Loogle!

Result

Found 14 declarations mentioning ExteriorAlgebra.map.

• ExteriorAlgebra.map 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) : ExteriorAlgebra R M →ₐ[R] ExteriorAlgebra R N
• ExteriorAlgebra.map_id 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] : ExteriorAlgebra.map LinearMap.id = AlgHom.id R (ExteriorAlgebra R M)
• ExteriorAlgebra.map_surjective_iff 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} : Function.Surjective ⇑(ExteriorAlgebra.map f) ↔ Function.Surjective ⇑f
• ExteriorAlgebra.map_injective 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} (hf : ∃ g, g ∘ₗ f = LinearMap.id) : Function.Injective ⇑(ExteriorAlgebra.map f)
• ExteriorAlgebra.map_comp_map 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} {N' : Type u5} [AddCommGroup N] [Module R N] [AddCommGroup N'] [Module R N'] (f : M →ₗ[R] N) (g : N →ₗ[R] N') : (ExteriorAlgebra.map g).comp (ExteriorAlgebra.map f) = ExteriorAlgebra.map (g ∘ₗ f)
• ExteriorAlgebra.map_injective_field 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {K : Type u_1} {E : Type u_2} {F : Type u_3} [Field K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {f : E →ₗ[K] F} (hf : LinearMap.ker f = ⊥) : Function.Injective ⇑(ExteriorAlgebra.map f)
• ExteriorAlgebra.map_comp_ιMulti 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : ℕ} (f : M →ₗ[R] N) : (ExteriorAlgebra.map f).toLinearMap.compAlternatingMap (ExteriorAlgebra.ιMulti R n) = (ExteriorAlgebra.ιMulti R n).compLinearMap f
• ExteriorAlgebra.leftInverse_map_iff 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {f : M →ₗ[R] N} {g : N →ₗ[R] M} : Function.LeftInverse ⇑(ExteriorAlgebra.map g) ⇑(ExteriorAlgebra.map f) ↔ Function.LeftInverse ⇑g ⇑f
• ExteriorAlgebra.map_comp_ι 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) : (ExteriorAlgebra.map f).toLinearMap ∘ₗ ExteriorAlgebra.ι R = ExteriorAlgebra.ι R ∘ₗ f
• ExteriorAlgebra.ιInv_comp_map 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) : ExteriorAlgebra.ιInv ∘ₗ (ExteriorAlgebra.map f).toLinearMap = f ∘ₗ ExteriorAlgebra.ιInv
• ExteriorAlgebra.map_apply_ιMulti 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M) : (ExteriorAlgebra.map f) ((ExteriorAlgebra.ιMulti R n) m) = (ExteriorAlgebra.ιMulti R n) (⇑f ∘ m)
• ExteriorAlgebra.map_apply_ι 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) (m : M) : (ExteriorAlgebra.map f) ((ExteriorAlgebra.ι R) m) = (ExteriorAlgebra.ι R) (f m)
• ExteriorAlgebra.toTrivSqZeroExt_comp_map 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module Rᵐᵒᵖ N] [IsCentralScalar R N] (f : M →ₗ[R] N) : ExteriorAlgebra.toTrivSqZeroExt.comp (ExteriorAlgebra.map f) = (TrivSqZeroExt.map f).comp ExteriorAlgebra.toTrivSqZeroExt
• ExteriorAlgebra.ι_range_map_map 📋 Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
  {R : Type u1} [CommRing R] {M : Type u2} [AddCommGroup M] [Module R M] {N : Type u4} [AddCommGroup N] [Module R N] (f : M →ₗ[R] N) : Submodule.map (ExteriorAlgebra.map f).toLinearMap (LinearMap.range (ExteriorAlgebra.ι R)) = Submodule.map (ExteriorAlgebra.ι R) (LinearMap.range f)

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