Loogle!

Result

Found 3662 declarations mentioning AddMonoidHom. Of these, 12 have a name containing "image".

• AddMonoidHom.closure_preimage_le 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (s : Set N) : AddSubgroup.closure (⇑f ⁻¹' s) ≤ AddSubgroup.comap f (AddSubgroup.closure s)
• AddSubgroup.closure_preimage_eq_top 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] (s : Set G) : AddSubgroup.closure (⇑(AddSubgroup.closure s).subtype ⁻¹' s) = ⊤
• Finset.imageAddMonoidHom 📋 Mathlib.Algebra.Group.Pointwise.Finset.Basic
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] [AddZeroClass α] [AddZeroClass β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) : Finset α →+ Finset β
• Finset.imageAddMonoidHom_apply 📋 Mathlib.Algebra.Group.Pointwise.Finset.Basic
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] [AddZeroClass α] [AddZeroClass β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) (a✝ : Finset α) : (Finset.imageAddMonoidHom f) a✝ = (Finset.imageAddHom f).toFun a✝
• Finset.imageAddMonoidHom.eq_1 📋 Mathlib.Algebra.Group.Pointwise.Finset.Basic
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [DecidableEq α] [DecidableEq β] [AddZeroClass α] [AddZeroClass β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) : Finset.imageAddMonoidHom f = { toFun := (Finset.imageAddHom f).toFun, map_zero' := ⋯, map_add' := ⋯ }
• AddSubgroup.map_centralizer_le_centralizer_image 📋 Mathlib.GroupTheory.Subgroup.Centralizer
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (s : Set G) (f : G →+ G') : AddSubgroup.map f (AddSubgroup.centralizer s) ≤ AddSubgroup.centralizer (⇑f '' s)
• AddMonoidHom.map_finsum_of_preimage_zero 📋 Mathlib.Algebra.BigOperators.Finprod
  {M : Type u_2} {N : Type u_3} {α : Sort u_4} [AddCommMonoid M] [AddCommMonoid N] (f : M →+ N) (hf : ∀ (x : M), f x = 0 → x = 0) (g : α → M) : f (∑ᶠ (i : α), g i) = ∑ᶠ (i : α), f (g i)
• image_norm_nonempty 📋 Mathlib.Analysis.Normed.Group.Quotient
  {M : Type u_1} [SeminormedAddCommGroup M] {S : AddSubgroup M} (x : M ⧸ S) : (norm '' {m | (QuotientAddGroup.mk' S) m = x}).Nonempty
• AddMonoidHom.preimage_vadd_setₛₗ 📋 Mathlib.GroupTheory.GroupAction.Pointwise
  {M : Type u_1} {N : Type u_2} {α : Type u_4} {β : Type u_5} [AddMonoid M] [AddMonoid N] [AddAction M α] [AddAction N β] {c : M} {F : Type u_6} (σ : M →+ N) [FunLike F α β] [AddActionSemiHomClass F (⇑σ) α β] (f : F) (hc : IsAddUnit c) (t : Set β) : ⇑f ⁻¹' (σ c +ᵥ t) = c +ᵥ ⇑f ⁻¹' t
• Behrend.threeAPFree_image_sphere 📋 Mathlib.Combinatorics.Additive.AP.Three.Behrend
  {n d k : ℕ} : ThreeAPFree ↑(Finset.image (⇑(Behrend.map (2 * d - 1))) (Behrend.sphere n d k))
• AddMonoidHom.preErgodic_of_dense_iUnion_preimage_zero 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [SecondCountableTopology G] [MeasurableSpace G] [BorelSpace G] {μ : MeasureTheory.Measure G} [MeasureTheory.IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsAddLeftInvariant] (f : G →+ G) (hf : Dense (⋃ n, (⇑f)^[n] ⁻¹' 0)) : PreErgodic (⇑f) μ
• AddMonoidHom.ergodic_of_dense_iUnion_preimage_zero 📋 Mathlib.Dynamics.Ergodic.Action.OfMinimal
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [SecondCountableTopology G] [MeasurableSpace G] [BorelSpace G] [CompactSpace G] {μ : MeasureTheory.Measure G} [μ.IsAddHaarMeasure] (f : G →+ G) (hf : Dense (⋃ n, (⇑f)^[n] ⁻¹' 0)) (hcont : Continuous ⇑f) (hsurj : Function.Surjective ⇑f) : Ergodic (⇑f) μ

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