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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 Submonoid.op {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
Submonoid Mᵐᵒᵖ

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

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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

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          @[simp]
          theorem Submonoid.coe_op {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
          ↑x.op = MulOpposite.unop ⁻¹' ↑x
          source
          @[simp]
          theorem AddSubmonoid.coe_op {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid M) :
          ↑x.op = AddOpposite.unop ⁻¹' ↑x
          source
          @[simp]
          theorem Submonoid.mem_op {M : Type u_2} [MulOneClass M] {x : Mᵐᵒᵖ} {S : Submonoid M} :
          x ∈ S.op ↔ MulOpposite.unop x ∈ S
          source
          @[simp]
          theorem AddSubmonoid.mem_op {M : Type u_2} [AddZeroClass M] {x : Mᵃᵒᵖ} {S : AddSubmonoid M} :
          x ∈ S.op ↔ AddOpposite.unop x ∈ S
          source
          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

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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

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

                  Lattice results #

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

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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.

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                          @[simp]
                          theorem Submonoid.opEquiv_symm_apply {M : Type u_2} [MulOneClass M] (x : Submonoid Mᵐᵒᵖ) :
                          (RelIso.symm opEquiv) x = x.unop
                          source
                          @[simp]
                          theorem AddSubmonoid.opEquiv_symm_apply {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid Mᵃᵒᵖ) :
                          (RelIso.symm opEquiv) x = x.unop
                          source
                          @[simp]
                          theorem AddSubmonoid.opEquiv_apply {M : Type u_2} [AddZeroClass M] (x : AddSubmonoid M) :
                          opEquiv x = x.op
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                          @[simp]
                          theorem Submonoid.opEquiv_apply {M : Type u_2} [MulOneClass M] (x : Submonoid M) :
                          opEquiv x = x.op
                          source
                          theorem Submonoid.op_injective {M : Type u_2} [MulOneClass M] :
                          Function.Injective Submonoid.op
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                          theorem AddSubmonoid.op_injective {M : Type u_2} [AddZeroClass M] :
                          Function.Injective AddSubmonoid.op
                          source
                          theorem Submonoid.unop_injective {M : Type u_2} [MulOneClass M] :
                          Function.Injective Submonoid.unop
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                          theorem AddSubmonoid.unop_injective {M : Type u_2} [AddZeroClass M] :
                          Function.Injective AddSubmonoid.unop
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                          @[simp]
                          theorem Submonoid.op_inj {M : Type u_2} [MulOneClass M] {S T : Submonoid M} :
                          S.op = T.op ↔ S = T
                          source
                          @[simp]
                          theorem AddSubmonoid.op_inj {M : Type u_2} [AddZeroClass M] {S T : AddSubmonoid M} :
                          S.op = T.op ↔ S = T
                          source
                          @[simp]
                          theorem Submonoid.unop_inj {M : Type u_2} [MulOneClass M] {S T : Submonoid Mᵐᵒᵖ} :
                          S.unop = T.unop ↔ S = T
                          source
                          @[simp]
                          theorem AddSubmonoid.unop_inj {M : Type u_2} [AddZeroClass M] {S T : AddSubmonoid Mᵃᵒᵖ} :
                          S.unop = T.unop ↔ S = T
                          source
                          @[simp]
                          theorem Submonoid.op_bot {M : Type u_2} [MulOneClass M] :
                          ⊥.op = ⊥
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                          @[simp]
                          theorem AddSubmonoid.op_bot {M : Type u_2} [AddZeroClass M] :
                          ⊥.op = ⊥
                          source
                          @[simp]
                          theorem Submonoid.op_eq_bot {M : Type u_2} [MulOneClass M] {S : Submonoid M} :
                          S.op = ⊥ ↔ S = ⊥
                          source
                          @[simp]
                          theorem AddSubmonoid.op_eq_bot {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid M} :
                          S.op = ⊥ ↔ S = ⊥
                          source
                          @[simp]
                          theorem Submonoid.unop_bot {M : Type u_2} [MulOneClass M] :
                          ⊥.unop = ⊥
                          source
                          @[simp]
                          theorem AddSubmonoid.unop_bot {M : Type u_2} [AddZeroClass M] :
                          ⊥.unop = ⊥
                          source
                          @[simp]
                          theorem Submonoid.unop_eq_bot {M : Type u_2} [MulOneClass M] {S : Submonoid Mᵐᵒᵖ} :
                          S.unop = ⊥ ↔ S = ⊥
                          source
                          @[simp]
                          theorem AddSubmonoid.unop_eq_bot {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid Mᵃᵒᵖ} :
                          S.unop = ⊥ ↔ S = ⊥
                          source
                          @[simp]
                          theorem Submonoid.op_top {M : Type u_2} [MulOneClass M] :
                          ⊤.op = ⊤
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                          @[simp]
                          theorem AddSubmonoid.op_top {M : Type u_2} [AddZeroClass M] :
                          ⊤.op = ⊤
                          source
                          @[simp]
                          theorem Submonoid.op_eq_top {M : Type u_2} [MulOneClass M] {S : Submonoid M} :
                          S.op = ⊤ ↔ S = ⊤
                          source
                          @[simp]
                          theorem AddSubmonoid.op_eq_top {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid M} :
                          S.op = ⊤ ↔ S = ⊤
                          source
                          @[simp]
                          theorem Submonoid.unop_top {M : Type u_2} [MulOneClass M] :
                          ⊤.unop = ⊤
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                          @[simp]
                          theorem AddSubmonoid.unop_top {M : Type u_2} [AddZeroClass M] :
                          ⊤.unop = ⊤
                          source
                          @[simp]
                          theorem Submonoid.unop_eq_top {M : Type u_2} [MulOneClass M] {S : Submonoid Mᵐᵒᵖ} :
                          S.unop = ⊤ ↔ S = ⊤
                          source
                          @[simp]
                          theorem AddSubmonoid.unop_eq_top {M : Type u_2} [AddZeroClass M] {S : AddSubmonoid Mᵃᵒᵖ} :
                          S.unop = ⊤ ↔ S = ⊤
                          source
                          theorem Submonoid.op_sup {M : Type u_2} [MulOneClass M] (S₁ S₂ : Submonoid M) :
                          (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
                          source
                          theorem AddSubmonoid.op_sup {M : Type u_2} [AddZeroClass M] (S₁ S₂ : AddSubmonoid M) :
                          (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
                          source
                          theorem Submonoid.unop_sup {M : Type u_2} [MulOneClass M] (S₁ S₂ : Submonoid Mᵐᵒᵖ) :
                          (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
                          source
                          theorem AddSubmonoid.unop_sup {M : Type u_2} [AddZeroClass M] (S₁ S₂ : AddSubmonoid Mᵃᵒᵖ) :
                          (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
                          source
                          theorem Submonoid.op_inf {M : Type u_2} [MulOneClass M] (S₁ S₂ : Submonoid M) :
                          (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
                          source
                          theorem AddSubmonoid.op_inf {M : Type u_2} [AddZeroClass M] (S₁ S₂ : AddSubmonoid M) :
                          (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
                          source
                          theorem Submonoid.unop_inf {M : Type u_2} [MulOneClass M] (S₁ S₂ : Submonoid Mᵐᵒᵖ) :
                          (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
                          source
                          theorem AddSubmonoid.unop_inf {M : Type u_2} [AddZeroClass M] (S₁ S₂ : AddSubmonoid Mᵃᵒᵖ) :
                          (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
                          source
                          theorem Submonoid.op_sSup {M : Type u_2} [MulOneClass M] (S : Set (Submonoid M)) :
                          (sSup S).op = sSup (Submonoid.unop ⁻¹' S)
                          source
                          theorem AddSubmonoid.op_sSup {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid M)) :
                          (sSup S).op = sSup (AddSubmonoid.unop ⁻¹' S)
                          source
                          theorem Submonoid.unop_sSup {M : Type u_2} [MulOneClass M] (S : Set (Submonoid Mᵐᵒᵖ)) :
                          (sSup S).unop = sSup (Submonoid.op ⁻¹' S)
                          source
                          theorem AddSubmonoid.unop_sSup {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid Mᵃᵒᵖ)) :
                          (sSup S).unop = sSup (AddSubmonoid.op ⁻¹' S)
                          source
                          theorem Submonoid.op_sInf {M : Type u_2} [MulOneClass M] (S : Set (Submonoid M)) :
                          (sInf S).op = sInf (Submonoid.unop ⁻¹' S)
                          source
                          theorem AddSubmonoid.op_sInf {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid M)) :
                          (sInf S).op = sInf (AddSubmonoid.unop ⁻¹' S)
                          source
                          theorem Submonoid.unop_sInf {M : Type u_2} [MulOneClass M] (S : Set (Submonoid Mᵐᵒᵖ)) :
                          (sInf S).unop = sInf (Submonoid.op ⁻¹' S)
                          source
                          theorem AddSubmonoid.unop_sInf {M : Type u_2} [AddZeroClass M] (S : Set (AddSubmonoid Mᵃᵒᵖ)) :
                          (sInf S).unop = sInf (AddSubmonoid.op ⁻¹' S)
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          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
                          source
                          theorem Submonoid.op_closure {M : Type u_2} [MulOneClass M] (s : Set M) :
                          (closure s).op = closure (MulOpposite.unop ⁻¹' s)
                          source
                          theorem AddSubmonoid.op_closure {M : Type u_2} [AddZeroClass M] (s : Set M) :
                          (closure s).op = closure (AddOpposite.unop ⁻¹' s)
                          source
                          theorem Submonoid.unop_closure {M : Type u_2} [MulOneClass M] (s : Set Mᵐᵒᵖ) :
                          (closure s).unop = closure (MulOpposite.op ⁻¹' s)
                          source
                          theorem AddSubmonoid.unop_closure {M : Type u_2} [AddZeroClass M] (s : Set Mᵃᵒᵖ) :
                          (closure s).unop = closure (AddOpposite.op ⁻¹' s)
                          source
                          def Submonoid.equivOp {M : Type u_2} [MulOneClass M] (H : Submonoid M) :
                          ↥H ≃ ↥H.op

                          Bijection between a submonoid H and its opposite.

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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.

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                                Instances For
                                  source
                                  @[simp]
                                  theorem Submonoid.equivOp_apply_coe {M : Type u_2} [MulOneClass M] (H : Submonoid M) (a : ↥H) :
                                  ↑(H.equivOp a) = MulOpposite.op ↑a
                                  source
                                  @[simp]
                                  theorem AddSubmonoid.equivOp_apply_coe {M : Type u_2} [AddZeroClass M] (H : AddSubmonoid M) (a : ↥H) :
                                  ↑(H.equivOp a) = AddOpposite.op ↑a
                                  source
                                  @[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
                                  source
                                  @[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