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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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  • One or more equations did not get rendered due to their size.
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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