Loogle!

Result

Found 11 declarations mentioning IsLocalRing.ResidueField.map.

• IsLocalRing.ResidueField.map_id 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} [CommRing R] [IsLocalRing R] : IsLocalRing.ResidueField.map (RingHom.id R) = RingHom.id (IsLocalRing.ResidueField R)
• IsLocalRing.ResidueField.map 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] : IsLocalRing.ResidueField R →+* IsLocalRing.ResidueField S
• IsLocalRing.ResidueField.map_id_apply 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} [CommRing R] [IsLocalRing R] (x : IsLocalRing.ResidueField R) : (IsLocalRing.ResidueField.map (RingHom.id R)) x = x
• IsLocalRing.ResidueField.map_comp_residue 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] : (IsLocalRing.ResidueField.map f).comp (IsLocalRing.residue R) = (IsLocalRing.residue S).comp f
• IsLocalRing.ResidueField.map.congr_simp 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f f✝ : R →+* S) (e_f : f = f✝) [IsLocalHom f] : IsLocalRing.ResidueField.map f = IsLocalRing.ResidueField.map f✝
• IsLocalRing.ResidueField.map_comp 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] (f : T →+* R) (g : R →+* S) [IsLocalHom f] [IsLocalHom g] : IsLocalRing.ResidueField.map (g.comp f) = (IsLocalRing.ResidueField.map g).comp (IsLocalRing.ResidueField.map f)
• IsLocalRing.ResidueField.map_residue 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R →+* S) [IsLocalHom f] (r : R) : (IsLocalRing.ResidueField.map f) ((IsLocalRing.residue R) r) = (IsLocalRing.residue S) (f r)
• IsLocalRing.ResidueField.map_map 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] [CommRing T] [IsLocalRing T] (f : R →+* S) (g : S →+* T) (x : IsLocalRing.ResidueField R) [IsLocalHom f] [IsLocalHom g] : (IsLocalRing.ResidueField.map g) ((IsLocalRing.ResidueField.map f) x) = (IsLocalRing.ResidueField.map (g.comp f)) x
• IsLocalRing.ResidueField.mapEquiv_apply 📋 Mathlib.RingTheory.LocalRing.ResidueField.Basic
  {R : Type u_1} {S : Type u_2} [CommRing R] [IsLocalRing R] [CommRing S] [IsLocalRing S] (f : R ≃+* S) (a : IsLocalRing.ResidueField R) : (IsLocalRing.ResidueField.mapEquiv f) a = (IsLocalRing.ResidueField.map ↑f) a
• AlgebraicGeometry.LocallyRingedSpace.residueFieldMap.eq_1 📋 Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField
  {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x : ↑X.toTopCat) : AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f x = CommRingCat.ofHom (IsLocalRing.ResidueField.map (CommRingCat.Hom.hom (AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x)))
• AlgebraicGeometry.Scheme.Hom.residueFieldMap.eq_1 📋 Mathlib.AlgebraicGeometry.ResidueField
  {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) (x : ↥X) : f.residueFieldMap x = CommRingCat.ofHom (IsLocalRing.ResidueField.map (CommRingCat.Hom.hom (f.stalkMap x)))

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. 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.

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.

This is Loogle revision ee8c038 serving mathlib revision 7a9e177