Mul-opposite subgroups

Tags

subgroup, subgroups

theorem AddSubgroup.opposite.proof_2 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : ∀ {x : Gᵃᵒᵖ}, AddOpposite.unop x ∈ H → -AddOpposite.unop x ∈ H

theorem AddSubgroup.opposite.proof_5 {G : Type u_1} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) : ∀ {x : G}, AddOpposite.op x ∈ H → -AddOpposite.op x ∈ H

theorem AddSubgroup.opposite.proof_1 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : ∀ {a b : Gᵃᵒᵖ}, a ∈ AddOpposite.unop ⁻¹' ↑H → b ∈ AddOpposite.unop ⁻¹' ↑H → AddOpposite.unop b + AddOpposite.unop a ∈ H

theorem AddSubgroup.opposite.proof_4 {G : Type u_1} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) : 0 ∈ H

def AddSubgroup.opposite {G : Type u_1} [AddGroup G] : AddSubgroup G ≃ AddSubgroup Gᵃᵒᵖ

An additive subgroup H of G determines an additive subgroup H.opposite of the opposite additive group Gᵃᵒᵖ.

theorem AddSubgroup.opposite.proof_6 {G : Type u_1} [AddGroup G] : ∀ (x : AddSubgroup G), (fun H => { toAddSubmonoid := { toAddSubsemigroup := { carrier := AddOpposite.op ⁻¹' ↑H, add_mem' := (_ : ∀ {a b : G}, a ∈ AddOpposite.op ⁻¹' ↑H → b ∈ AddOpposite.op ⁻¹' ↑H → { unop' := b } + { unop' := a } ∈ H) }, zero_mem' := (_ : 0 ∈ H) }, neg_mem' := (_ : ∀ {x : G}, AddOpposite.op x ∈ H → -AddOpposite.op x ∈ H) }) ((fun H => { toAddSubmonoid := { toAddSubsemigroup := { carrier := AddOpposite.unop ⁻¹' ↑H, add_mem' := (_ : ∀ {a b : Gᵃᵒᵖ}, a ∈ AddOpposite.unop ⁻¹' ↑H → b ∈ AddOpposite.unop ⁻¹' ↑H → AddOpposite.unop b + AddOpposite.unop a ∈ H) }, zero_mem' := (_ : 0 ∈ H) }, neg_mem' := (_ : ∀ {x : Gᵃᵒᵖ}, AddOpposite.unop x ∈ H → -AddOpposite.unop x ∈ H) }) x) = x

theorem AddSubgroup.opposite.proof_7 {G : Type u_1} [AddGroup G] : ∀ (x : AddSubgroup Gᵃᵒᵖ), (fun H => { toAddSubmonoid := { toAddSubsemigroup := { carrier := AddOpposite.unop ⁻¹' ↑H, add_mem' := (_ : ∀ {a b : Gᵃᵒᵖ}, a ∈ AddOpposite.unop ⁻¹' ↑H → b ∈ AddOpposite.unop ⁻¹' ↑H → AddOpposite.unop b + AddOpposite.unop a ∈ H) }, zero_mem' := (_ : 0 ∈ H) }, neg_mem' := (_ : ∀ {x : Gᵃᵒᵖ}, AddOpposite.unop x ∈ H → -AddOpposite.unop x ∈ H) }) ((fun H => { toAddSubmonoid := { toAddSubsemigroup := { carrier := AddOpposite.op ⁻¹' ↑H, add_mem' := (_ : ∀ {a b : G}, a ∈ AddOpposite.op ⁻¹' ↑H → b ∈ AddOpposite.op ⁻¹' ↑H → { unop' := b } + { unop' := a } ∈ H) }, zero_mem' := (_ : 0 ∈ H) }, neg_mem' := (_ : ∀ {x : G}, AddOpposite.op x ∈ H → -AddOpposite.op x ∈ H) }) x) = x

theorem AddSubgroup.opposite.proof_3 {G : Type u_1} [AddGroup G] (H : AddSubgroup Gᵃᵒᵖ) : ∀ {a b : G}, a ∈ AddOpposite.op ⁻¹' ↑H → b ∈ AddOpposite.op ⁻¹' ↑H → { unop' := b } + { unop' := a } ∈ H

def Subgroup.opposite {G : Type u_1} [Group G] : Subgroup G ≃ Subgroup Gᵐᵒᵖ

A subgroup H of G determines a subgroup H.opposite of the opposite group Gᵐᵒᵖ.

theorem AddSubgroup.oppositeEquiv.proof_1 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : ∀ (x : G), x ∈ H ↔ x ∈ H

def AddSubgroup.oppositeEquiv {G : Type u_1} [AddGroup G] (H : AddSubgroup G) : { x // x ∈ H } ≃ { x // x ∈ ↑AddSubgroup.opposite H }

Bijection between an additive subgroup H and its opposite.

@[simp]

theorem Subgroup.oppositeEquiv_symm_apply_coe {G : Type u_1} [Group G] (H : Subgroup G) (b : { b // (fun x => x ∈ ↑Subgroup.opposite H) b }) : ↑(↑(Subgroup.oppositeEquiv H).symm b) = MulOpposite.unop ↑b

@[simp]

theorem AddSubgroup.oppositeEquiv_symm_apply_coe {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (b : { b // (fun x => x ∈ ↑AddSubgroup.opposite H) b }) : ↑(↑(AddSubgroup.oppositeEquiv H).symm b) = AddOpposite.unop ↑b

@[simp]

theorem AddSubgroup.oppositeEquiv_apply_coe {G : Type u_1} [AddGroup G] (H : AddSubgroup G) (a : { a // (fun x => x ∈ H) a }) : ↑(↑(AddSubgroup.oppositeEquiv H) a) = AddOpposite.op ↑a

@[simp]

theorem Subgroup.oppositeEquiv_apply_coe {G : Type u_1} [Group G] (H : Subgroup G) (a : { a // (fun x => x ∈ H) a }) : ↑(↑(Subgroup.oppositeEquiv H) a) = MulOpposite.op ↑a

def Subgroup.oppositeEquiv {G : Type u_1} [Group G] (H : Subgroup G) : { x // x ∈ H } ≃ { x // x ∈ ↑Subgroup.opposite H }

Bijection between a subgroup H and its opposite.

instance AddSubgroup.instEncodableSubtypeAddOppositeMemAddSubgroupAddGroupInstMembershipInstSetLikeAddSubgroupCoeEquivInstFunLikeEquivOpposite {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Encodable { x // x ∈ H }] : Encodable { x // x ∈ ↑AddSubgroup.opposite H }

instance Subgroup.instEncodableSubtypeMulOppositeMemSubgroupGroupInstMembershipInstSetLikeSubgroupCoeEquivInstFunLikeEquivOpposite {G : Type u_1} [Group G] (H : Subgroup G) [Encodable { x // x ∈ H }] : Encodable { x // x ∈ ↑Subgroup.opposite H }

instance AddSubgroup.instCountableSubtypeAddOppositeMemAddSubgroupAddGroupInstMembershipInstSetLikeAddSubgroupCoeEquivInstFunLikeEquivOpposite {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Countable { x // x ∈ H }] : Countable { x // x ∈ ↑AddSubgroup.opposite H }

theorem AddSubgroup.instCountableSubtypeAddOppositeMemAddSubgroupAddGroupInstMembershipInstSetLikeAddSubgroupCoeEquivInstFunLikeEquivOpposite.proof_1 {G : Type u_1} [AddGroup G] (H : AddSubgroup G) [Countable { x // x ∈ H }] : Countable { x // x ∈ ↑AddSubgroup.opposite H }

instance Subgroup.instCountableSubtypeMulOppositeMemSubgroupGroupInstMembershipInstSetLikeSubgroupCoeEquivInstFunLikeEquivOpposite {G : Type u_1} [Group G] (H : Subgroup G) [Countable { x // x ∈ H }] : Countable { x // x ∈ ↑Subgroup.opposite H }

theorem AddSubgroup.vadd_opposite_add {G : Type u_1} [AddGroup G] {H : AddSubgroup G} (x : G) (g : G) (h : { x // x ∈ ↑AddSubgroup.opposite H }) : h +ᵥ (g + x) = g + (h +ᵥ x)

theorem Subgroup.smul_opposite_mul {G : Type u_1} [Group G] {H : Subgroup G} (x : G) (g : G) (h : { x // x ∈ ↑Subgroup.opposite H }) : h • (g * x) = g * h • x