Loogle!

Result

Found 14 declarations mentioning Matrix.GeneralLinearGroup.map.

• Matrix.GeneralLinearGroup.map 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) : GL n R →* GL n S
• Matrix.GeneralLinearGroup.map_id 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : Matrix.GeneralLinearGroup.map (RingHom.id R) = MonoidHom.id (GL n R)
• Matrix.GeneralLinearGroup.map.eq_1 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) : Matrix.GeneralLinearGroup.map f = Units.map ↑f.mapMatrix
• Matrix.GeneralLinearGroup.map_comp 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (f : T →+* R) (g : R →+* S) : Matrix.GeneralLinearGroup.map (g.comp f) = (Matrix.GeneralLinearGroup.map g).comp (Matrix.GeneralLinearGroup.map f)
• Matrix.GeneralLinearGroup.map_one 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) : (Matrix.GeneralLinearGroup.map f) 1 = 1
• Matrix.GeneralLinearGroup.map_apply 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (i j : n) (g : GL n R) : ↑((Matrix.GeneralLinearGroup.map f) g) i j = f (↑g i j)
• Matrix.GeneralLinearGroup.val_map_apply 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (u : (Matrix n n R)ˣ) : ↑((Matrix.GeneralLinearGroup.map f) u) = (↑u).map ⇑f
• Matrix.GeneralLinearGroup.map_inv 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (Matrix.GeneralLinearGroup.map f) g⁻¹ = ((Matrix.GeneralLinearGroup.map f) g)⁻¹
• Matrix.GeneralLinearGroup.map_inv_mul_map 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (Matrix.GeneralLinearGroup.map f) g⁻¹ * (Matrix.GeneralLinearGroup.map f) g = 1
• Matrix.GeneralLinearGroup.map_mul_map_inv 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (Matrix.GeneralLinearGroup.map f) g * (Matrix.GeneralLinearGroup.map f) g⁻¹ = 1
• Matrix.GeneralLinearGroup.map_mul 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g h : GL n R) : (Matrix.GeneralLinearGroup.map f) (g * h) = (Matrix.GeneralLinearGroup.map f) g * (Matrix.GeneralLinearGroup.map f) h
• Matrix.GeneralLinearGroup.map_comp_apply 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (f : T →+* R) (g : R →+* S) (x : GL n T) : ((Matrix.GeneralLinearGroup.map g).comp (Matrix.GeneralLinearGroup.map f)) x = (Matrix.GeneralLinearGroup.map g) ((Matrix.GeneralLinearGroup.map f) x)
• Matrix.GeneralLinearGroup.map_det 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
  {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : Matrix.GeneralLinearGroup.det ((Matrix.GeneralLinearGroup.map f) g) = (Units.map ↑f) (Matrix.GeneralLinearGroup.det g)
• Matrix.GeneralLinearGroup.map_swap 📋 Mathlib.LinearAlgebra.Matrix.Swap
  {R : Type u_1} {n : Type u_2} [CommRing R] [DecidableEq n] [Fintype n] {S : Type u_3} [CommRing S] (f : R →+* S) (i j : n) : (Matrix.GeneralLinearGroup.map f) (Matrix.GeneralLinearGroup.swap R i j) = Matrix.GeneralLinearGroup.swap S i j

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