Field structure on the multiplicative/additive opposite

instance AddOpposite.ratCast (α : Type u_1) [RatCast α] : RatCast αᵃᵒᵖ

Equations

• AddOpposite.ratCast α = { ratCast := fun (n : ℚ) => AddOpposite.op ↑n }

instance MulOpposite.ratCast (α : Type u_1) [RatCast α] : RatCast αᵐᵒᵖ

Equations

• MulOpposite.ratCast α = { ratCast := fun (n : ℚ) => MulOpposite.op ↑n }

@[simp]

theorem AddOpposite.op_ratCast {α : Type u_1} [RatCast α] (q : ℚ) : AddOpposite.op ↑q = ↑q

@[simp]

theorem MulOpposite.op_ratCast {α : Type u_1} [RatCast α] (q : ℚ) : MulOpposite.op ↑q = ↑q

@[simp]

theorem AddOpposite.unop_ratCast {α : Type u_1} [RatCast α] (q : ℚ) : AddOpposite.unop ↑q = ↑q

@[simp]

theorem MulOpposite.unop_ratCast {α : Type u_1} [RatCast α] (q : ℚ) : MulOpposite.unop ↑q = ↑q

instance MulOpposite.divisionSemiring (α : Type u_1) [DivisionSemiring α] : DivisionSemiring αᵐᵒᵖ

Equations

• MulOpposite.divisionSemiring α = let __src := MulOpposite.groupWithZero α; let __src_1 := MulOpposite.semiring α; DivisionSemiring.mk ⋯ GroupWithZero.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance MulOpposite.divisionRing (α : Type u_1) [DivisionRing α] : DivisionRing αᵐᵒᵖ

Equations

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

instance MulOpposite.semifield (α : Type u_1) [Semifield α] : Semifield αᵐᵒᵖ

Equations

• MulOpposite.semifield α = let __src := MulOpposite.divisionSemiring α; let __src_1 := MulOpposite.commSemiring α; Semifield.mk ⋯ DivisionSemiring.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance MulOpposite.field (α : Type u_1) [Field α] : Field αᵐᵒᵖ

Equations

• MulOpposite.field α = let __src := MulOpposite.divisionRing α; let __src_1 := MulOpposite.commRing α; Field.mk ⋯ DivisionRing.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionRing.qsmul ⋯

instance AddOpposite.divisionSemiring {α : Type u_1} [DivisionSemiring α] : DivisionSemiring αᵃᵒᵖ

Equations

• AddOpposite.divisionSemiring = let __src := AddOpposite.groupWithZero α; let __src_1 := AddOpposite.semiring α; DivisionSemiring.mk ⋯ GroupWithZero.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance AddOpposite.divisionRing {α : Type u_1} [DivisionRing α] : DivisionRing αᵃᵒᵖ

Equations

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

instance AddOpposite.semifield {α : Type u_1} [Semifield α] : Semifield αᵃᵒᵖ

Equations

• AddOpposite.semifield = let __src := AddOpposite.divisionSemiring; let __src_1 := AddOpposite.commSemiring α; Semifield.mk ⋯ DivisionSemiring.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance AddOpposite.field {α : Type u_1} [Field α] : Field αᵃᵒᵖ

Equations

• AddOpposite.field = let __src := AddOpposite.divisionRing; let __src_1 := AddOpposite.commRing α; Field.mk ⋯ DivisionRing.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionRing.qsmul ⋯