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.

source
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

Equations
  • S.op = { toSubmonoid := S.op, add_mem' := ⋯, zero_mem' := ⋯ }
Instances For
    source
    @[simp]
    theorem Subsemiring.coe_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
    ↑S.op = MulOpposite.unop ⁻¹' ↑S
    source
    @[simp]
    theorem Subsemiring.op_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
    S.op.toSubmonoid = S.op
    source
    @[simp]
    theorem Subsemiring.mem_op {R : Type u_2} [NonAssocSemiring R] {x : Rᵐᵒᵖ} {S : Subsemiring R} :
    x ∈ S.op ↔ MulOpposite.unop x ∈ S
    source
    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

    Equations
    • S.unop = { toSubmonoid := S.unop, add_mem' := ⋯, zero_mem' := ⋯ }
    Instances For
      source
      @[simp]
      theorem Subsemiring.coe_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
      ↑S.unop = MulOpposite.op ⁻¹' ↑S
      source
      @[simp]
      theorem Subsemiring.unop_toSubmonoid {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
      S.unop.toSubmonoid = S.unop
      source
      @[simp]
      theorem Subsemiring.mem_unop {R : Type u_2} [NonAssocSemiring R] {x : R} {S : Subsemiring Rᵐᵒᵖ} :
      x ∈ S.unop ↔ MulOpposite.op x ∈ S
      source
      @[simp]
      theorem Subsemiring.unop_op {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
      S.op.unop = S
      source
      @[simp]
      theorem Subsemiring.op_unop {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
      S.unop.op = S

      Lattice results #

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

      Equations
      • Subsemiring.opEquiv = { toFun := Subsemiring.op, invFun := Subsemiring.unop, left_inv := ⋯, right_inv := ⋯, map_rel_iff' := ⋯ }
      Instances For
        source
        @[simp]
        theorem Subsemiring.opEquiv_symm_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring Rᵐᵒᵖ) :
        (RelIso.symm Subsemiring.opEquiv) S = S.unop
        source
        @[simp]
        theorem Subsemiring.opEquiv_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
        Subsemiring.opEquiv S = S.op
        source
        theorem Subsemiring.op_injective {R : Type u_2} [NonAssocSemiring R] :
        Function.Injective Subsemiring.op
        source
        theorem Subsemiring.unop_injective {R : Type u_2} [NonAssocSemiring R] :
        Function.Injective Subsemiring.unop
        source
        @[simp]
        theorem Subsemiring.op_inj {R : Type u_2} [NonAssocSemiring R] {S T : Subsemiring R} :
        S.op = T.op ↔ S = T
        source
        @[simp]
        theorem Subsemiring.unop_inj {R : Type u_2} [NonAssocSemiring R] {S T : Subsemiring Rᵐᵒᵖ} :
        S.unop = T.unop ↔ S = T
        source
        @[simp]
        theorem Subsemiring.op_bot {R : Type u_2} [NonAssocSemiring R] :
        ⊥.op = ⊥
        source
        @[simp]
        theorem Subsemiring.op_eq_bot {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R} :
        S.op = ⊥ ↔ S = ⊥
        source
        @[simp]
        theorem Subsemiring.unop_bot {R : Type u_2} [NonAssocSemiring R] :
        ⊥.unop = ⊥
        source
        @[simp]
        theorem Subsemiring.unop_eq_bot {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring Rᵐᵒᵖ} :
        S.unop = ⊥ ↔ S = ⊥
        source
        @[simp]
        theorem Subsemiring.op_top {R : Type u_2} [NonAssocSemiring R] :
        ⊤.op = ⊤
        source
        @[simp]
        theorem Subsemiring.op_eq_top {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring R} :
        S.op = ⊤ ↔ S = ⊤
        source
        @[simp]
        theorem Subsemiring.unop_top {R : Type u_2} [NonAssocSemiring R] :
        ⊤.unop = ⊤
        source
        @[simp]
        theorem Subsemiring.unop_eq_top {R : Type u_2} [NonAssocSemiring R] {S : Subsemiring Rᵐᵒᵖ} :
        S.unop = ⊤ ↔ S = ⊤
        source
        theorem Subsemiring.op_sup {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R) :
        (S₁ ⊔ S₂).op = S₁.op ⊔ S₂.op
        source
        theorem Subsemiring.unop_sup {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring Rᵐᵒᵖ) :
        (S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop
        source
        theorem Subsemiring.op_inf {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring R) :
        (S₁ ⊓ S₂).op = S₁.op ⊓ S₂.op
        source
        theorem Subsemiring.unop_inf {R : Type u_2} [NonAssocSemiring R] (S₁ S₂ : Subsemiring Rᵐᵒᵖ) :
        (S₁ ⊓ S₂).unop = S₁.unop ⊓ S₂.unop
        source
        theorem Subsemiring.op_sSup {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R)) :
        (sSup S).op = sSup (Subsemiring.unop ⁻¹' S)
        source
        theorem Subsemiring.unop_sSup {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring Rᵐᵒᵖ)) :
        (sSup S).unop = sSup (Subsemiring.op ⁻¹' S)
        source
        theorem Subsemiring.op_sInf {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring R)) :
        (sInf S).op = sInf (Subsemiring.unop ⁻¹' S)
        source
        theorem Subsemiring.unop_sInf {R : Type u_2} [NonAssocSemiring R] (S : Set (Subsemiring Rᵐᵒᵖ)) :
        (sInf S).unop = sInf (Subsemiring.op ⁻¹' S)
        source
        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
        source
        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
        source
        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
        source
        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
        source
        theorem Subsemiring.op_closure {R : Type u_2} [NonAssocSemiring R] (s : Set R) :
        (Subsemiring.closure s).op = Subsemiring.closure (MulOpposite.unop ⁻¹' s)
        source
        theorem Subsemiring.unop_closure {R : Type u_2} [NonAssocSemiring R] (s : Set Rᵐᵒᵖ) :
        (Subsemiring.closure s).unop = Subsemiring.closure (MulOpposite.op ⁻¹' s)
        source
        def Subsemiring.addEquivOp {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) :
        ↥S ≃+ ↥S.op

        Bijection between a subsemiring S and its opposite.

        Equations
        • S.addEquivOp = { toEquiv := S.equivOp, map_add' := ⋯ }
        Instances For
          source
          @[simp]
          theorem Subsemiring.addEquivOp_apply_coe {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a : ↥S.toSubmonoid) :
          ↑(S.addEquivOp a) = MulOpposite.op ↑a
          source
          @[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
          source
          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.

          Equations
          • S.ringEquivOpMop = { toEquiv := (S.addEquivOp.trans MulOpposite.opAddEquiv).toEquiv, map_mul' := ⋯, map_add' := ⋯ }
          Instances For
            source
            @[simp]
            theorem Subsemiring.ringEquivOpMop_apply {R : Type u_2} [NonAssocSemiring R] (S : Subsemiring R) (a✝ : ↥S) :
            S.ringEquivOpMop a✝ = MulOpposite.op (S.addEquivOp a✝)
            source
            @[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✝)
            source
            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.

            Equations
            • S.mopRingEquivOp = { toEquiv := (MulOpposite.opAddEquiv.symm.trans S.addEquivOp).toEquiv, map_mul' := ⋯, map_add' := ⋯ }
            Instances For
              source
              @[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✝)
              source
              @[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✝)