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Found 143 declarations mentioning AddSubgroup.map.

• AddSubgroup.map_id 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] (K : AddSubgroup G) : AddSubgroup.map (AddMonoidHom.id G) K = K
• AddSubgroup.map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) : AddSubgroup N
• AddSubgroup.map_bot 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) : AddSubgroup.map f ⊥ = ⊥
• AddSubgroup.addSubgroupOf_map_subtype 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] (H K : AddSubgroup G) : AddSubgroup.map K.subtype (H.addSubgroupOf K) = H ⊓ K
• AddSubgroup.le_comap_map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) : H ≤ AddSubgroup.comap f (AddSubgroup.map f H)
• AddSubgroup.map_comap_le 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) : AddSubgroup.map f (AddSubgroup.comap f H) ≤ H
• AddSubgroup.map_addSubgroupOf_eq_of_le 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H K : AddSubgroup G} (h : H ≤ K) : AddSubgroup.map K.subtype (H.addSubgroupOf K) = H
• AddSubgroup.gc_map_comap 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) : GaloisConnection (AddSubgroup.map f) (AddSubgroup.comap f)
• AddSubgroup.map_iSup 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} (f : G →+ N) (s : ι → AddSubgroup G) : AddSubgroup.map f (iSup s) = ⨆ i, AddSubgroup.map f (s i)
• AddSubgroup.map_isAddCommutative 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') [IsAddCommutative ↥H] : IsAddCommutative ↥(AddSubgroup.map f H)
• AddSubgroup.map_isCommutative 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') [IsAddCommutative ↥H] : IsAddCommutative ↥(AddSubgroup.map f H)
• AddSubgroup.map_zero_eq_bot 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] : AddSubgroup.map 0 K = ⊥
• AddSubgroup.map_inf_le 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) : AddSubgroup.map f (H ⊓ K) ≤ AddSubgroup.map f H ⊓ AddSubgroup.map f K
• AddSubgroup.map_mono 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K K' : AddSubgroup G} : K ≤ K' → AddSubgroup.map f K ≤ AddSubgroup.map f K'
• AddSubgroup.map_le_iff_le_comap 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {H : AddSubgroup N} : AddSubgroup.map f K ≤ H ↔ K ≤ AddSubgroup.comap f H
• AddMonoidHom.map_closure 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (s : Set G) : AddSubgroup.map f (AddSubgroup.closure s) = AddSubgroup.closure (⇑f '' s)
• AddSubgroup.map_sup 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) : AddSubgroup.map f (H ⊔ K) = AddSubgroup.map f H ⊔ AddSubgroup.map f K
• AddSubgroup.map_top_of_surjective 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (h : Function.Surjective ⇑f) : AddSubgroup.map f ⊤ = ⊤
• AddSubgroup.coe_map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G) : ↑(AddSubgroup.map f K) = ⇑f '' ↑K
• AddSubgroup.map_equiv_top 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {F : Type u_7} [EquivLike F G N] [AddEquivClass F G N] (f : F) : AddSubgroup.map ↑f ⊤ = ⊤
• AddSubgroup.map_map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {P : Type u_6} [AddGroup P] (g : N →+ P) (f : G →+ N) : AddSubgroup.map g (AddSubgroup.map f K) = AddSubgroup.map (g.comp f) K
• AddSubgroup.comap_equiv_eq_map_symm' 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G) : AddSubgroup.comap f.toAddMonoidHom K = AddSubgroup.map f.symm.toAddMonoidHom K
• AddSubgroup.map_equiv_eq_comap_symm' 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G) : AddSubgroup.map f.toAddMonoidHom K = AddSubgroup.comap f.symm.toAddMonoidHom K
• AddSubgroup.mem_map_of_mem 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {K : AddSubgroup G} {x : G} (hx : x ∈ K) : f x ∈ AddSubgroup.map f K
• AddSubgroup.map_iInf 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {ι : Sort u_7} [Nonempty ι] (f : G →+ N) (hf : Function.Injective ⇑f) (s : ι → AddSubgroup G) : AddSubgroup.map f (iInf s) = ⨅ i, AddSubgroup.map f (s i)
• AddSubgroup.map_inf 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f) : AddSubgroup.map f (H ⊓ K) = AddSubgroup.map f H ⊓ AddSubgroup.map f K
• AddSubgroup.map_inf_eq 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H K : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f) : AddSubgroup.map f (H ⊓ K) = AddSubgroup.map f H ⊓ AddSubgroup.map f K
• AddSubgroup.mem_map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {K : AddSubgroup G} {y : N} : y ∈ AddSubgroup.map f K ↔ ∃ x ∈ K, f x = y
• AddSubgroup.apply_coe_mem_map 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (K : AddSubgroup G) (x : ↥K) : f ↑x ∈ AddSubgroup.map f K
• AddSubgroup.map_toAddSubmonoid 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (K : AddSubgroup G) : (AddSubgroup.map f K).toAddSubmonoid = AddSubmonoid.map f K.toAddSubmonoid
• AddSubgroup.equivMapOfInjective 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f) : ↥H ≃+ ↥(AddSubgroup.map f H)
• AddSubgroup.map.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) : AddSubgroup.map f H = { carrier := ⇑f '' ↑H, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }
• AddSubgroup.mem_map_iff_mem 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective ⇑f) {K : AddSubgroup G} {x : G} : f x ∈ AddSubgroup.map f K ↔ x ∈ K
• AddEquiv.coe_mapAddSubgroup 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_5} [AddGroup H] (e : G ≃+ H) : ⇑e.mapAddSubgroup = AddSubgroup.map e.toAddMonoidHom
• AddSubgroup.mem_map_equiv 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G ≃+ N} {K : AddSubgroup G} {x : N} : x ∈ AddSubgroup.map f.toAddMonoidHom K ↔ f.symm x ∈ K
• AddMonoidHom.addSubgroupMap 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) : ↥H →+ ↥(AddSubgroup.map f H)
• AddSubgroup.map_eq_comap_of_inverse 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {g : N →+ G} (hl : Function.LeftInverse ⇑g ⇑f) (hr : Function.RightInverse ⇑g ⇑f) (H : AddSubgroup G) : AddSubgroup.map f H = AddSubgroup.comap g H
• AddSubgroup.equivMapOfInjective.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f) : H.equivMapOfInjective f hf = { toEquiv := Equiv.Set.image (⇑f) (↑H) hf, map_add' := ⋯ }
• AddMonoidHom.addSubgroupMap.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) : f.addSubgroupMap H = f.addSubmonoidMap H.toAddSubmonoid
• AddEquiv.mapAddSubgroup_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_6} [AddGroup H] (f : G ≃+ H) (H✝ : AddSubgroup G) : f.mapAddSubgroup H✝ = AddSubgroup.map (↑f) H✝
• AddEquiv.mapAddSubgroup_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_6} [AddGroup H] (f : G ≃+ H) (H✝ : AddSubgroup H) : (RelIso.symm f.mapAddSubgroup) H✝ = AddSubgroup.map (↑f.symm) H✝
• AddEquiv.mapAddSubgroup.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {H : Type u_6} [AddGroup H] (f : G ≃+ H) : f.mapAddSubgroup = { toFun := AddSubgroup.map ↑f, invFun := AddSubgroup.map ↑f.symm, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
• AddSubgroup.comap_equiv_eq_map_symm 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : N ≃+ G) (K : AddSubgroup G) : AddSubgroup.comap (↑f) K = AddSubgroup.map (↑f.symm) K
• AddSubgroup.map_equiv_eq_comap_symm 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G ≃+ N) (K : AddSubgroup G) : AddSubgroup.map (↑f) K = AddSubgroup.comap (↑f.symm) K
• AddSubgroup.map_symm_eq_iff_map_eq 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] (K : AddSubgroup G) {N : Type u_5} [AddGroup N] {H : AddSubgroup N} {e : G ≃+ N} : AddSubgroup.map (↑e.symm) H = K ↔ AddSubgroup.map (↑e) K = H
• AddEquiv.addSubgroupMap 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) : ↥H ≃+ ↥(AddSubgroup.map (↑e) H)
• AddSubgroup.coe_equivMapOfInjective_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) (hf : Function.Injective ⇑f) (h : ↥H) : ↑((H.equivMapOfInjective f hf) h) = f ↑h
• AddEquiv.addSubgroupMap.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) : e.addSubgroupMap H = e.addSubmonoidMap H.toAddSubmonoid
• AddMonoidHom.addSubgroupMap_surjective 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) : Function.Surjective ⇑(f.addSubgroupMap H)
• AddSubgroup.equivMapOfInjective_coe_addEquiv 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (e : G ≃+ G') : H.equivMapOfInjective ↑e ⋯ = e.addSubgroupMap H
• AddMonoidHom.addSubgroupMap_apply_coe 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H : AddSubgroup G) (x : ↥H.toAddSubmonoid) : ↑((f.addSubgroupMap H) x) = f ↑x
• AddEquiv.coe_addSubgroupMap_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) (g : ↥H) : ↑((e.addSubgroupMap H) g) = e ↑g
• AddEquiv.addSubgroupMap_symm_apply 📋 Mathlib.Algebra.Group.Subgroup.Map
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (e : G ≃+ G') (H : AddSubgroup G) (g : ↥(AddSubgroup.map (↑e) H)) : (e.addSubgroupMap H).symm g = ⟨e.symm ↑g, ⋯⟩
• AddMonoidHom.range_eq_map 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) : f.range = AddSubgroup.map f ⊤
• AddSubgroup.map_comap_eq 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup N) : AddSubgroup.map f (AddSubgroup.comap f H) = f.range ⊓ H
• AddSubgroup.map_le_range 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) : AddSubgroup.map f H ≤ f.range
• AddSubgroup.map_comap_eq_self 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup N} (h : H ≤ f.range) : AddSubgroup.map f (AddSubgroup.comap f H) = H
• AddSubgroup.comap_map_eq 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) (H : AddSubgroup G) : AddSubgroup.comap f (AddSubgroup.map f H) = H ⊔ f.ker
• AddSubgroup.map_subtype_le 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (K : AddSubgroup ↥H) : AddSubgroup.map H.subtype K ≤ H
• AddSubgroup.comap_map_eq_self 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup G} (h : f.ker ≤ H) : AddSubgroup.comap f (AddSubgroup.map f H) = H
• AddSubgroup.map_eq_bot_iff 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} : AddSubgroup.map f H = ⊥ ↔ H ≤ f.ker
• AddSubgroup.map_injective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (h : Function.Injective ⇑f) : Function.Injective (AddSubgroup.map f)
• AddSubgroup.comap_map_eq_self_of_injective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (h : Function.Injective ⇑f) (H : AddSubgroup G) : AddSubgroup.comap f (AddSubgroup.map f H) = H
• AddSubgroup.map_comap_eq_self_of_surjective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (h : Function.Surjective ⇑f) (H : AddSubgroup N) : AddSubgroup.map f (AddSubgroup.comap f H) = H
• AddMonoidHom.map_range 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} {P : Type u_6} [AddGroup N] [AddGroup P] (g : N →+ P) (f : G →+ N) : AddSubgroup.map g f.range = (g.comp f).range
• AddMonoidHom.range_comp 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} {P : Type u_6} [AddGroup N] [AddGroup P] (g : N →+ P) (f : G →+ N) : (g.comp f).range = AddSubgroup.map g f.range
• AddMonoidHom.restrict_range 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (K : AddSubgroup G) (f : G →+ N) : (f.restrict K).range = AddSubgroup.map f K
• AddSubgroup.map_eq_range_iff 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H : AddSubgroup G} : AddSubgroup.map f H = f.range ↔ Codisjoint H f.ker
• AddMonoidHom.range.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) : f.range = (AddSubgroup.map f ⊤).copy (Set.range ⇑f) ⋯
• AddSubgroup.map_eq_bot_iff_of_injective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} (hf : Function.Injective ⇑f) : AddSubgroup.map f H = ⊥ ↔ H = ⊥
• AddSubgroup.map_eq_map_iff 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} : AddSubgroup.map f H = AddSubgroup.map f K ↔ H ⊔ f.ker = K ⊔ f.ker
• AddSubgroup.map_le_map_iff 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} : AddSubgroup.map f H ≤ AddSubgroup.map f K ↔ H ≤ K ⊔ f.ker
• AddSubgroup.map_injective_of_ker_le 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (f : G →+ N) {H K : AddSubgroup G} (hH : f.ker ≤ H) (hK : f.ker ≤ K) (hf : AddSubgroup.map f H = AddSubgroup.map f K) : H = K
• AddSubgroup.map_le_map_iff_of_injective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective ⇑f) {H K : AddSubgroup G} : AddSubgroup.map f H ≤ AddSubgroup.map f K ↔ H ≤ K
• AddSubgroup.map_lt_map_iff_of_injective 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} (hf : Function.Injective ⇑f) {H K : AddSubgroup G} : AddSubgroup.map f H < AddSubgroup.map f K ↔ H < K
• AddSubgroup.map_le_map_iff' 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {f : G →+ N} {H K : AddSubgroup G} : AddSubgroup.map f H ≤ AddSubgroup.map f K ↔ H ⊔ f.ker ≤ K ⊔ f.ker
• AddSubgroup.ker_addSubgroupMap 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G →+ N) : (f.addSubgroupMap H).ker = f.ker.addSubgroupOf H
• AddSubgroup.map_subtype_le_map_subtype 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {G' : AddSubgroup G} {H K : AddSubgroup ↥G'} : AddSubgroup.map G'.subtype H ≤ AddSubgroup.map G'.subtype K ↔ H ≤ K
• AddSubgroup.map_subtype_lt_map_subtype 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] {G' : AddSubgroup G} {H K : AddSubgroup ↥G'} : AddSubgroup.map G'.subtype H < AddSubgroup.map G'.subtype K ↔ H < K
• AddSubgroup.MapSubtype.orderIso.eq_1 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : AddSubgroup.MapSubtype.orderIso H = { toFun := fun H' => ⟨AddSubgroup.map H.subtype H', ⋯⟩, invFun := fun sH' => (↑sH').addSubgroupOf H, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
• AddSubgroup.MapSubtype.orderIso_apply_coe 📋 Mathlib.Algebra.Group.Subgroup.Ker
  {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (H' : AddSubgroup ↥H) : ↑((AddSubgroup.MapSubtype.orderIso H) H') = AddSubgroup.map H.subtype H'
• Submodule.map_toAddSubgroup 📋 Mathlib.Algebra.Module.Submodule.Map
  {R : Type u_1} {M : Type u_5} {M₂ : Type u_7} [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup M₂] [Module R M₂] (f : M →ₗ[R] M₂) (p : Submodule R M) : (Submodule.map f p).toAddSubgroup = AddSubgroup.map (↑f) p.toAddSubgroup
• AddMonoidHom.coe_toMultiplicative_map 📋 Mathlib.Algebra.Module.Submodule.Map
  {G : Type u_10} {G₂ : Type u_11} [AddGroup G] [AddGroup G₂] (f : G →+ G₂) (s : AddSubgroup G) : Subgroup.map (AddMonoidHom.toMultiplicative f) (AddSubgroup.toSubgroup s) = AddSubgroup.toSubgroup (AddSubgroup.map f s)
• MonoidHom.coe_toAdditive_map 📋 Mathlib.Algebra.Module.Submodule.Map
  {G : Type u_10} {G₂ : Type u_11} [Group G] [Group G₂] (f : G →* G₂) (s : Subgroup G) : AddSubgroup.map (MonoidHom.toAdditive f) (Subgroup.toAddSubgroup s) = Subgroup.toAddSubgroup (Subgroup.map f s)
• AddMonoidHom.coe_toIntLinearMap_map 📋 Mathlib.Algebra.Module.Submodule.Map
  {A : Type u_10} {A₂ : Type u_11} [AddCommGroup A] [AddCommGroup A₂] (f : A →+ A₂) (s : AddSubgroup A) : Submodule.map f.toIntLinearMap (AddSubgroup.toIntSubmodule s) = AddSubgroup.toIntSubmodule (AddSubgroup.map f s)
• AddSubgroup.characteristic_iff_map_eq 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : H.Characteristic ↔ ∀ (ϕ : G ≃+ G), AddSubgroup.map ϕ.toAddMonoidHom H = H
• AddSubgroup.le_normalizer_map 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {N : Type u_5} [AddGroup N] (f : G →+ N) : AddSubgroup.map f H.normalizer ≤ (AddSubgroup.map f H).normalizer
• AddSubgroup.characteristic_iff_le_map 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : H.Characteristic ↔ ∀ (ϕ : G ≃+ G), H ≤ AddSubgroup.map ϕ.toAddMonoidHom H
• AddSubgroup.characteristic_iff_map_le 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} : H.Characteristic ↔ ∀ (ϕ : G ≃+ G), AddSubgroup.map ϕ.toAddMonoidHom H ≤ H
• AddSubgroup.map_equiv_normalizer_eq 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) (f : G ≃+ N) : AddSubgroup.map f.toAddMonoidHom H.normalizer = (AddSubgroup.map f.toAddMonoidHom H).normalizer
• AddSubgroup.Normal.map 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} (h : H.Normal) (f : G →+ N) (hf : Function.Surjective ⇑f) : (AddSubgroup.map f H).Normal
• AddSubgroup.map_normalizer_eq_of_bijective 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] (H : AddSubgroup G) {f : G →+ N} (hf : Function.Bijective ⇑f) : AddSubgroup.map f H.normalizer = (AddSubgroup.map f H).normalizer
• AddSubgroup.le_prod_iff 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} {J : AddSubgroup (G × N)} : J ≤ H.prod K ↔ AddSubgroup.map (AddMonoidHom.fst G N) J ≤ H ∧ AddSubgroup.map (AddMonoidHom.snd G N) J ≤ K
• AddSubgroup.prod_le_iff 📋 Mathlib.Algebra.Group.Subgroup.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_5} [AddGroup N] {H : AddSubgroup G} {K : AddSubgroup N} {J : AddSubgroup (G × N)} : H.prod K ≤ J ↔ AddSubgroup.map (AddMonoidHom.inl G N) H ≤ J ∧ AddSubgroup.map (AddMonoidHom.inr G N) K ≤ J
• AddMonoidHom.map_zmultiples 📋 Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
  {G : Type u_1} [AddGroup G] {N : Type u_3} [AddGroup N] (f : G →+ N) (x : G) : AddSubgroup.map f (AddSubgroup.zmultiples x) = AddSubgroup.zmultiples (f x)
• 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)
• AddSubgroup.pointwise_smul_def 📋 Mathlib.Algebra.GroupWithZero.Subgroup
  {M : Type u_3} {A : Type u_4} [Monoid M] [AddGroup A] [DistribMulAction M A] {a : M} (S : AddSubgroup A) : a • S = AddSubgroup.map ((DistribMulAction.toAddMonoidEnd M A) a) S
• QuotientAddGroup.map_mk'_self 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] : AddSubgroup.map (QuotientAddGroup.mk' N) N = ⊥
• QuotientAddGroup.ker_lift 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] {M : Type x} [AddMonoid M] (φ : G →+ M) (HN : N ≤ φ.ker) : (QuotientAddGroup.lift N φ HN).ker = AddSubgroup.map (QuotientAddGroup.mk' N) φ.ker
• QuotientAddGroup.ker_map 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] {H : Type v} [AddGroup H] (M : AddSubgroup H) [M.Normal] (f : G →+ H) (h : N ≤ AddSubgroup.comap f M) : (QuotientAddGroup.map N M f h).ker = AddSubgroup.map (QuotientAddGroup.mk' N) (AddSubgroup.comap f M)
• QuotientAddGroup.congr 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [G'.Normal] [H'.Normal] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : G ⧸ G' ≃+ H ⧸ H'
• QuotientAddGroup.congr.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Defs
  {G : Type u} [AddGroup G] {H : Type v} [AddGroup H] (G' : AddSubgroup G) (H' : AddSubgroup H) [G'.Normal] [H'.Normal] (e : G ≃+ H) (he : AddSubgroup.map (↑e) G' = H') : QuotientAddGroup.congr G' H' e he = { toFun := ⇑(QuotientAddGroup.map G' H' ↑e ⋯), invFun := ⇑(QuotientAddGroup.map H' G' ↑e.symm ⋯), left_inv := ⋯, right_inv := ⋯, map_add' := ⋯ }
• AddSubgroup.card_map_of_injective 📋 Mathlib.Algebra.Group.Subgroup.Finite
  {G : Type u_1} [AddGroup G] {H : Type u_3} [AddGroup H] {K : AddSubgroup G} {f : G →+ H} (hf : Function.Injective ⇑f) : Nat.card ↥(AddSubgroup.map f K) = Nat.card ↥K
• AddSubgroup.pi_le_iff 📋 Mathlib.Algebra.Group.Subgroup.Finite
  {η : Type u_3} {f : η → Type u_4} [(i : η) → AddGroup (f i)] [DecidableEq η] [Finite η] {H : (i : η) → AddSubgroup (f i)} {J : AddSubgroup ((i : η) → f i)} : AddSubgroup.pi Set.univ H ≤ J ↔ ∀ (i : η), AddSubgroup.map (AddMonoidHom.single f i) (H i) ≤ J
• AddSubgroup.card_subtype 📋 Mathlib.Algebra.Group.Subgroup.Finite
  {G : Type u_1} [AddGroup G] (K : AddSubgroup G) (L : AddSubgroup ↥K) : Nat.card ↥(AddSubgroup.map K.subtype L) = Nat.card ↥L
• QuotientAddGroup.map_normal 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] : (AddSubgroup.map (QuotientAddGroup.mk' N) M).Normal
• QuotientAddGroup.comap_map_mk' 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N H : AddSubgroup G) [N.Normal] : AddSubgroup.comap (QuotientAddGroup.mk' N) (AddSubgroup.map (QuotientAddGroup.mk' N) H) = N ⊔ H
• QuotientAddGroup.strictMono_comap_prod_map 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] : StrictMono fun H => (AddSubgroup.comap N.subtype H, AddSubgroup.map (QuotientAddGroup.mk' N) H)
• QuotientAddGroup.quotientQuotientEquivQuotientAux 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) : (G ⧸ N) ⧸ AddSubgroup.map (QuotientAddGroup.mk' N) M →+ G ⧸ M
• QuotientAddGroup.quotientQuotientEquivQuotient 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) : (G ⧸ N) ⧸ AddSubgroup.map (QuotientAddGroup.mk' N) M ≃+ G ⧸ M
• QuotientAddGroup.quotientQuotientEquivQuotientAux.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) : QuotientAddGroup.quotientQuotientEquivQuotientAux N M h = QuotientAddGroup.lift (AddSubgroup.map (QuotientAddGroup.mk' N) M) (QuotientAddGroup.map N M (AddMonoidHom.id G) h) ⋯
• QuotientAddGroup.comapMk'OrderIso.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [hn : N.Normal] : QuotientAddGroup.comapMk'OrderIso N = { toFun := fun H' => ⟨AddSubgroup.comap (QuotientAddGroup.mk' N) H', ⋯⟩, invFun := fun H => AddSubgroup.map (QuotientAddGroup.mk' N) ↑H, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
• QuotientAddGroup.quotientQuotientEquivQuotient.eq_1 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) : QuotientAddGroup.quotientQuotientEquivQuotient N M h = (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h).toAddEquiv (QuotientAddGroup.map M (AddSubgroup.map (QuotientAddGroup.mk' N) M) (QuotientAddGroup.mk' N) ⋯) ⋯ ⋯
• QuotientAddGroup.quotientQuotientEquivQuotientAux_mk_mk 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) (x : G) : (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) ↑↑x = ↑x
• QuotientAddGroup.quotientQuotientEquivQuotientAux_mk 📋 Mathlib.GroupTheory.QuotientGroup.Basic
  {G : Type u} [AddGroup G] (N : AddSubgroup G) [nN : N.Normal] (M : AddSubgroup G) [nM : M.Normal] (h : N ≤ M) (x : G ⧸ N) : (QuotientAddGroup.quotientQuotientEquivQuotientAux N M h) ↑x = (QuotientAddGroup.map N M (AddMonoidHom.id G) h) x
• AddAction.stabilizer_vadd_eq_stabilizer_map_conj 📋 Mathlib.GroupTheory.GroupAction.Basic
  {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] (g : G) (a : α) : AddAction.stabilizer G (g +ᵥ a) = AddSubgroup.map (AddEquiv.toAddMonoidHom (Additive.toMul (AddAut.conj g))) (AddAction.stabilizer G a)
• AddAction.stabilizerEquivStabilizer.eq_1 📋 Mathlib.GroupTheory.GroupAction.Basic
  {G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {g : G} {a b : α} (hg : b = g +ᵥ a) : AddAction.stabilizerEquivStabilizer hg = (AddEquiv.addSubgroupMap (Additive.toMul (AddAut.conj g)) (AddAction.stabilizer G a)).trans (AddEquiv.addSubgroupCongr ⋯)
• AddSubgroup.relIndex_comap 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G' →+ G) (K : AddSubgroup G') : (AddSubgroup.comap f H).relIndex K = H.relIndex (AddSubgroup.map f K)
• AddSubgroup.relIndex_ker 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (K : AddSubgroup G) (f : G →+ G') : f.ker.relIndex K = Nat.card ↥(AddSubgroup.map f K)
• AddSubgroup.card_map_dvd 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') : Nat.card ↥(AddSubgroup.map f H) ∣ Nat.card ↥H
• AddSubgroup.dvd_index_map 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : f.ker ≤ H) : H.index ∣ (AddSubgroup.map f H).index
• AddSubgroup.index_map_of_bijective 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] {f : G →+ G'} (hf : Function.Bijective ⇑f) (H : AddSubgroup G) : (AddSubgroup.map f H).index = H.index
• AddSubgroup.index_map 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (f : G →+ G') : (AddSubgroup.map f H).index = (H ⊔ f.ker).index * f.range.index
• AddSubgroup.index_map_dvd 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : Function.Surjective ⇑f) : (AddSubgroup.map f H).index ∣ H.index
• AddSubgroup.index_map_subtype 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (K : AddSubgroup ↥H) : (AddSubgroup.map H.subtype K).index = K.index * H.index
• AddSubgroup.relIndex_map_map_of_injective 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] {f : G →+ G'} (H K : AddSubgroup G) (hf : Function.Injective ⇑f) : (AddSubgroup.map f H).relIndex (AddSubgroup.map f K) = H.relIndex K
• AddSubgroup.index_map_of_injective 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf : Function.Injective ⇑f) : (AddSubgroup.map f H).index = H.index * f.range.index
• AddSubgroup.relIndex_map_map 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (f : G →+ G') (H K : AddSubgroup G) : (AddSubgroup.map f H).relIndex (AddSubgroup.map f K) = (H ⊔ f.ker).relIndex (K ⊔ f.ker)
• AddSubgroup.index_map_eq 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) {f : G →+ G'} (hf1 : Function.Surjective ⇑f) (hf2 : f.ker ≤ H) : (AddSubgroup.map f H).index = H.index
• AddSubgroup.index_map_equiv 📋 Mathlib.GroupTheory.Index
  {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddGroup G'] (H : AddSubgroup G) (e : G ≃+ G') : (AddSubgroup.map (↑e) H).index = H.index
• DenseRange.topologicalClosure_map_addSubgroup 📋 Mathlib.Topology.Algebra.Group.Basic
  {G : Type w} {H : Type x} [TopologicalSpace G] [AddGroup G] [IsTopologicalAddGroup G] [AddGroup H] [TopologicalSpace H] [IsTopologicalAddGroup H] {f : G →+ H} (hf : Continuous ⇑f) (hf' : DenseRange ⇑f) {s : AddSubgroup G} (hs : s.topologicalClosure = ⊤) : (AddSubgroup.map f s).topologicalClosure = ⊤
• NormedAddGroupHom.comp_range 📋 Mathlib.Analysis.Normed.Group.Hom
  {V₁ : Type u_3} {V₂ : Type u_4} {V₃ : Type u_5} [SeminormedAddCommGroup V₁] [SeminormedAddCommGroup V₂] [SeminormedAddCommGroup V₃] (f : NormedAddGroupHom V₁ V₂) (g : NormedAddGroupHom V₂ V₃) : (g.comp f).range = AddSubgroup.map g.toAddMonoidHom f.range
• AddAction.map_stabilizer_le 📋 Mathlib.Algebra.Pointwise.Stabilizer
  {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] (f : G →+ H) (s : Set G) : AddSubgroup.map f (AddAction.stabilizer G s) ≤ AddAction.stabilizer H (⇑f '' s)
• AddSubgroup.goursatFst.eq_1 📋 Mathlib.GroupTheory.Goursat
  {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] (I : AddSubgroup (G × H)) : I.goursatFst = AddSubgroup.map ((AddMonoidHom.fst G H).comp I.subtype) ((AddMonoidHom.snd G H).comp I.subtype).ker
• AddSubgroup.goursatSnd.eq_1 📋 Mathlib.GroupTheory.Goursat
  {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] (I : AddSubgroup (G × H)) : I.goursatSnd = AddSubgroup.map ((AddMonoidHom.snd G H).comp I.subtype) ((AddMonoidHom.fst G H).comp I.subtype).ker
• AddSubgroup.goursat 📋 Mathlib.GroupTheory.Goursat
  {G : Type u_1} {H : Type u_2} [AddGroup G] [AddGroup H] {I : AddSubgroup (G × H)} : ∃ G' H' M N, ∃ (x : M.Normal) (x_1 : N.Normal), ∃ e, I = AddSubgroup.map (G'.subtype.prodMap H'.subtype) (AddSubgroup.comap ((QuotientAddGroup.mk' M).prodMap (QuotientAddGroup.mk' N)) e.toAddMonoidHom.graph)
• AddAction.IsBlock.of_addSubgroup_of_conjugate 📋 Mathlib.GroupTheory.GroupAction.Blocks
  {G : Type u_3} [AddGroup G] {X : Type u_4} [AddAction G X] {B : Set X} {H : AddSubgroup G} (hB : AddAction.IsBlock (↥H) B) (g : G) : AddAction.IsBlock (↥(AddSubgroup.map (AddEquiv.toAddMonoidHom (Additive.toMul (AddAut.conj g))) H)) (g +ᵥ B)
• SubAddAction.fixingAddSubgroup_vadd_eq_fixingAddSubgroup_map_conj 📋 Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
  {M : Type u_1} {α : Type u_2} [AddGroup M] [AddAction M α] (s : Set α) (g : M) : fixingAddSubgroup M (g +ᵥ s) = AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (fixingAddSubgroup M s)
• SubAddAction.fixingAddSubgroup_map_conj_eq 📋 Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
  {M : Type u_1} {α : Type u_2} [AddGroup M] [AddAction M α] {s t : Set α} {g : M} (hg : g +ᵥ t = s) : AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (fixingAddSubgroup M t) = fixingAddSubgroup M s
• SubAddAction.fixingAddSubgroupEquivFixingAddSubgroup.eq_1 📋 Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
  {M : Type u_1} {α : Type u_2} [AddGroup M] [AddAction M α] {s t : Set α} {g : M} (hg : g +ᵥ t = s) : SubAddAction.fixingAddSubgroupEquivFixingAddSubgroup hg = (AddEquiv.addSubgroupMap (AddAut.conj g) (fixingAddSubgroup M t)).trans (AddEquiv.addSubgroupCongr ⋯)
• SubAddAction.fixingAddSubgroup_of_insert 📋 Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup
  {M : Type u_1} {α : Type u_2} [AddGroup M] [AddAction M α] (a : α) (s : Set ↥(SubAddAction.ofStabilizer M a)) : fixingAddSubgroup M (insert a ((fun x => ↑x) '' s)) = AddSubgroup.map (AddAction.stabilizer M a).subtype (fixingAddSubgroup (↥(AddAction.stabilizer M a)) s)
• NumberField.Units.dirichletUnitTheorem.map_logEmbedding_sup_torsion 📋 Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
  {K : Type u_1} [Field K] [NumberField K] (s : AddSubgroup (Additive (NumberField.RingOfIntegers K)ˣ)) : AddSubgroup.map (NumberField.Units.logEmbedding K) (s ⊔ Subgroup.toAddSubgroup (NumberField.Units.torsion K)) = AddSubgroup.map (NumberField.Units.logEmbedding K) s
• NumberField.Units.span_basisOfIsMaxRank 📋 Mathlib.NumberTheory.NumberField.Units.Regulator
  {K : Type u_1} [Field K] [NumberField K] {u : Fin (NumberField.Units.rank K) → (NumberField.RingOfIntegers K)ˣ} (hu : NumberField.Units.IsMaxRank u) : (Submodule.span ℤ (Set.range ⇑(NumberField.Units.basisOfIsMaxRank hu))).toAddSubgroup = AddSubgroup.map (NumberField.Units.logEmbedding K) (Subgroup.toAddSubgroup (Subgroup.closure (Set.range u)))

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 ee8c038 serving mathlib revision 7a9e177