Documentation

Mathlib.Algebra.Ring.Subsemiring.MulOpposite

Subsemiring of opposite semirings #

For every semiring R, we construct an equivalence between subsemirings of R and that of Rᵐᵒᵖ.

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def Subsemiring.op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
Subsemiring Rᵐᵒᵖ

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

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      @[simp]
      theorem Subsemiring.coe_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
      ↑S.op = MulOpposite.unop ⁻¹' ↑S
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      @[simp]
      theorem Subsemiring.op_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
      S.op.toSubmonoid = S.op
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      @[simp]
      theorem Subsemiring.mem_op {R : Type u_2} [NonAssocSemiring R] {x : Rᵐᵒᵖ} {S : Subsemiring R} :
      x ∈ S.op ↔ MulOpposite.unop x ∈ S
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      def Subsemiring.unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
      Subsemiring R

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

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          @[simp]
          theorem Subsemiring.coe_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
          ↑S.unop = MulOpposite.op ⁻¹' ↑S
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          @[simp]
          theorem Subsemiring.unop_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
          S.unop.toSubmonoid = S.unop
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          @[simp]
          theorem Subsemiring.mem_unop {R : Type u_2} [NonAssocSemiring R] {x : R} {S : Subsemiring Rᵐᵒᵖ} :
          x ∈ S.unop ↔ MulOpposite.op x ∈ S
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          @[simp]
          theorem Subsemiring.unop_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
          S.op.unop = S
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          @[simp]
          theorem Subsemiring.op_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
          S.unop.op = S

          Lattice results #

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

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

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

              Bijection between a subsemiring S and its opposite.

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

                  Bijection between a subsemiring S and MulOpposite of its opposite.

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

                      Bijection between MulOpposite of a subsemiring S and its opposite.

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