Loogle!

Result

Found 31 declarations mentioning Subsemiring.map.

• Subsemiring.map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : Subsemiring S
• Subsemiring.map_id 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} [NonAssocSemiring R] (s : Subsemiring R) : Subsemiring.map (RingHom.id R) s = s
• RingHom.rangeS_eq_map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : f.rangeS = Subsemiring.map f ⊤
• Subsemiring.map_bot 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : Subsemiring.map f ⊥ = ⊥
• Subsemiring.map_comap_eq 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (t : Subsemiring S) : Subsemiring.map f (Subsemiring.comap f t) = t ⊓ f.rangeS
• RingHom.map_closureS 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Set R) : Subsemiring.map f (Subsemiring.closure s) = Subsemiring.closure (⇑f '' s)
• RingHom.map_rangeS 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (g : S →+* T) (f : R →+* S) : Subsemiring.map g f.rangeS = (g.comp f).rangeS
• Subsemiring.comap_map_eq_self_of_injective 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} (hf : Function.Injective ⇑f) (s : Subsemiring R) : Subsemiring.comap f (Subsemiring.map f s) = s
• Subsemiring.map_comap_eq_self_of_surjective 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} (hf : Function.Surjective ⇑f) (t : Subsemiring S) : Subsemiring.map f (Subsemiring.comap f t) = t
• Subsemiring.gc_map_comap 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : GaloisConnection (Subsemiring.map f) (Subsemiring.comap f)
• Subsemiring.map_map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} {T : Type w} [NonAssocSemiring R] [NonAssocSemiring S] [NonAssocSemiring T] (s : Subsemiring R) (g : S →+* T) (f : R →+* S) : Subsemiring.map g (Subsemiring.map f s) = Subsemiring.map (g.comp f) s
• Subsemiring.coe_map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : ↑(Subsemiring.map f s) = ⇑f '' ↑s
• Subsemiring.map_comap_eq_self 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {t : Subsemiring S} (h : t ≤ f.rangeS) : Subsemiring.map f (Subsemiring.comap f t) = t
• Subsemiring.map_iSup 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} (f : R →+* S) (s : ι → Subsemiring R) : Subsemiring.map f (iSup s) = ⨆ i, Subsemiring.map f (s i)
• Subsemiring.map_iInf 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {ι : Sort u_1} [Nonempty ι] (f : R →+* S) (hf : Function.Injective ⇑f) (s : ι → Subsemiring R) : Subsemiring.map f (iInf s) = ⨅ i, Subsemiring.map f (s i)
• Subsemiring.map_le_iff_le_comap 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {t : Subsemiring S} : Subsemiring.map f s ≤ t ↔ s ≤ Subsemiring.comap f t
• Subsemiring.map_inf 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s t : Subsemiring R) (f : R →+* S) (hf : Function.Injective ⇑f) : Subsemiring.map f (s ⊓ t) = Subsemiring.map f s ⊓ Subsemiring.map f t
• Subsemiring.mem_map 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R →+* S} {s : Subsemiring R} {y : S} : y ∈ Subsemiring.map f s ↔ ∃ x ∈ s, f x = y
• Subsemiring.map_sup 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s t : Subsemiring R) (f : R →+* S) : Subsemiring.map f (s ⊔ t) = Subsemiring.map f s ⊔ Subsemiring.map f t
• Subsemiring.map_toSubmonoid 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (s : Subsemiring R) : (Subsemiring.map f s).toSubmonoid = { carrier := ⇑f '' ↑s, mul_mem' := ⋯, one_mem' := ⋯ }
• RingHom.rangeS_toSubmonoid 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) : f.rangeS.toSubmonoid = { carrier := Set.range ⇑f, mul_mem' := ⋯, one_mem' := ⋯ }
• Subsemiring.equivMapOfInjective 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective ⇑f) : ↥s ≃+* ↥(Subsemiring.map f s)
• Subsemiring.mem_map_equiv 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] {f : R ≃+* S} {K : Subsemiring R} {x : S} : x ∈ Subsemiring.map (↑f) K ↔ f.symm x ∈ K
• Subsemiring.comap_equiv_eq_map_symm 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring S) : Subsemiring.comap (↑f) K = Subsemiring.map (↑f.symm) K
• Subsemiring.map_equiv_eq_comap_symm 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (K : Subsemiring R) : Subsemiring.map (↑f) K = Subsemiring.comap (↑f.symm) K
• Subsemiring.coe_equivMapOfInjective_apply 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (s : Subsemiring R) (f : R →+* S) (hf : Function.Injective ⇑f) (x : ↥s) : ↑((s.equivMapOfInjective f hf) x) = f ↑x
• RingEquiv.subsemiringMap 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) : ↥s ≃+* ↥(Subsemiring.map (↑e) s)
• RingEquiv.subsemiringMap_apply_coe 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (x : ↥s) : ↑((e.subsemiringMap s) x) = e ↑x
• RingEquiv.subsemiringMap_symm_apply_coe 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R : Type u} {S : Type v} [NonAssocSemiring R] [NonAssocSemiring S] (e : R ≃+* S) (s : Subsemiring R) (x : ↥(Subsemiring.map e.toRingHom s)) : ↑((e.subsemiringMap s).symm x) = e.symm ↑x
• Subalgebra.map_toSubsemiring 📋 Mathlib.Algebra.Algebra.Subalgebra.Basic
  {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (f : A →ₐ[R] B) (S : Subalgebra R A) : (Subalgebra.map f S).toSubsemiring = Subsemiring.map (↑f) S.toSubsemiring
• Subsemiring.pointwise_smul_def 📋 Mathlib.Algebra.Ring.Subsemiring.Pointwise
  {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] {a : M} (S : Subsemiring R) : a • S = Subsemiring.map (MulSemiringAction.toRingHom M R a) S

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