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Mathlib.Algebra.Group.Units.Opposite

Units in multiplicative and additive opposites #

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def Units.opEquiv {M : Type u_2} [Monoid M] :
Mᵐᵒᵖˣ ≃* Mˣᵐᵒᵖ

The units of the opposites are equivalent to the opposites of the units.

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      def AddUnits.opEquiv {M : Type u_2} [AddMonoid M] :
      AddUnits Mᵃᵒᵖ ≃+ (AddUnits M)ᵃᵒᵖ

      The additive units of the additive opposites are equivalent to the additive opposites of the additive units.

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          @[simp]
          theorem Units.coe_unop_opEquiv {M : Type u_2} [Monoid M] (u : Mᵐᵒᵖˣ) :
          ↑(MulOpposite.unop (opEquiv u)) = MulOpposite.unop ↑u
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          @[simp]
          theorem AddUnits.coe_unop_opEquiv {M : Type u_2} [AddMonoid M] (u : AddUnits Mᵃᵒᵖ) :
          ↑(AddOpposite.unop (opEquiv u)) = AddOpposite.unop ↑u
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          @[simp]
          theorem Units.coe_opEquiv_symm {M : Type u_2} [Monoid M] (u : Mˣᵐᵒᵖ) :
          ↑(opEquiv.symm u) = MulOpposite.op ↑(MulOpposite.unop u)
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          @[simp]
          theorem AddUnits.coe_opEquiv_symm {M : Type u_2} [AddMonoid M] (u : (AddUnits M)ᵃᵒᵖ) :
          ↑(opEquiv.symm u) = AddOpposite.op ↑(AddOpposite.unop u)
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          theorem IsUnit.op {M : Type u_2} [Monoid M] {m : M} (h : IsUnit m) :
          IsUnit (MulOpposite.op m)
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          theorem IsAddUnit.op {M : Type u_2} [AddMonoid M] {m : M} (h : IsAddUnit m) :
          IsAddUnit (AddOpposite.op m)
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          theorem IsUnit.unop {M : Type u_2} [Monoid M] {m : Mᵐᵒᵖ} (h : IsUnit m) :
          IsUnit (MulOpposite.unop m)
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          theorem IsAddUnit.unop {M : Type u_2} [AddMonoid M] {m : Mᵃᵒᵖ} (h : IsAddUnit m) :
          IsAddUnit (AddOpposite.unop m)
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          @[simp]
          theorem isUnit_op {M : Type u_2} [Monoid M] {m : M} :
          IsUnit (MulOpposite.op m) ↔ IsUnit m
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          @[simp]
          theorem isAddUnit_op {M : Type u_2} [AddMonoid M] {m : M} :
          IsAddUnit (AddOpposite.op m) ↔ IsAddUnit m
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          @[simp]
          theorem isUnit_unop {M : Type u_2} [Monoid M] {m : Mᵐᵒᵖ} :
          IsUnit (MulOpposite.unop m) ↔ IsUnit m
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          @[simp]
          theorem isAddUnit_unop {M : Type u_2} [AddMonoid M] {m : Mᵃᵒᵖ} :
          IsAddUnit (AddOpposite.unop m) ↔ IsAddUnit m