Loogle!

Result

Found 16 declarations mentioning QuotientGroup.map.

• QuotientGroup.map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) : G ⧸ N →* H ⧸ M
• QuotientGroup.ker_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) : (QuotientGroup.map N M f h).ker = Subgroup.map (QuotientGroup.mk' N) (Subgroup.comap f M)
• QuotientGroup.map_id 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (h : N ≤ Subgroup.comap (MonoidHom.id G) N := ⋯) : QuotientGroup.map N N (MonoidHom.id G) h = MonoidHom.id (G ⧸ N)
• QuotientGroup.map.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) : QuotientGroup.map N M f h = QuotientGroup.lift N ((QuotientGroup.mk' M).comp f) ⋯
• QuotientGroup.map_id_apply 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (h : N ≤ Subgroup.comap (MonoidHom.id G) N := ⋯) (x : G ⧸ N) : (QuotientGroup.map N N (MonoidHom.id G) h) x = x
• QuotientGroup.map_mk 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ↑x = ↑(f x)
• QuotientGroup.map_surjective_of_surjective 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (hf : Function.Surjective (QuotientGroup.mk ∘ ⇑f)) (h : N ≤ Subgroup.comap f M) : Function.Surjective ⇑(QuotientGroup.map N M f h)
• QuotientGroup.map_mk' 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] (M : Subgroup H) [M.Normal] (f : G →* H) (h : N ≤ Subgroup.comap f M) (x : G) : (QuotientGroup.map N M f h) ((QuotientGroup.mk' N) x) = ↑(f x)
• QuotientGroup.map_comp_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] {I : Type u_1} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : N ≤ Subgroup.comap (g.comp f) O := ⋯) : (QuotientGroup.map M O g hg).comp (QuotientGroup.map N M f hf) = QuotientGroup.map N O (g.comp f) hgf
• QuotientGroup.map_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [Group H] {I : Type u_1} [Group I] (M : Subgroup H) (O : Subgroup I) [M.Normal] [O.Normal] (f : G →* H) (g : H →* I) (hf : N ≤ Subgroup.comap f M) (hg : M ≤ Subgroup.comap g O) (hgf : N ≤ Subgroup.comap (g.comp f) O := ⋯) (x : G ⧸ N) : (QuotientGroup.map M O g hg) ((QuotientGroup.map N M f hf) x) = (QuotientGroup.map N O (g.comp f) hgf) x
• QuotientGroup.congr.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [Group G] {H : Type v} [Group H] (G' : Subgroup G) (H' : Subgroup H) [G'.Normal] [H'.Normal] (e : G ≃* H) (he : Subgroup.map (↑e) G' = H') : QuotientGroup.congr G' H' e he = { toFun := ⇑(QuotientGroup.map G' H' ↑e ⋯), invFun := ⇑(QuotientGroup.map H' G' ↑e.symm ⋯), left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯ }
• QuotientGroup.quotientQuotientEquivQuotientAux.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (h : N ≤ M) : QuotientGroup.quotientQuotientEquivQuotientAux N M h = QuotientGroup.lift (Subgroup.map (QuotientGroup.mk' N) M) (QuotientGroup.map N M (MonoidHom.id G) h) ⋯
• QuotientGroup.quotientMapSubgroupOfOfLe.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [Group G] {A' A B' B : Subgroup G} [_hAN : (A'.subgroupOf A).Normal] [_hBN : (B'.subgroupOf B).Normal] (h' : A' ≤ B') (h : A ≤ B) : QuotientGroup.quotientMapSubgroupOfOfLe h' h = QuotientGroup.map (A'.subgroupOf A) (B'.subgroupOf B) (Subgroup.inclusion h) ⋯
• QuotientGroup.quotientQuotientEquivQuotient.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (h : N ≤ M) : QuotientGroup.quotientQuotientEquivQuotient N M h = (QuotientGroup.quotientQuotientEquivQuotientAux N M h).toMulEquiv (QuotientGroup.map M (Subgroup.map (QuotientGroup.mk' N) M) (QuotientGroup.mk' N) ⋯) ⋯ ⋯
• QuotientGroup.quotientQuotientEquivQuotientAux_mk 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] (M : Subgroup G) [nM : M.Normal] (h : N ≤ M) (x : G ⧸ N) : (QuotientGroup.quotientQuotientEquivQuotientAux N M h) ↑x = (QuotientGroup.map N M (MonoidHom.id G) h) x
• IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroMod.eq_1 📋 Mathlib.RingTheory.DedekindDomain.SelmerGroup
  {R : Type u} [CommRing R] [IsDedekindDomain R] {K : Type v} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℕ) : v.valuationOfNeZeroMod n = (AddEquiv.toMultiplicative (Int.quotientZMultiplesNatEquivZMod n)).toMonoidHom.comp (QuotientGroup.map (powMonoidHom n).range (AddSubgroup.toSubgroup (AddSubgroup.zmultiples ↑n)) v.valuationOfNeZero ⋯)

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 19971e9 serving mathlib revision bce1d65