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Mathlib.Algebra.Group.Submonoid.MulOpposite

Submonoid of opposite monoids #

For every monoid M, we construct an equivalence between submonoids of M and that of Mᵐᵒᵖ.

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def AddSubmonoid.op {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid M) :
AddSubmonoid Mᵃᵒᵖ

Pull an additive submonoid back to an opposite submonoid along AddOpposite.unop

Equations
  • x.op = { carrier := AddOpposite.unop ⁻¹' ↑x, add_mem' := ⋯, zero_mem' := ⋯ }
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@[simp]
theorem AddSubmonoid.op_coe {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid M) :
↑x.op = AddOpposite.unop ⁻¹' ↑x
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@[simp]
theorem Submonoid.op_coe {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
↑x.op = MulOpposite.unop ⁻¹' ↑x
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def Submonoid.op {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
Submonoid Mᵐᵒᵖ

Pull a submonoid back to an opposite submonoid along MulOpposite.unop

Equations
  • x.op = { carrier := MulOpposite.unop ⁻¹' ↑x, mul_mem' := ⋯, one_mem' := ⋯ }
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@[simp]
theorem AddSubmonoid.mem_op {M : Type u_2} [AddZeroClass M] {x : Mᵃᵒᵖ} {S : AddSubmonoid M} :
x ∈ S.op ↔ AddOpposite.unop x ∈ S
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@[simp]
theorem Submonoid.mem_op {M : Type u_2} [MulOneClass M] {x : Mᵐᵒᵖ} {S : Submonoid M} :
x ∈ S.op ↔ MulOpposite.unop x ∈ S
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def AddSubmonoid.unop {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid Mᵃᵒᵖ) :
AddSubmonoid M

Pull an opposite additive submonoid back to a submonoid along AddOpposite.op

Equations
  • x.unop = { carrier := AddOpposite.op ⁻¹' ↑x, add_mem' := ⋯, zero_mem' := ⋯ }
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@[simp]
theorem Submonoid.unop_coe {M : Type u_2} [MulOneClass M] (x : Submonoid Mᵐᵒᵖ) :
↑x.unop = MulOpposite.op ⁻¹' ↑x
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@[simp]
theorem AddSubmonoid.unop_coe {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid Mᵃᵒᵖ) :
↑x.unop = AddOpposite.op ⁻¹' ↑x
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def Submonoid.unop {M : Type u_2} [MulOneClass M] (x : Submonoid Mᵐᵒᵖ) :
Submonoid M

Pull an opposite submonoid back to a submonoid along MulOpposite.op

Equations
  • x.unop = { carrier := MulOpposite.op ⁻¹' ↑x, mul_mem' := ⋯, one_mem' := ⋯ }
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@[simp]
theorem AddSubmonoid.mem_unop {M : Type u_2} [AddZeroClass M] {x : M} {S : AddSubmonoid Mᵃᵒᵖ} :
x ∈ S.unop ↔ AddOpposite.op x ∈ S
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@[simp]
theorem Submonoid.mem_unop {M : Type u_2} [MulOneClass M] {x : M} {S : Submonoid Mᵐᵒᵖ} :
x ∈ S.unop ↔ MulOpposite.op x ∈ S
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@[simp]
theorem AddSubmonoid.unop_op {M : Type u_2} [AddZeroClass M] (S : AddSubmonoid M) :
S.op.unop = S
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@[simp]
theorem Submonoid.unop_op {M : Type u_2} [MulOneClass M] (S : Submonoid M) :
S.op.unop = S
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@[simp]
theorem AddSubmonoid.op_unop {M : Type u_2} [AddZeroClass M] (S : AddSubmonoid Mᵃᵒᵖ) :
S.unop.op = S
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@[simp]
theorem Submonoid.op_unop {M : Type u_2} [MulOneClass M] (S : Submonoid Mᵐᵒᵖ) :
S.unop.op = S

Lattice results #

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theorem AddSubmonoid.op_le_iff {M : Type u_2} [AddZeroClass M] {S₁ : AddSubmonoid M} {S₂ : AddSubmonoid Mᵃᵒᵖ} :
S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
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theorem Submonoid.op_le_iff {M : Type u_2} [MulOneClass M] {S₁ : Submonoid M} {S₂ : Submonoid Mᵐᵒᵖ} :
S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
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theorem AddSubmonoid.le_op_iff {M : Type u_2} [AddZeroClass M] {S₁ : AddSubmonoid Mᵃᵒᵖ} {S₂ : AddSubmonoid M} :
S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
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theorem Submonoid.le_op_iff {M : Type u_2} [MulOneClass M] {S₁ : Submonoid Mᵐᵒᵖ} {S₂ : Submonoid M} :
S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
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@[simp]
theorem AddSubmonoid.op_le_op_iff {M : Type u_2} [AddZeroClass M] {S₁ : AddSubmonoid M} {S₂ : AddSubmonoid M} :
S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
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@[simp]
theorem Submonoid.op_le_op_iff {M : Type u_2} [MulOneClass M] {S₁ : Submonoid M} {S₂ : Submonoid M} :
S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
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@[simp]
theorem AddSubmonoid.unop_le_unop_iff {M : Type u_2} [AddZeroClass M] {S₁ : AddSubmonoid Mᵃᵒᵖ} {S₂ : AddSubmonoid Mᵃᵒᵖ} :
S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
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@[simp]
theorem Submonoid.unop_le_unop_iff {M : Type u_2} [MulOneClass M] {S₁ : Submonoid Mᵐᵒᵖ} {S₂ : Submonoid Mᵐᵒᵖ} :
S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
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def AddSubmonoid.opEquiv {M : Type u_2} [AddZeroClass M] :
AddSubmonoid M ≃o AddSubmonoid Mᵃᵒᵖ

A additive submonoid H of G determines an additive submonoid H.op of the opposite group Gᵐᵒᵖ.

Equations
  • AddSubmonoid.opEquiv = { toFun := AddSubmonoid.op, invFun := AddSubmonoid.unop, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
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@[simp]
theorem AddSubmonoid.opEquiv_apply {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid M) :
AddSubmonoid.opEquiv x = x.op
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@[simp]
theorem Submonoid.opEquiv_apply {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
Submonoid.opEquiv x = x.op
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@[simp]
theorem Submonoid.opEquiv_symm_apply {M : Type u_2} [MulOneClass M] (x : Submonoid Mᵐᵒᵖ) :
(RelIso.symm Submonoid.opEquiv) x = x.unop
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@[simp]
theorem AddSubmonoid.opEquiv_symm_apply {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid Mᵃᵒᵖ) :
(RelIso.symm AddSubmonoid.opEquiv) x = x.unop
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def Submonoid.opEquiv {M : Type u_2} [MulOneClass M] :
Submonoid M ≃o Submonoid Mᵐᵒᵖ

A submonoid H of G determines a submonoid H.op of the opposite group Gᵐᵒᵖ.

Equations
  • Submonoid.opEquiv = { toFun := Submonoid.op, invFun := Submonoid.unop, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
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theorem AddSubmonoid.op_injective {M : Type u_2} [AddZeroClass M] :
Function.Injective AddSubmonoid.op
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theorem Submonoid.op_injective {M : Type u_2} [MulOneClass M] :
Function.Injective Submonoid.op
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theorem AddSubmonoid.unop_injective {M : Type u_2} [AddZeroClass M] :
Function.Injective AddSubmonoid.unop
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theorem Submonoid.unop_injective {M : Type u_2} [MulOneClass M] :
Function.Injective Submonoid.unop
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@[simp]
theorem AddSubmonoid.op_inj {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid M} {T : AddSubmonoid M} :
S.op = T.op ↔ S = T
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@[simp]
theorem Submonoid.op_inj {M : Type u_2} [MulOneClass M] {S : Submonoid M} {T : Submonoid M} :
S.op = T.op ↔ S = T
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@[simp]
theorem AddSubmonoid.unop_inj {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid Mᵃᵒᵖ} {T : AddSubmonoid Mᵃᵒᵖ} :
S.unop = T.unop ↔ S = T
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@[simp]
theorem Submonoid.unop_inj {M : Type u_2} [MulOneClass M] {S : Submonoid Mᵐᵒᵖ} {T : Submonoid Mᵐᵒᵖ} :
S.unop = T.unop ↔ S = T
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@[simp]
theorem AddSubmonoid.op_bot {M : Type u_2} [AddZeroClass M] :
⊥.op = ⊥
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@[simp]
theorem Submonoid.op_bot {M : Type u_2} [MulOneClass M] :
⊥.op = ⊥
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@[simp]
theorem AddSubmonoid.op_eq_bot {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid M} :
S.op = ⊥ ↔ S = ⊥
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@[simp]
theorem Submonoid.op_eq_bot {M : Type u_2} [MulOneClass M] {S : Submonoid M} :
S.op = ⊥ ↔ S = ⊥
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@[simp]
theorem AddSubmonoid.unop_bot {M : Type u_2} [AddZeroClass M] :
⊥.unop = ⊥
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@[simp]
theorem Submonoid.unop_bot {M : Type u_2} [MulOneClass M] :
⊥.unop = ⊥
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@[simp]
theorem AddSubmonoid.unop_eq_bot {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid Mᵃᵒᵖ} :
S.unop = ⊥ ↔ S = ⊥
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@[simp]
theorem Submonoid.unop_eq_bot {M : Type u_2} [MulOneClass M] {S : Submonoid Mᵐᵒᵖ} :
S.unop = ⊥ ↔ S = ⊥
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@[simp]
theorem AddSubmonoid.op_top {M : Type u_2} [AddZeroClass M] :
⊤.op = ⊤
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@[simp]
theorem Submonoid.op_top {M : Type u_2} [MulOneClass M] :
⊤.op = ⊤
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@[simp]
theorem AddSubmonoid.op_eq_top {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid M} :
S.op = ⊤ ↔ S = ⊤
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@[simp]
theorem Submonoid.op_eq_top {M : Type u_2} [MulOneClass M] {S : Submonoid M} :
S.op = ⊤ ↔ S = ⊤
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@[simp]
theorem AddSubmonoid.unop_top {M : Type u_2} [AddZeroClass M] :
⊤.unop = ⊤
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@[simp]
theorem Submonoid.unop_top {M : Type u_2} [MulOneClass M] :
⊤.unop = ⊤
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@[simp]
theorem AddSubmonoid.unop_eq_top {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid Mᵃᵒᵖ} :
S.unop = ⊤ ↔ S = ⊤
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@[simp]
theorem Submonoid.unop_eq_top {M : Type u_2} [MulOneClass M] {S : Submonoid Mᵐᵒᵖ} :
S.unop = ⊤ ↔ S = ⊤
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theorem AddSubmonoid.op_sup {M : Type u_2} [AddZeroClass M] (S₁ : AddSubmonoid M) (S₂ : AddSubmonoid M) :
(S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
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theorem Submonoid.op_sup {M : Type u_2} [MulOneClass M] (S₁ : Submonoid M) (S₂ : Submonoid M) :
(S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
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theorem AddSubmonoid.unop_sup {M : Type u_2} [AddZeroClass M] (S₁ : AddSubmonoid Mᵃᵒᵖ) (S₂ : AddSubmonoid Mᵃᵒᵖ) :
(S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
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theorem Submonoid.unop_sup {M : Type u_2} [MulOneClass M] (S₁ : Submonoid Mᵐᵒᵖ) (S₂ : Submonoid Mᵐᵒᵖ) :
(S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
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theorem AddSubmonoid.op_inf {M : Type u_2} [AddZeroClass M] (S₁ : AddSubmonoid M) (S₂ : AddSubmonoid M) :
(S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
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theorem Submonoid.op_inf {M : Type u_2} [MulOneClass M] (S₁ : Submonoid M) (S₂ : Submonoid M) :
(S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
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theorem AddSubmonoid.unop_inf {M : Type u_2} [AddZeroClass M] (S₁ : AddSubmonoid Mᵃᵒᵖ) (S₂ : AddSubmonoid Mᵃᵒᵖ) :
(S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
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theorem Submonoid.unop_inf {M : Type u_2} [MulOneClass M] (S₁ : Submonoid Mᵐᵒᵖ) (S₂ : Submonoid Mᵐᵒᵖ) :
(S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
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theorem AddSubmonoid.op_sSup {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid M)) :
(sSup S).op = sSup (AddSubmonoid.unop ⁻¹' S)
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theorem Submonoid.op_sSup {M : Type u_2} [MulOneClass M] (S : Set (Submonoid M)) :
(sSup S).op = sSup (Submonoid.unop ⁻¹' S)
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theorem AddSubmonoid.unop_sSup {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid Mᵃᵒᵖ)) :
(sSup S).unop = sSup (AddSubmonoid.op ⁻¹' S)
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theorem Submonoid.unop_sSup {M : Type u_2} [MulOneClass M] (S : Set (Submonoid Mᵐᵒᵖ)) :
(sSup S).unop = sSup (Submonoid.op ⁻¹' S)
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theorem AddSubmonoid.op_sInf {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid M)) :
(sInf S).op = sInf (AddSubmonoid.unop ⁻¹' S)
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theorem Submonoid.op_sInf {M : Type u_2} [MulOneClass M] (S : Set (Submonoid M)) :
(sInf S).op = sInf (Submonoid.unop ⁻¹' S)
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theorem AddSubmonoid.unop_sInf {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid Mᵃᵒᵖ)) :
(sInf S).unop = sInf (AddSubmonoid.op ⁻¹' S)
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theorem Submonoid.unop_sInf {M : Type u_2} [MulOneClass M] (S : Set (Submonoid Mᵐᵒᵖ)) :
(sInf S).unop = sInf (Submonoid.op ⁻¹' S)
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theorem AddSubmonoid.op_iSup {ι : Sort u_1} {M : Type u_2} [AddZeroClass M] (S : ι → AddSubmonoid M) :
(iSup S).op = ⨆ (i : ι), (S i).op
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theorem Submonoid.op_iSup {ι : Sort u_1} {M : Type u_2} [MulOneClass M] (S : ι → Submonoid M) :
(iSup S).op = ⨆ (i : ι), (S i).op
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theorem AddSubmonoid.unop_iSup {ι : Sort u_1} {M : Type u_2} [AddZeroClass M] (S : ι → AddSubmonoid Mᵃᵒᵖ) :
(iSup S).unop = ⨆ (i : ι), (S i).unop
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theorem Submonoid.unop_iSup {ι : Sort u_1} {M : Type u_2} [MulOneClass M] (S : ι → Submonoid Mᵐᵒᵖ) :
(iSup S).unop = ⨆ (i : ι), (S i).unop
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theorem AddSubmonoid.op_iInf {ι : Sort u_1} {M : Type u_2} [AddZeroClass M] (S : ι → AddSubmonoid M) :
(iInf S).op = ⨅ (i : ι), (S i).op
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theorem Submonoid.op_iInf {ι : Sort u_1} {M : Type u_2} [MulOneClass M] (S : ι → Submonoid M) :
(iInf S).op = ⨅ (i : ι), (S i).op
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theorem AddSubmonoid.unop_iInf {ι : Sort u_1} {M : Type u_2} [AddZeroClass M] (S : ι → AddSubmonoid Mᵃᵒᵖ) :
(iInf S).unop = ⨅ (i : ι), (S i).unop
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theorem Submonoid.unop_iInf {ι : Sort u_1} {M : Type u_2} [MulOneClass M] (S : ι → Submonoid Mᵐᵒᵖ) :
(iInf S).unop = ⨅ (i : ι), (S i).unop
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theorem AddSubmonoid.op_closure {M : Type u_2} [AddZeroClass M] (s : Set M) :
(AddSubmonoid.closure s).op = AddSubmonoid.closure (AddOpposite.unop ⁻¹' s)
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theorem Submonoid.op_closure {M : Type u_2} [MulOneClass M] (s : Set M) :
(Submonoid.closure s).op = Submonoid.closure (MulOpposite.unop ⁻¹' s)
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theorem AddSubmonoid.unop_closure {M : Type u_2} [AddZeroClass M] (s : Set Mᵃᵒᵖ) :
(AddSubmonoid.closure s).unop = AddSubmonoid.closure (AddOpposite.op ⁻¹' s)
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theorem Submonoid.unop_closure {M : Type u_2} [MulOneClass M] (s : Set Mᵐᵒᵖ) :
(Submonoid.closure s).unop = Submonoid.closure (MulOpposite.op ⁻¹' s)
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def AddSubmonoid.equivOp {M : Type u_2} [AddZeroClass M] (H : AddSubmonoid M) :
↥H ≃ ↥H.op

Bijection between an additive submonoid H and its opposite.

Equations
  • H.equivOp = AddOpposite.opEquiv.subtypeEquiv ⋯
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@[simp]
theorem Submonoid.equivOp_symm_apply_coe {M : Type u_2} [MulOneClass M] (H : Submonoid M) (b : ↥H.op) :
↑(H.equivOp.symm b) = MulOpposite.unop ↑b
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@[simp]
theorem AddSubmonoid.equivOp_symm_apply_coe {M : Type u_2} [AddZeroClass M] (H : AddSubmonoid M) (b : ↥H.op) :
↑(H.equivOp.symm b) = AddOpposite.unop ↑b
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@[simp]
theorem Submonoid.equivOp_apply_coe {M : Type u_2} [MulOneClass M] (H : Submonoid M) (a : ↥H) :
↑(H.equivOp a) = MulOpposite.op ↑a
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@[simp]
theorem AddSubmonoid.equivOp_apply_coe {M : Type u_2} [AddZeroClass M] (H : AddSubmonoid M) (a : ↥H) :
↑(H.equivOp a) = AddOpposite.op ↑a
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def Submonoid.equivOp {M : Type u_2} [MulOneClass M] (H : Submonoid M) :
↥H ≃ ↥H.op

Bijection between a submonoid H and its opposite.

Equations
  • H.equivOp = MulOpposite.opEquiv.subtypeEquiv ⋯