Loogle!

Result

Found 23 declarations mentioning LocalizedModule.map.

• LocalizedModule.map 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] : (M →ₗ[R] N) →ₗ[R] LocalizedModule S M →ₗ[Localization S] LocalizedModule S N
• LocalizedModule.map_id 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] : (LocalizedModule.map S) LinearMap.id = LinearMap.id
• LocalizedModule.map_injective 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (l : M →ₗ[R] N) (hl : Function.Injective ⇑l) : Function.Injective ⇑((LocalizedModule.map S) l)
• LocalizedModule.map_surjective 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (l : M →ₗ[R] N) (hl : Function.Surjective ⇑l) : Function.Surjective ⇑((LocalizedModule.map S) l)
• LocalizedModule.map_mk 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (x : M) (y : ↥S) : ((LocalizedModule.map S) f) (LocalizedModule.mk x y) = LocalizedModule.mk (f x) y
• IsLocalizedModule.map_bijective_iff_localizedModuleMap_bijective 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} {M : Type u_2} {N : Type u_3} {M' : Type u_4} {N' : Type u_5} [CommSemiring R] {S : Submonoid R} [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N'] (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [IsLocalizedModule S g₁] [IsLocalizedModule S g₂] {l : M →ₗ[R] N} : Function.Bijective ⇑((IsLocalizedModule.map S g₁ g₂) l) ↔ Function.Bijective ⇑((LocalizedModule.map S) l)
• IsLocalizedModule.map_injective_iff_localizedModuleMap_injective 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} {M : Type u_2} {N : Type u_3} {M' : Type u_4} {N' : Type u_5} [CommSemiring R] {S : Submonoid R} [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N'] (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [IsLocalizedModule S g₁] [IsLocalizedModule S g₂] {l : M →ₗ[R] N} : Function.Injective ⇑((IsLocalizedModule.map S g₁ g₂) l) ↔ Function.Injective ⇑((LocalizedModule.map S) l)
• IsLocalizedModule.map_surjective_iff_localizedModuleMap_surjective 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} {M : Type u_2} {N : Type u_3} {M' : Type u_4} {N' : Type u_5} [CommSemiring R] {S : Submonoid R} [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid M'] [Module R M'] [AddCommMonoid N'] [Module R N'] (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [IsLocalizedModule S g₁] [IsLocalizedModule S g₂] {l : M →ₗ[R] N} : Function.Surjective ⇑((IsLocalizedModule.map S g₁ g₂) l) ↔ Function.Surjective ⇑((LocalizedModule.map S) l)
• LocalizedModule.coe_map_eq 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] {S : Submonoid R} {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_5} [AddCommMonoid N] [Module R N] {M' : Type u_3} {N' : Type u_4} [AddCommMonoid M'] [AddCommMonoid N'] [Module R M'] [Module R N'] (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [IsLocalizedModule S g₁] [IsLocalizedModule S g₂] (l : M →ₗ[R] N) : ⇑((LocalizedModule.map S) l) = ⇑(IsLocalizedModule.iso S g₂).symm ∘ ⇑((IsLocalizedModule.map S g₁ g₂) l) ∘ ⇑(IsLocalizedModule.iso S g₁)
• LocalizedModule.restrictScalars_map_eq 📋 Mathlib.RingTheory.Localization.Module
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_5} [AddCommMonoid N] [Module R N] {M' : Type u_3} {N' : Type u_4} [AddCommMonoid M'] [AddCommMonoid N'] [Module R M'] [Module R N'] (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [IsLocalizedModule S g₁] [IsLocalizedModule S g₂] (l : M →ₗ[R] N) : ↑R ((LocalizedModule.map S) l) = ↑(IsLocalizedModule.iso S g₂).symm ∘ₗ (IsLocalizedModule.map S g₁ g₂) l ∘ₗ ↑(IsLocalizedModule.iso S g₁)
• bijective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Bijective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Bijective ⇑f
• injective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Injective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Injective ⇑f
• surjective_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Surjective ⇑((LocalizedModule.map J.primeCompl) f)) : Function.Surjective ⇑f
• bijective_of_localized_span 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (s : Set R) (spn : Ideal.span s = ⊤) (f : M →ₗ[R] N) (h : ∀ (r : ↑s), Function.Bijective ⇑((LocalizedModule.map (Submonoid.powers ↑r)) f)) : Function.Bijective ⇑f
• injective_of_localized_span 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (s : Set R) (spn : Ideal.span s = ⊤) (f : M →ₗ[R] N) (h : ∀ (r : ↑s), Function.Injective ⇑((LocalizedModule.map (Submonoid.powers ↑r)) f)) : Function.Injective ⇑f
• surjective_of_localized_span 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (s : Set R) (spn : Ideal.span s = ⊤) (f : M →ₗ[R] N) (h : ∀ (r : ↑s), Function.Surjective ⇑((LocalizedModule.map (Submonoid.powers ↑r)) f)) : Function.Surjective ⇑f
• exact_of_localized_maximal 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} {L : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid L] [Module R L] (f : M →ₗ[R] N) (g : N →ₗ[R] L) (h : ∀ (J : Ideal R) [inst : J.IsMaximal], Function.Exact ⇑((LocalizedModule.map J.primeCompl) f) ⇑((LocalizedModule.map J.primeCompl) g)) : Function.Exact ⇑f ⇑g
• exact_of_localized_span 📋 Mathlib.RingTheory.LocalProperties.Exactness
  {R : Type u_1} {M : Type u_2} {N : Type u_3} {L : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid L] [Module R L] (s : Set R) (spn : Ideal.span s = ⊤) (f : M →ₗ[R] N) (g : N →ₗ[R] L) (h : ∀ (r : ↑s), Function.Exact ⇑((LocalizedModule.map (Submonoid.powers ↑r)) f) ⇑((LocalizedModule.map (Submonoid.powers ↑r)) g)) : Function.Exact ⇑f ⇑g
• LocalizedModule.map.eq_1 📋 Mathlib.Algebra.Module.FinitePresentation
  {R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] : LocalizedModule.map S = IsLocalizedModule.mapExtendScalars S (LocalizedModule.mkLinearMap S M) (LocalizedModule.mkLinearMap S N) (Localization S)
• instIsLocalizedModuleLinearMapIdLocalizationLocalizedModuleMapOfFinitePresentation 📋 Mathlib.Algebra.Module.FinitePresentation
  {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (S : Submonoid R) [Module.FinitePresentation R M] : IsLocalizedModule S (LocalizedModule.map S)
• exists_bijective_map_powers 📋 Mathlib.Algebra.Module.FinitePresentation
  {R : Type u_3} {M : Type u_4} {N : Type u_5} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (S : Submonoid R) {M' : Type u_1} [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] {N' : Type u_2} [AddCommGroup N'] [Module R N'] (g : N →ₗ[R] N') [IsLocalizedModule S g] [Module.Finite R M] [Module.FinitePresentation R N] (l : M →ₗ[R] N) (hf : Function.Bijective ⇑((IsLocalizedModule.map S f g) l)) : ∃ r ∈ S, ∀ (t : R), r ∣ t → Function.Bijective ⇑((LocalizedModule.map (Submonoid.powers t)) l)
• Module.FinitePresentation.exists_notMem_bijective 📋 Mathlib.Algebra.Module.FinitePresentation
  {R : Type u_5} {M : Type u_6} {N : Type u_7} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] [Module.Finite R M] [Module.FinitePresentation R N] (f : M →ₗ[R] N) (p : Ideal R) [p.IsPrime] {Mₚ : Type u_3} {Nₚ : Type u_4} [AddCommGroup Mₚ] [AddCommGroup Nₚ] [Module R Mₚ] [Module R Nₚ] (fM : M →ₗ[R] Mₚ) (fN : N →ₗ[R] Nₚ) [IsLocalizedModule p.primeCompl fM] [IsLocalizedModule p.primeCompl fN] (hf : Function.Bijective ⇑((IsLocalizedModule.map p.primeCompl fM fN) f)) : ∃ g ∉ p, Function.Bijective ⇑((LocalizedModule.map (Submonoid.powers g)) f)
• LinearMap.split_surjective_of_localization_maximal 📋 Mathlib.RingTheory.LocalProperties.Projective
  {R : Type u_1} {N : Type u_2} {M : Type uM} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) [Module.FinitePresentation R N] (H : ∀ (I : Ideal R) (x : I.IsMaximal), ∃ g, (LocalizedModule.map I.primeCompl) f ∘ₗ g = LinearMap.id) : ∃ g, f ∘ₗ g = LinearMap.id

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 36960b0 serving mathlib revision 9a4cf1d