Loogle!

Result

Found 32 declarations mentioning LieIdeal.map.

• LieIdeal.map 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) : LieIdeal R L'
• LieHom.idealRange_eq_map 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') : f.idealRange = LieIdeal.map f ⊤
• LieIdeal.map_sup_ker_eq_map 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} : LieIdeal.map f (I ⊔ f.ker) = LieIdeal.map f I
• LieHom.map_le_idealRange 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) : LieIdeal.map f I ≤ f.idealRange
• LieIdeal.map_comap_eq 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {J : LieIdeal R L'} (h : f.IsIdealMorphism) : LieIdeal.map f (LieIdeal.comap f J) = f.idealRange ⊓ J
• LieIdeal.map_sup_ker_eq_map' 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} : LieIdeal.map f I ⊔ LieIdeal.map f f.ker = LieIdeal.map f I
• LieIdeal.comap_map_le 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} : I ≤ LieIdeal.comap f (LieIdeal.map f I)
• LieIdeal.map_comap_le 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {J : LieIdeal R L'} : LieIdeal.map f (LieIdeal.comap f J) ≤ J
• LieIdeal.map_mono 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} : Monotone (LieIdeal.map f)
• LieIdeal.map_sup 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I I₂ : LieIdeal R L} : LieIdeal.map f (I ⊔ I₂) = LieIdeal.map f I ⊔ LieIdeal.map f I₂
• LieIdeal.bot_of_map_eq_bot 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} (h₁ : Function.Injective ⇑f) (h₂ : LieIdeal.map f I = ⊥) : I = ⊥
• LieIdeal.gc_map_comap 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') : GaloisConnection (LieIdeal.map f) (LieIdeal.comap f)
• LieIdeal.mem_map 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} {x : L} (hx : x ∈ I) : f x ∈ LieIdeal.map f I
• LieIdeal.map_eq_bot_iff 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} : LieIdeal.map f I = ⊥ ↔ I ≤ f.ker
• LieIdeal.map_of_image 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} {J : LieIdeal R L'} (h : ⇑f '' ↑I = ↑J) : LieIdeal.map f I = J
• LieIdeal.map_le_iff_le_comap 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} {J : LieIdeal R L'} : LieIdeal.map f I ≤ J ↔ I ≤ LieIdeal.comap f J
• LieIdeal.comap_map_eq 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} (h : ↑(LieIdeal.map f I) = ⇑f '' ↑I) : LieIdeal.comap f (LieIdeal.map f I) = I ⊔ f.ker
• LieIdeal.map_le 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) (J : LieIdeal R L') : LieIdeal.map f I ≤ J ↔ ⇑f '' ↑I ⊆ ↑J
• LieIdeal.map.eq_1 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) : LieIdeal.map f I = LieSubmodule.lieSpan R L' ↑(Submodule.map (↑f) (LieIdeal.toLieSubalgebra R L I).toSubmodule)
• LieIdeal.mem_map_of_surjective 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} {y : L'} (h₁ : Function.Surjective ⇑f) (h₂ : y ∈ LieIdeal.map f I) : ∃ x, f ↑x = y
• LieIdeal.coe_map_of_surjective 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] {f : L →ₗ⁅R⁆ L'} {I : LieIdeal R L} (h : Function.Surjective ⇑f) : ↑(LieIdeal.map f I) = Submodule.map ↑f ↑I
• LieIdeal.map_coeSubmodule 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) (h : ↑(LieIdeal.map f I) = ⇑f '' ↑I) : ↑(LieIdeal.map f I) = Submodule.map ↑f ↑I
• LieIdeal.map_toSubmodule 📋 Mathlib.Algebra.Lie.Ideal
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieRing L'] [LieAlgebra R L'] [LieAlgebra R L] (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) (h : ↑(LieIdeal.map f I) = ⇑f '' ↑I) : ↑(LieIdeal.map f I) = Submodule.map ↑f ↑I
• LieIdeal.map_bracket_eq 📋 Mathlib.Algebra.Lie.IdealOperations
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') {I₁ I₂ : LieIdeal R L} (h : Function.Surjective ⇑f) : LieIdeal.map f ⁅I₁, I₂⁆ = ⁅LieIdeal.map f I₁, LieIdeal.map f I₂⁆
• LieIdeal.map_bracket_le 📋 Mathlib.Algebra.Lie.IdealOperations
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (f : L →ₗ⁅R⁆ L') {I₁ I₂ : LieIdeal R L} : LieIdeal.map f ⁅I₁, I₂⁆ ≤ ⁅LieIdeal.map f I₁, LieIdeal.map f I₂⁆
• LieIdeal.map_comap_bracket_eq 📋 Mathlib.Algebra.Lie.IdealOperations
  {R : Type u} {L : Type v} {L' : Type w₂} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] {f : L →ₗ⁅R⁆ L'} {J₁ J₂ : LieIdeal R L'} (h : f.IsIdealMorphism) : LieIdeal.map f ⁅LieIdeal.comap f J₁, LieIdeal.comap f J₂⁆ = ⁅f.idealRange ⊓ J₁, f.idealRange ⊓ J₂⁆
• LieIdeal.map_comap_incl 📋 Mathlib.Algebra.Lie.IdealOperations
  {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {I₁ I₂ : LieIdeal R L} : LieIdeal.map I₁.incl (LieIdeal.comap I₁.incl I₂) = I₁ ⊓ I₂
• LieIdeal.derivedSeries_map_eq 📋 Mathlib.Algebra.Lie.Solvable
  {R : Type u} {L : Type v} {L' : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] {f : L' →ₗ⁅R⁆ L} (k : ℕ) (h : Function.Surjective ⇑f) : LieIdeal.map f (LieAlgebra.derivedSeries R L' k) = LieAlgebra.derivedSeries R L k
• LieIdeal.derivedSeries_map_le 📋 Mathlib.Algebra.Lie.Solvable
  {R : Type u} {L : Type v} {L' : Type w₁} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] {f : L' →ₗ⁅R⁆ L} (k : ℕ) : LieIdeal.map f (LieAlgebra.derivedSeries R L' k) ≤ LieAlgebra.derivedSeries R L k
• LieIdeal.derivedSeries_eq_derivedSeriesOfIdeal_map 📋 Mathlib.Algebra.Lie.Solvable
  {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L) (k : ℕ) : LieIdeal.map I.incl (LieAlgebra.derivedSeries R (↥I) k) = LieAlgebra.derivedSeriesOfIdeal R L k I
• LieIdeal.lowerCentralSeries_map_eq 📋 Mathlib.Algebra.Lie.Nilpotent
  {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h : Function.Surjective ⇑f) : LieIdeal.map f (LieModule.lowerCentralSeries R L L k) = LieModule.lowerCentralSeries R L' L' k
• LieIdeal.map_lowerCentralSeries_le 📋 Mathlib.Algebra.Lie.Nilpotent
  {R : Type u} {L : Type v} {L' : Type w} [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L'] (k : ℕ) {f : L →ₗ⁅R⁆ L'} : LieIdeal.map f (LieModule.lowerCentralSeries R L L k) ≤ LieModule.lowerCentralSeries R L' L' k

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 40fea08