasExponent_iff 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) (L : Type u_3) [CommRing K] [Ring L] [Algebra K L] : IsPurelyInseparable.HasExponent K L ↔ ∃ e, ∀ (a : L), a ^ ringExpChar K ^ e ∈ (algebraMap K L).range
• IsPurelyInseparable.exponent_def 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) {L : Type u_3} [CommRing K] [Ring L] [Algebra K L] [IsPurelyInseparable.HasExponent K L] (a : L) : a ^ ringExpChar K ^ IsPurelyInseparable.exponent K L ∈ (algebraMap K L).range
• IsPurelyInseparable.hasExponent_iff' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) (L : Type u_3) [CommRing K] [Ring L] [Algebra K L] (p : ℕ) [ExpChar K p] : IsPurelyInseparable.HasExponent K L ↔ ∃ e, ∀ (a : L), a ^ p ^ e ∈ (algebraMap K L).range
• IsPurelyInseparable.exponent_def' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) {L : Type u_3} [CommRing K] [Ring L] [Algebra K L] [IsPurelyInseparable.HasExponent K L] (p : ℕ) [ExpChar K p] (a : L) : a ^ p ^ IsPurelyInseparable.exponent K L ∈ (algebraMap K L).range
• IsPurelyInseparable.elemExponent_eq_zero_of_mem_range 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] {a : L} (h : a ∈ (algebraMap K L).range) : IsPurelyInseparable.elemExponent K a = 0
• IsPurelyInseparable.exponent_min 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [CommRing K] [Ring L] [Algebra K L] [IsPurelyInseparable.HasExponent K L] {e : ℕ} (h : e < IsPurelyInseparable.exponent K L) : ∃ a, a ^ ringExpChar K ^ e ∉ (algebraMap K L).range
• IsPurelyInseparable.exponent_min' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [CommRing K] [Ring L] [Algebra K L] [IsPurelyInseparable.HasExponent K L] (p : ℕ) [ExpChar K p] {e : ℕ} (h : e < IsPurelyInseparable.exponent K L) : ∃ a, a ^ p ^ e ∉ (algebraMap K L).range
• IsPurelyInseparable.elemExponent_def 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] (a : L) : a ^ ringExpChar K ^ IsPurelyInseparable.elemExponent K a ∈ (algebraMap K L).range
• IsPurelyInseparable.elemExponent_def' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] (p : ℕ) [ExpChar K p] (a : L) : a ^ p ^ IsPurelyInseparable.elemExponent K a ∈ (algebraMap K L).range
• IsPurelyInseparable.elemExponent_le_of_pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] {a : L} {n : ℕ} (h : a ^ ringExpChar K ^ n ∈ (algebraMap K L).range) : IsPurelyInseparable.elemExponent K a ≤ n
• IsPurelyInseparable.elemExponent_min 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] {a : L} {n : ℕ} (h : n < IsPurelyInseparable.elemExponent K a) : a ^ ringExpChar K ^ n ∉ (algebraMap K L).range
• IsPurelyInseparable.elemExponent_le_of_pow_mem' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  {K : Type u_2} {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] (p : ℕ) [ExpChar K p] {a : L} {n : ℕ} (h : a ^ p ^ n ∈ (algebraMap K L).range) : IsPurelyInseparable.elemExponent K a ≤ n
• IsPurelyInseparable.elemExponent_min' 📋 Mathlib.FieldTheory.PurelyInseparable.Exponent
  (K : Type u_2) {L : Type u_3} [Field K] [Field L] [Algebra K L] [IsPurelyInseparable K L] (p : ℕ) [ExpChar K p] {a : L} {n : ℕ} (h : n < IsPurelyInseparable.elemExponent K a) : a ^ p ^ n ∉ (algebraMap K L).range
• mem_perfectClosure_iff 📋 Mathlib.FieldTheory.PurelyInseparable.PerfectClosure
  {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] {x : E} : x ∈ perfectClosure F E ↔ ∃ n, x ^ ringExpChar F ^ n ∈ (algebraMap F E).range
• mem_perfectClosure_iff_pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.PerfectClosure
  {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (q : ℕ) [ExpChar F q] {x : E} : x ∈ perfectClosure F E ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• IntermediateField.isPurelyInseparable_adjoin_iff_pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.PerfectClosure
  (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q] {S : Set E} : IsPurelyInseparable F ↥(IntermediateField.adjoin F S) ↔ ∀ x ∈ S, ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• IntermediateField.isPurelyInseparable_adjoin_simple_iff_pow_mem 📋 Mathlib.FieldTheory.PurelyInseparable.PerfectClosure
  (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] (q : ℕ) [hF : ExpChar F q] {x : E} : IsPurelyInseparable F ↥F⟮x⟯ ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).range
• RatFunc.Luroth.generator_spec 📋 Mathlib.FieldTheory.RatFunc.Luroth
  {K : Type u_1} [Field K] {E : IntermediateField K (RatFunc K)} (h : E ≠ ⊥) : RatFunc.Luroth.generator E ∉ (algebraMap K (RatFunc K)).range
• NumberField.canonicalEmbedding.mem_span_latticeBasis 📋 Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
  (K : Type u_1) [Field K] [NumberField K] {x : (K →+* ℂ) → ℂ} : x ∈ Submodule.span ℤ (Set.range ⇑(NumberField.canonicalEmbedding.latticeBasis K)) ↔ x ∈ ((NumberField.canonicalEmbedding K).comp (algebraMap (NumberField.RingOfIntegers K) K)).range
• Submodule.mem_traceDual 📋 Mathlib.RingTheory.DedekindDomain.Different
  {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] {I : Submodule B L} {x : L} : x ∈ Submodule.traceDual A K I ↔ ∀ a ∈ I, ((Algebra.traceForm K L) x) a ∈ (algebraMap A K).range
• FractionalIdeal.mem_dual 📋 Mathlib.RingTheory.DedekindDomain.Different
  {A : Type u_1} {K : Type u_2} {L : Type u} {B : Type u_3} [CommRing A] [Field K] [CommRing B] [Field L] [Algebra A K] [Algebra B L] [Algebra A B] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsScalarTower A B L] [IsDomain A] [IsFractionRing A K] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsIntegralClosure B A L] [IsFractionRing B L] [IsIntegrallyClosed A] [IsDedekindDomain B] {I : FractionalIdeal (nonZeroDivisors B) L} (hI : I ≠ 0) {x : L} : x ∈ FractionalIdeal.dual A K I ↔ ∀ a ∈ I, ((Algebra.traceForm K L) x) a ∈ (algebraMap A K).range
• PrimeSpectrum.isHomeomorph_comap 📋 Mathlib.RingTheory.Spectrum.Prime.Homeomorph
  {R : Type u_3} {S : Type u_4} [CommRing R] [CommRing S] (f : R →+* S) (H : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range) (hker : RingHom.ker f ≤ nilradical R) : IsHomeomorph (PrimeSpectrum.comap f)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59