Loogle!

Result

Found 9 declarations mentioning SemidirectProduct.map.

• SemidirectProduct.map 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) : N₁ ⋊[φ₁] G₁ →* N₂ ⋊[φ₂] G₂
• SemidirectProduct.map_comp_inl 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) : (SemidirectProduct.map fn fg h).comp SemidirectProduct.inl = SemidirectProduct.inl.comp fn
• SemidirectProduct.map_comp_inr 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) : (SemidirectProduct.map fn fg h).comp SemidirectProduct.inr = SemidirectProduct.inr.comp fg
• SemidirectProduct.rightHom_comp_map 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) : SemidirectProduct.rightHom.comp (SemidirectProduct.map fn fg h) = fg.comp SemidirectProduct.rightHom
• SemidirectProduct.map_left 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) (g : N₁ ⋊[φ₁] G₁) : ((SemidirectProduct.map fn fg h) g).left = fn g.left
• SemidirectProduct.map_right 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) (g : N₁ ⋊[φ₁] G₁) : ((SemidirectProduct.map fn fg h) g).right = fg g.right
• SemidirectProduct.map.eq_1 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) : SemidirectProduct.map fn fg h = { toFun := fun x => { left := fn x.left, right := fg x.right }, map_one' := ⋯, map_mul' := ⋯ }
• SemidirectProduct.map_inl 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) (n : N₁) : (SemidirectProduct.map fn fg h) (SemidirectProduct.inl n) = SemidirectProduct.inl (fn n)
• SemidirectProduct.map_inr 📋 Mathlib.GroupTheory.SemidirectProduct
  {N₁ : Type u_4} {G₁ : Type u_5} {N₂ : Type u_6} {G₂ : Type u_7} [Group N₁] [Group G₁] [Group N₂] [Group G₂] {φ₁ : G₁ →* MulAut N₁} {φ₂ : G₂ →* MulAut N₂} (fn : N₁ →* N₂) (fg : G₁ →* G₂) (h : ∀ (g : G₁), fn.comp (MulEquiv.toMonoidHom (φ₁ g)) = (MulEquiv.toMonoidHom (φ₂ (fg g))).comp fn) (g : G₁) : (SemidirectProduct.map fn fg h) (SemidirectProduct.inr g) = SemidirectProduct.inr (fg g)

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