Documentation

Mathlib.Algebra.Ring.Subring.MulOpposite

Subring of opposite rings #

For every ring R, we construct an equivalence between subrings of R and that of Rᵐᵒᵖ.

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def Subring.op {R : Type u_2} [Ring R] (S : Subring R) :
Subring Rᵐᵒᵖ

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

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      @[simp]
      theorem Subring.coe_op {R : Type u_2} [Ring R] (S : Subring R) :
      ↑S.op = MulOpposite.unop ⁻¹' ↑S
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      @[simp]
      theorem Subring.op_toSubsemiring {R : Type u_2} [Ring R] (S : Subring R) :
      S.op.toSubsemiring = S.op
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      @[simp]
      theorem Subring.mem_op {R : Type u_2} [Ring R] {x : Rᵐᵒᵖ} {S : Subring R} :
      x ∈ S.op ↔ MulOpposite.unop x ∈ S
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      def Subring.unop {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ) :
      Subring R

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

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          @[simp]
          theorem Subring.coe_unop {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ) :
          ↑S.unop = MulOpposite.op ⁻¹' ↑S
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          @[simp]
          theorem Subring.unop_toSubsemiring {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ) :
          S.unop.toSubsemiring = S.unop
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          @[simp]
          theorem Subring.mem_unop {R : Type u_2} [Ring R] {x : R} {S : Subring Rᵐᵒᵖ} :
          x ∈ S.unop ↔ MulOpposite.op x ∈ S
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          @[simp]
          theorem Subring.unop_op {R : Type u_2} [Ring R] (S : Subring R) :
          S.op.unop = S
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          @[simp]
          theorem Subring.op_unop {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ) :
          S.unop.op = S

          Lattice results #

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          theorem Subring.op_le_iff {R : Type u_2} [Ring R] {S₁ : Subring R} {S₂ : Subring Rᵐᵒᵖ} :
          S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop
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          theorem Subring.le_op_iff {R : Type u_2} [Ring R] {S₁ : Subring Rᵐᵒᵖ} {S₂ : Subring R} :
          S₁ ≤ S₂.op ↔ S₁.unop ≤ S₂
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          @[simp]
          theorem Subring.op_le_op_iff {R : Type u_2} [Ring R] {S₁ S₂ : Subring R} :
          S₁.op ≤ S₂.op ↔ S₁ ≤ S₂
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          @[simp]
          theorem Subring.unop_le_unop_iff {R : Type u_2} [Ring R] {S₁ S₂ : Subring Rᵐᵒᵖ} :
          S₁.unop ≤ S₂.unop ↔ S₁ ≤ S₂
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          def Subring.opEquiv {R : Type u_2} [Ring R] :
          Subring R ≃o Subring Rᵐᵒᵖ

          A subring S of R determines a subring S.op of the opposite ring Rᵐᵒᵖ.

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              theorem Subring.opEquiv_symm_apply {R : Type u_2} [Ring R] (S : Subring Rᵐᵒᵖ) :
              (RelIso.symm opEquiv) S = S.unop
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              @[simp]
              theorem Subring.opEquiv_apply {R : Type u_2} [Ring R] (S : Subring R) :
              opEquiv S = S.op
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              theorem Subring.op_injective {R : Type u_2} [Ring R] :
              Function.Injective Subring.op
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              theorem Subring.unop_injective {R : Type u_2} [Ring R] :
              Function.Injective Subring.unop
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              @[simp]
              theorem Subring.op_inj {R : Type u_2} [Ring R] {S T : Subring R} :
              S.op = T.op ↔ S = T
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              @[simp]
              theorem Subring.unop_inj {R : Type u_2} [Ring R] {S T : Subring Rᵐᵒᵖ} :
              S.unop = T.unop ↔ S = T
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              @[simp]
              theorem Subring.op_bot {R : Type u_2} [Ring R] :
              ⊥.op = ⊥
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              @[simp]
              theorem Subring.op_eq_bot {R : Type u_2} [Ring R] {S : Subring R} :
              S.op = ⊥ ↔ S = ⊥
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              @[simp]
              theorem Subring.unop_bot {R : Type u_2} [Ring R] :
              ⊥.unop = ⊥
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              @[simp]
              theorem Subring.unop_eq_bot {R : Type u_2} [Ring R] {S : Subring Rᵐᵒᵖ} :
              S.unop = ⊥ ↔ S = ⊥
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              @[simp]
              theorem Subring.op_top {R : Type u_2} [Ring R] :
              ⊤.op = ⊤
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              @[simp]
              theorem Subring.op_eq_top {R : Type u_2} [Ring R] {S : Subring R} :
              S.op = ⊤ ↔ S = ⊤
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              @[simp]
              theorem Subring.unop_top {R : Type u_2} [Ring R] :
              ⊤.unop = ⊤
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              @[simp]
              theorem Subring.unop_eq_top {R : Type u_2} [Ring R] {S : Subring Rᵐᵒᵖ} :
              S.unop = ⊤ ↔ S = ⊤
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              theorem Subring.op_sup {R : Type u_2} [Ring R] (S₁ S₂ : Subring R) :
              (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
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              theorem Subring.unop_sup {R : Type u_2} [Ring R] (S₁ S₂ : Subring Rᵐᵒᵖ) :
              (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
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              theorem Subring.op_inf {R : Type u_2} [Ring R] (S₁ S₂ : Subring R) :
              (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
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              theorem Subring.unop_inf {R : Type u_2} [Ring R] (S₁ S₂ : Subring Rᵐᵒᵖ) :
              (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
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              theorem Subring.op_sSup {R : Type u_2} [Ring R] (S : Set (Subring R)) :
              (sSup S).op = sSup (Subring.unop ⁻¹' S)
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              theorem Subring.unop_sSup {R : Type u_2} [Ring R] (S : Set (Subring Rᵐᵒᵖ)) :
              (sSup S).unop = sSup (Subring.op ⁻¹' S)
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              theorem Subring.op_sInf {R : Type u_2} [Ring R] (S : Set (Subring R)) :
              (sInf S).op = sInf (Subring.unop ⁻¹' S)
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              theorem Subring.unop_sInf {R : Type u_2} [Ring R] (S : Set (Subring Rᵐᵒᵖ)) :
              (sInf S).unop = sInf (Subring.op ⁻¹' S)
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              theorem Subring.op_iSup {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring R) :
              (iSup S).op = ⨆ (i : ι), (S i).op
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              theorem Subring.unop_iSup {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring Rᵐᵒᵖ) :
              (iSup S).unop = ⨆ (i : ι), (S i).unop
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              theorem Subring.op_iInf {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring R) :
              (iInf S).op = ⨅ (i : ι), (S i).op
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              theorem Subring.unop_iInf {ι : Sort u_1} {R : Type u_2} [Ring R] (S : ι → Subring Rᵐᵒᵖ) :
              (iInf S).unop = ⨅ (i : ι), (S i).unop
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              theorem Subring.op_closure {R : Type u_2} [Ring R] (s : Set R) :
              (closure s).op = closure (MulOpposite.unop ⁻¹' s)
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              theorem Subring.unop_closure {R : Type u_2} [Ring R] (s : Set Rᵐᵒᵖ) :
              (closure s).unop = closure (MulOpposite.op ⁻¹' s)
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              def Subring.addEquivOp {R : Type u_2} [Ring R] (S : Subring R) :
              ↥S ≃+ ↥S.op

              Bijection between a subring S and its opposite.

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                  theorem Subring.addEquivOp_symm_apply_coe {R : Type u_2} [Ring R] (S : Subring R) (b : ↥S.op) :
                  ↑(S.addEquivOp.symm b) = MulOpposite.unop ↑b
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                  @[simp]
                  theorem Subring.addEquivOp_apply_coe {R : Type u_2} [Ring R] (S : Subring R) (a : ↥S.toSubmonoid) :
                  ↑(S.addEquivOp a) = MulOpposite.op ↑a
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                  def Subring.ringEquivOpMop {R : Type u_2} [Ring R] (S : Subring R) :
                  ↥S ≃+* (↥S.op)ᵐᵒᵖ

                  Bijection between a subring S and MulOpposite of its opposite.

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                      theorem Subring.ringEquivOpMop_apply {R : Type u_2} [Ring R] (S : Subring R) (a✝ : ↥S.toSubsemiring) :
                      S.ringEquivOpMop a✝ = MulOpposite.op (S.addEquivOp a✝)
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                      @[simp]
                      theorem Subring.ringEquivOpMop_symm_apply_coe {R : Type u_2} [Ring R] (S : Subring R) (a✝ : (↥S.op)ᵐᵒᵖ) :
                      ↑(S.ringEquivOpMop.symm a✝) = MulOpposite.unop ↑(MulOpposite.unop a✝)
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                      def Subring.mopRingEquivOp {R : Type u_2} [Ring R] (S : Subring R) :
                      (↥S)ᵐᵒᵖ ≃+* ↥S.op

                      Bijection between MulOpposite of a subring S and its opposite.

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                          theorem Subring.mopRingEquivOp_apply_coe {R : Type u_2} [Ring R] (S : Subring R) (a✝ : (↥S.toSubsemiring)ᵐᵒᵖ) :
                          ↑(S.mopRingEquivOp a✝) = MulOpposite.op ↑(MulOpposite.unop a✝)
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                          @[simp]
                          theorem Subring.mopRingEquivOp_symm_apply {R : Type u_2} [Ring R] (S : Subring R) (a✝ : ↥S.op) :
                          S.mopRingEquivOp.symm a✝ = MulOpposite.op (S.addEquivOp.symm a✝)