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Mathlib.Algebra.Order.GroupWithZero.Unbundled.OrderIso

Multiplication by a positive element as an order isomorphism #

def OrderIso.mulLeft₀ {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] (a : G₀) (ha : 0 < a) :
G₀ ≃o G₀

Equiv.mulLeft₀ as an order isomorphism.

Equations
    Instances For
      @[simp]
      theorem OrderIso.mulLeft₀_symm_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) :
      @[simp]
      theorem OrderIso.mulLeft₀_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) :
      (mulLeft₀ a ha) x = a * x
      theorem OrderIso.mulLeft₀_symm {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [PosMulReflectLT G₀] (a : G₀) (ha : 0 < a) :
      def OrderIso.mulRight₀ {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) :
      G₀ ≃o G₀

      Equiv.mulRight₀ as an order isomorphism.

      Equations
        Instances For
          @[simp]
          theorem OrderIso.mulRight₀_symm_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) :
          @[simp]
          theorem OrderIso.mulRight₀_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x : G₀) :
          (mulRight₀ a ha) x = x * a
          theorem OrderIso.mulRight₀_symm {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) :
          def OrderIso.divRight₀ {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) :
          G₀ ≃o G₀

          Equiv.divRight₀ as an order isomorphism.

          Equations
            Instances For
              @[simp]
              theorem OrderIso.divRight₀_symm_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x✝ : G₀) :
              (RelIso.symm (divRight₀ a ha)) x✝ = x✝ * a
              @[simp]
              theorem OrderIso.divRight₀_apply {G₀ : Type u_1} [GroupWithZero G₀] [PartialOrder G₀] [MulPosReflectLT G₀] (a : G₀) (ha : 0 < a) (x✝ : G₀) :
              (divRight₀ a ha) x✝ = x✝ / a