File: Mathlib/Algebra/Group/Subgroup/Order.lean

full_name: abs_mem_iff
code: @[simp] theorem abs_mem_iff {S G} [AddGroup G] [LinearOrder G] {_ : SetLike S G}
    [NegMemClass S G] {H : S} {x : G} : |x| ∈ H ↔ x ∈ H 
goal: ∀ {x : SetLike S G} [inst : NegMemClass S G] {H : S} {x_1 : G}, |x_1| ∈ H ↔ x_1 ∈ H

full_name: instIsModularLatticeSubgroup
code: @[to_additive]
instance : IsModularLattice (Subgroup C) 
goal: IsModularLattice (Subgroup C)

full_name: instIsModularLatticeAddSubgroup
code: @[to_additive]
instance : IsModularLattice (Subgroup C) 
goal: IsModularLattice (AddSubgroup C)

full_name: Subgroup.NormalizerCondition.normal_of_coatom
code: /-- In a group that satisfies the normalizer condition, every maximal subgroup is normal -/
theorem NormalizerCondition.normal_of_coatom (hnc : NormalizerCondition G) (hmax : IsCoatom H) :
    H.Normal 
goal: H.Normal

full_name: Subgroup.isCoatom_comap
code: @[simp]
theorem isCoatom_comap {H : Type*} [Group H] (f : G ≃* H) {K : Subgroup H} :
    IsCoatom (Subgroup.comap (f : G →* H) K) ↔ IsCoatom K 
goal: IsCoatom (Subgroup.comap (↑f) K) ↔ IsCoatom K

full_name: Subgroup.isCoatom_map
code: @[simp]
theorem isCoatom_map (f : G ≃* H) {K : Subgroup G} :
    IsCoatom (Subgroup.map (f : G →* H) K) ↔ IsCoatom K 
goal: IsCoatom (Subgroup.map (↑f) K) ↔ IsCoatom K

full_name: Subgroup.isCoatom_comap_of_surjective
code: lemma isCoatom_comap_of_surjective
    {H : Type*} [Group H] {φ : G →* H} (hφ : Function.Surjective φ)
    {M : Subgroup H} (hM : IsCoatom M) : IsCoatom (M.comap φ) 
goal: IsCoatom (Subgroup.comap φ M)

full_name: SubgroupClass.toOrderedCommGroup
code: /-- A subgroup of an `OrderedCommGroup` is an `OrderedCommGroup`. -/
@[to_additive "An additive subgroup of an `AddOrderedCommGroup` is an `AddOrderedCommGroup`."]
instance (priority := 75) toOrderedCommGroup [OrderedCommGroup G]
    [SubgroupClass S G] (H : S) : OrderedCommGroup H 
goal: OrderedCommGroup ↥H

full_name: AddSubgroupClass.toOrderedAddCommGroup
code: /-- A subgroup of an `OrderedCommGroup` is an `OrderedCommGroup`. -/
@[to_additive "An additive subgroup of an `AddOrderedCommGroup` is an `AddOrderedCommGroup`."]
instance (priority := 75) toOrderedCommGroup [OrderedCommGroup G]
    [SubgroupClass S G] (H : S) : OrderedCommGroup H 
goal: OrderedAddCommGroup ↥H

full_name: AddSubgroupClass.toLinearOrderedAddCommGroup
code: /-- A subgroup of a `LinearOrderedCommGroup` is a `LinearOrderedCommGroup`. -/
@[to_additive
      "An additive subgroup of a `LinearOrderedAddCommGroup` is a
        `LinearOrderedAddCommGroup`."]
instance (priority := 75) toLinearOrderedCommGroup [LinearOrderedCommGroup G]
    [SubgroupClass S G] (H : S) : LinearOrderedCommGroup H 
goal: LinearOrderedAddCommGroup ↥H

full_name: SubgroupClass.toLinearOrderedCommGroup
code: /-- A subgroup of a `LinearOrderedCommGroup` is a `LinearOrderedCommGroup`. -/
@[to_additive
      "An additive subgroup of a `LinearOrderedAddCommGroup` is a
        `LinearOrderedAddCommGroup`."]
instance (priority := 75) toLinearOrderedCommGroup [LinearOrderedCommGroup G]
    [SubgroupClass S G] (H : S) : LinearOrderedCommGroup H 
goal: LinearOrderedCommGroup ↥H

full_name: AddSubgroup.toOrderedAddCommGroup
code: /-- A subgroup of an `OrderedCommGroup` is an `OrderedCommGroup`. -/
@[to_additive "An `AddSubgroup` of an `AddOrderedCommGroup` is an `AddOrderedCommGroup`."]
instance toOrderedCommGroup [OrderedCommGroup G] (H : Subgroup G) :
    OrderedCommGroup H 
goal: OrderedAddCommGroup ↥H

full_name: Subgroup.toOrderedCommGroup
code: /-- A subgroup of an `OrderedCommGroup` is an `OrderedCommGroup`. -/
@[to_additive "An `AddSubgroup` of an `AddOrderedCommGroup` is an `AddOrderedCommGroup`."]
instance toOrderedCommGroup [OrderedCommGroup G] (H : Subgroup G) :
    OrderedCommGroup H 
goal: OrderedCommGroup ↥H

full_name: AddSubgroup.toLinearOrderedAddCommGroup
code: /-- A subgroup of a `LinearOrderedCommGroup` is a `LinearOrderedCommGroup`. -/
@[to_additive
      "An `AddSubgroup` of a `LinearOrderedAddCommGroup` is a
        `LinearOrderedAddCommGroup`."]
instance toLinearOrderedCommGroup [LinearOrderedCommGroup G] (H : Subgroup G) :
    LinearOrderedCommGroup H 
goal: LinearOrderedAddCommGroup ↥H

full_name: Subgroup.toLinearOrderedCommGroup
code: /-- A subgroup of a `LinearOrderedCommGroup` is a `LinearOrderedCommGroup`. -/
@[to_additive
      "An `AddSubgroup` of a `LinearOrderedAddCommGroup` is a
        `LinearOrderedAddCommGroup`."]
instance toLinearOrderedCommGroup [LinearOrderedCommGroup G] (H : Subgroup G) :
    LinearOrderedCommGroup H 
goal: LinearOrderedCommGroup ↥H