Multiplicative opposite and algebraic operations on it

In this file we define MulOpposite α = αᵐᵒᵖ to be the multiplicative opposite of α. It inherits all additive algebraic structures on α (in other files), and reverses the order of multipliers in multiplicative structures, i.e., op (x * y) = op y * op x, where MulOpposite.op is the canonical map from α to αᵐᵒᵖ.

We also define AddOpposite α = αᵃᵒᵖ to be the additive opposite of α. It inherits all multiplicative algebraic structures on α (in other files), and reverses the order of summands in additive structures, i.e. op (x + y) = op y + op x, where AddOpposite.op is the canonical map from α to αᵃᵒᵖ.

Notation

• αᵐᵒᵖ = MulOpposite α
• αᵃᵒᵖ = AddOpposite α

Implementation notes

In mathlib3 αᵐᵒᵖ was just a type synonym for α, marked irreducible after the API was developed. In mathlib4 we use a structure with one field, because it is not possible to change the reducibility of a declaration after its definition, and because Lean 4 has definitional eta reduction for structures (Lean 3 does not).

Tags

multiplicative opposite, additive opposite

structure PreOpposite (α : Type u_3) : Type u_3

Auxiliary type to implement MulOpposite and AddOpposite.

It turns out to be convenient to have MulOpposite α = AddOpposite α true by definition, in the same way that it is convenient to have Additive α = α; this means that we also get the defeq AddOpposite (Additive α) = MulOpposite α, which is convenient when working with quotients.

This is a compromise between making MulOpposite α = AddOpposite α = α (what we had in Lean 3) and having no defeqs within those three types (which we had as of mathlib4#1036).

• op' :: (
• unop' : α

  The element of α represented by x : PreOpposite α.

• )

def MulOpposite (α : Type u_3) : Type u_3

Multiplicative opposite of a type. This type inherits all additive structures on α and reverses left and right in multiplication.

Equations

• αᵐᵒᵖ = PreOpposite α

def AddOpposite (α : Type u_3) : Type u_3

Additive opposite of a type. This type inherits all multiplicative structures on α and reverses left and right in addition.

Equations

• αᵃᵒᵖ = PreOpposite α

def MulOpposite.op {α : Type u_1} : α → αᵐᵒᵖ

The element of MulOpposite α that represents x : α.

Equations

• MulOpposite.op = PreOpposite.op'

def AddOpposite.op {α : Type u_1} : α → αᵃᵒᵖ

The element of αᵃᵒᵖ that represents x : α.

Equations

• AddOpposite.op = PreOpposite.op'

def MulOpposite.unop {α : Type u_1} : αᵐᵒᵖ → α

The element of α represented by x : αᵐᵒᵖ.

Equations

• MulOpposite.unop = PreOpposite.unop'

def AddOpposite.unop {α : Type u_1} : αᵃᵒᵖ → α

The element of α represented by x : αᵃᵒᵖ.

Equations

• AddOpposite.unop = PreOpposite.unop'

@[simp]

theorem MulOpposite.unop_op {α : Type u_1} (x : α) : MulOpposite.unop (MulOpposite.op x) = x

@[simp]

theorem AddOpposite.unop_op {α : Type u_1} (x : α) : AddOpposite.unop (AddOpposite.op x) = x

@[simp]

theorem MulOpposite.op_unop {α : Type u_1} (x : αᵐᵒᵖ) : MulOpposite.op (MulOpposite.unop x) = x

@[simp]

theorem AddOpposite.op_unop {α : Type u_1} (x : αᵃᵒᵖ) : AddOpposite.op (AddOpposite.unop x) = x

@[simp]

theorem MulOpposite.op_comp_unop {α : Type u_1} : MulOpposite.op ∘ MulOpposite.unop = id

@[simp]

theorem AddOpposite.op_comp_unop {α : Type u_1} : AddOpposite.op ∘ AddOpposite.unop = id

@[simp]

theorem MulOpposite.unop_comp_op {α : Type u_1} : MulOpposite.unop ∘ MulOpposite.op = id

@[simp]

theorem AddOpposite.unop_comp_op {α : Type u_1} : AddOpposite.unop ∘ AddOpposite.op = id

def MulOpposite.rec' {α : Type u_1} {F : αᵐᵒᵖ → Sort u_3} (h : (X : α) → F (MulOpposite.op X)) (X : αᵐᵒᵖ) : F X

A recursor for MulOpposite. Use as induction x.

Equations

• MulOpposite.rec' h X = h (MulOpposite.unop X)

def AddOpposite.rec' {α : Type u_1} {F : αᵃᵒᵖ → Sort u_3} (h : (X : α) → F (AddOpposite.op X)) (X : αᵃᵒᵖ) : F X

A recursor for AddOpposite. Use as induction x.

Equations

• AddOpposite.rec' h X = h (AddOpposite.unop X)

def MulOpposite.opEquiv {α : Type u_1} : α ≃ αᵐᵒᵖ

The canonical bijection between α and αᵐᵒᵖ.

Equations

• MulOpposite.opEquiv = { toFun := MulOpposite.op, invFun := MulOpposite.unop, left_inv := ⋯, right_inv := ⋯ }

def AddOpposite.opEquiv {α : Type u_1} : α ≃ αᵃᵒᵖ

The canonical bijection between α and αᵃᵒᵖ.

Equations

• AddOpposite.opEquiv = { toFun := AddOpposite.op, invFun := AddOpposite.unop, left_inv := ⋯, right_inv := ⋯ }

@[simp]

theorem MulOpposite.opEquiv_apply {α : Type u_1} : ⇑MulOpposite.opEquiv = MulOpposite.op

@[simp]

theorem AddOpposite.opEquiv_symm_apply {α : Type u_1} : ⇑AddOpposite.opEquiv.symm = AddOpposite.unop

@[simp]

theorem AddOpposite.opEquiv_apply {α : Type u_1} : ⇑AddOpposite.opEquiv = AddOpposite.op

@[simp]

theorem MulOpposite.opEquiv_symm_apply {α : Type u_1} : ⇑MulOpposite.opEquiv.symm = MulOpposite.unop

theorem MulOpposite.op_bijective {α : Type u_1} : Function.Bijective MulOpposite.op

theorem AddOpposite.op_bijective {α : Type u_1} : Function.Bijective AddOpposite.op

theorem MulOpposite.unop_bijective {α : Type u_1} : Function.Bijective MulOpposite.unop

theorem AddOpposite.unop_bijective {α : Type u_1} : Function.Bijective AddOpposite.unop

theorem MulOpposite.op_injective {α : Type u_1} : Function.Injective MulOpposite.op

theorem AddOpposite.op_injective {α : Type u_1} : Function.Injective AddOpposite.op

theorem MulOpposite.op_surjective {α : Type u_1} : Function.Surjective MulOpposite.op

theorem AddOpposite.op_surjective {α : Type u_1} : Function.Surjective AddOpposite.op

theorem MulOpposite.unop_injective {α : Type u_1} : Function.Injective MulOpposite.unop

theorem AddOpposite.unop_injective {α : Type u_1} : Function.Injective AddOpposite.unop

theorem MulOpposite.unop_surjective {α : Type u_1} : Function.Surjective MulOpposite.unop

theorem AddOpposite.unop_surjective {α : Type u_1} : Function.Surjective AddOpposite.unop

@[simp]

theorem MulOpposite.op_inj {α : Type u_1} {x : α} {y : α} : MulOpposite.op x = MulOpposite.op y ↔ x = y

@[simp]

theorem AddOpposite.op_inj {α : Type u_1} {x : α} {y : α} : AddOpposite.op x = AddOpposite.op y ↔ x = y

@[simp]

theorem MulOpposite.unop_inj {α : Type u_1} {x : αᵐᵒᵖ} {y : αᵐᵒᵖ} : MulOpposite.unop x = MulOpposite.unop y ↔ x = y

@[simp]

theorem AddOpposite.unop_inj {α : Type u_1} {x : αᵃᵒᵖ} {y : αᵃᵒᵖ} : AddOpposite.unop x = AddOpposite.unop y ↔ x = y

instance MulOpposite.instNontrivial {α : Type u_1} [Nontrivial α] : Nontrivial αᵐᵒᵖ

Equations

• ⋯ = ⋯

instance AddOpposite.instNontrivial {α : Type u_1} [Nontrivial α] : Nontrivial αᵃᵒᵖ

Equations

• ⋯ = ⋯

instance MulOpposite.instInhabited {α : Type u_1} [Inhabited α] : Inhabited αᵐᵒᵖ

Equations

• MulOpposite.instInhabited = { default := MulOpposite.op default }

instance AddOpposite.instInhabited {α : Type u_1} [Inhabited α] : Inhabited αᵃᵒᵖ

Equations

• AddOpposite.instInhabited = { default := AddOpposite.op default }

instance MulOpposite.instSubsingleton {α : Type u_1} [Subsingleton α] : Subsingleton αᵐᵒᵖ

Equations

• ⋯ = ⋯

instance AddOpposite.instSubsingleton {α : Type u_1} [Subsingleton α] : Subsingleton αᵃᵒᵖ

Equations

• ⋯ = ⋯

instance MulOpposite.instUnique {α : Type u_1} [Unique α] : Unique αᵐᵒᵖ

Equations

• MulOpposite.instUnique = Unique.mk' αᵐᵒᵖ

theorem AddOpposite.instUnique.proof_1 {α : Type u_1} [Unique α] : Subsingleton αᵃᵒᵖ

instance AddOpposite.instUnique {α : Type u_1} [Unique α] : Unique αᵃᵒᵖ

Equations

• AddOpposite.instUnique = Unique.mk' αᵃᵒᵖ

instance MulOpposite.instIsEmpty {α : Type u_1} [IsEmpty α] : IsEmpty αᵐᵒᵖ

Equations

• ⋯ = ⋯

instance AddOpposite.instIsEmpty {α : Type u_1} [IsEmpty α] : IsEmpty αᵃᵒᵖ

Equations

• ⋯ = ⋯

instance MulOpposite.instDecidableEq {α : Type u_1} [DecidableEq α] : DecidableEq αᵐᵒᵖ

Equations

• MulOpposite.instDecidableEq = ⋯.decidableEq

instance AddOpposite.instDecidableEq {α : Type u_1} [DecidableEq α] : DecidableEq αᵃᵒᵖ

Equations

• AddOpposite.instDecidableEq = ⋯.decidableEq

instance MulOpposite.instZero {α : Type u_1} [Zero α] : Zero αᵐᵒᵖ

Equations

• MulOpposite.instZero = { zero := MulOpposite.op 0 }

instance MulOpposite.instOne {α : Type u_1} [One α] : One αᵐᵒᵖ

Equations

• MulOpposite.instOne = { one := MulOpposite.op 1 }

instance AddOpposite.instZero {α : Type u_1} [Zero α] : Zero αᵃᵒᵖ

Equations

• AddOpposite.instZero = { zero := AddOpposite.op 0 }

instance MulOpposite.instAdd {α : Type u_1} [Add α] : Add αᵐᵒᵖ

Equations

• MulOpposite.instAdd = { add := fun (x y : αᵐᵒᵖ) => MulOpposite.op (MulOpposite.unop x + MulOpposite.unop y) }

instance MulOpposite.instSub {α : Type u_1} [Sub α] : Sub αᵐᵒᵖ

Equations

• MulOpposite.instSub = { sub := fun (x y : αᵐᵒᵖ) => MulOpposite.op (MulOpposite.unop x - MulOpposite.unop y) }

instance MulOpposite.instNeg {α : Type u_1} [Neg α] : Neg αᵐᵒᵖ

Equations

• MulOpposite.instNeg = { neg := fun (x : αᵐᵒᵖ) => MulOpposite.op (-MulOpposite.unop x) }

instance MulOpposite.instInvolutiveNeg {α : Type u_1} [InvolutiveNeg α] : InvolutiveNeg αᵐᵒᵖ

Equations

• MulOpposite.instInvolutiveNeg = InvolutiveNeg.mk ⋯

instance MulOpposite.instMul {α : Type u_1} [Mul α] : Mul αᵐᵒᵖ

Equations

• MulOpposite.instMul = { mul := fun (x y : αᵐᵒᵖ) => MulOpposite.op (MulOpposite.unop y * MulOpposite.unop x) }

instance AddOpposite.instAdd {α : Type u_1} [Add α] : Add αᵃᵒᵖ

Equations

• AddOpposite.instAdd = { add := fun (x y : αᵃᵒᵖ) => AddOpposite.op (AddOpposite.unop y + AddOpposite.unop x) }

instance MulOpposite.instInv {α : Type u_1} [Inv α] : Inv αᵐᵒᵖ

Equations

• MulOpposite.instInv = { inv := fun (x : αᵐᵒᵖ) => MulOpposite.op (MulOpposite.unop x)⁻¹ }

instance AddOpposite.instNeg {α : Type u_1} [Neg α] : Neg αᵃᵒᵖ

Equations

• AddOpposite.instNeg = { neg := fun (x : αᵃᵒᵖ) => AddOpposite.op (-AddOpposite.unop x) }

instance MulOpposite.instInvolutiveInv {α : Type u_1} [InvolutiveInv α] : InvolutiveInv αᵐᵒᵖ

Equations

• MulOpposite.instInvolutiveInv = InvolutiveInv.mk ⋯

theorem AddOpposite.instInvolutiveNeg.proof_1 {α : Type u_1} [InvolutiveNeg α] : ∀ (x : αᵃᵒᵖ), - -x = x

instance AddOpposite.instInvolutiveNeg {α : Type u_1} [InvolutiveNeg α] : InvolutiveNeg αᵃᵒᵖ

Equations

• AddOpposite.instInvolutiveNeg = InvolutiveNeg.mk ⋯

instance MulOpposite.instSMul {α : Type u_1} {β : Type u_2} [SMul α β] : SMul α βᵐᵒᵖ

Equations

• MulOpposite.instSMul = { smul := fun (c : α) (x : βᵐᵒᵖ) => MulOpposite.op (c • MulOpposite.unop x) }

instance AddOpposite.instVAdd {α : Type u_1} {β : Type u_2} [VAdd α β] : VAdd α βᵃᵒᵖ

Equations

• AddOpposite.instVAdd = { vadd := fun (c : α) (x : βᵃᵒᵖ) => AddOpposite.op (c +ᵥ AddOpposite.unop x) }

@[simp]

theorem MulOpposite.op_zero {α : Type u_1} [Zero α] : MulOpposite.op 0 = 0

@[simp]

theorem MulOpposite.unop_zero {α : Type u_1} [Zero α] : MulOpposite.unop 0 = 0

@[simp]

theorem MulOpposite.op_one {α : Type u_1} [One α] : MulOpposite.op 1 = 1

@[simp]

theorem AddOpposite.op_zero {α : Type u_1} [Zero α] : AddOpposite.op 0 = 0

@[simp]

theorem MulOpposite.unop_one {α : Type u_1} [One α] : MulOpposite.unop 1 = 1

@[simp]

theorem AddOpposite.unop_zero {α : Type u_1} [Zero α] : AddOpposite.unop 0 = 0

@[simp]

theorem MulOpposite.op_add {α : Type u_1} [Add α] (x : α) (y : α) : MulOpposite.op (x + y) = MulOpposite.op x + MulOpposite.op y

@[simp]

theorem MulOpposite.unop_add {α : Type u_1} [Add α] (x : αᵐᵒᵖ) (y : αᵐᵒᵖ) : MulOpposite.unop (x + y) = MulOpposite.unop x + MulOpposite.unop y

@[simp]

theorem MulOpposite.op_neg {α : Type u_1} [Neg α] (x : α) : MulOpposite.op (-x) = -MulOpposite.op x

@[simp]

theorem MulOpposite.unop_neg {α : Type u_1} [Neg α] (x : αᵐᵒᵖ) : MulOpposite.unop (-x) = -MulOpposite.unop x

@[simp]

theorem MulOpposite.op_mul {α : Type u_1} [Mul α] (x : α) (y : α) : MulOpposite.op (x * y) = MulOpposite.op y * MulOpposite.op x

@[simp]

theorem AddOpposite.op_add {α : Type u_1} [Add α] (x : α) (y : α) : AddOpposite.op (x + y) = AddOpposite.op y + AddOpposite.op x

@[simp]

theorem MulOpposite.unop_mul {α : Type u_1} [Mul α] (x : αᵐᵒᵖ) (y : αᵐᵒᵖ) : MulOpposite.unop (x * y) = MulOpposite.unop y * MulOpposite.unop x

@[simp]

theorem AddOpposite.unop_add {α : Type u_1} [Add α] (x : αᵃᵒᵖ) (y : αᵃᵒᵖ) : AddOpposite.unop (x + y) = AddOpposite.unop y + AddOpposite.unop x

@[simp]

theorem MulOpposite.op_inv {α : Type u_1} [Inv α] (x : α) : MulOpposite.op x⁻¹ = (MulOpposite.op x)⁻¹

@[simp]

theorem AddOpposite.op_neg {α : Type u_1} [Neg α] (x : α) : AddOpposite.op (-x) = -AddOpposite.op x

@[simp]

theorem MulOpposite.unop_inv {α : Type u_1} [Inv α] (x : αᵐᵒᵖ) : MulOpposite.unop x⁻¹ = (MulOpposite.unop x)⁻¹

@[simp]

theorem AddOpposite.unop_neg {α : Type u_1} [Neg α] (x : αᵃᵒᵖ) : AddOpposite.unop (-x) = -AddOpposite.unop x

@[simp]

theorem MulOpposite.op_sub {α : Type u_1} [Sub α] (x : α) (y : α) : MulOpposite.op (x - y) = MulOpposite.op x - MulOpposite.op y

@[simp]

theorem MulOpposite.unop_sub {α : Type u_1} [Sub α] (x : αᵐᵒᵖ) (y : αᵐᵒᵖ) : MulOpposite.unop (x - y) = MulOpposite.unop x - MulOpposite.unop y

@[simp]

theorem MulOpposite.op_smul {α : Type u_1} {β : Type u_2} [SMul α β] (a : α) (b : β) : MulOpposite.op (a • b) = a • MulOpposite.op b

@[simp]

theorem AddOpposite.op_vadd {α : Type u_1} {β : Type u_2} [VAdd α β] (a : α) (b : β) : AddOpposite.op (a +ᵥ b) = a +ᵥ AddOpposite.op b

@[simp]

theorem MulOpposite.unop_smul {α : Type u_1} {β : Type u_2} [SMul α β] (a : α) (b : βᵐᵒᵖ) : MulOpposite.unop (a • b) = a • MulOpposite.unop b

@[simp]

theorem AddOpposite.unop_vadd {α : Type u_1} {β : Type u_2} [VAdd α β] (a : α) (b : βᵃᵒᵖ) : AddOpposite.unop (a +ᵥ b) = a +ᵥ AddOpposite.unop b

@[simp]

theorem MulOpposite.unop_eq_zero_iff {α : Type u_1} [Zero α] (a : αᵐᵒᵖ) : MulOpposite.unop a = 0 ↔ a = 0

@[simp]

theorem MulOpposite.op_eq_zero_iff {α : Type u_1} [Zero α] (a : α) : MulOpposite.op a = 0 ↔ a = 0

theorem MulOpposite.unop_ne_zero_iff {α : Type u_1} [Zero α] (a : αᵐᵒᵖ) : MulOpposite.unop a ≠ 0 ↔ a ≠ 0

theorem MulOpposite.op_ne_zero_iff {α : Type u_1} [Zero α] (a : α) : MulOpposite.op a ≠ 0 ↔ a ≠ 0

@[simp]

theorem MulOpposite.unop_eq_one_iff {α : Type u_1} [One α] (a : αᵐᵒᵖ) : MulOpposite.unop a = 1 ↔ a = 1

@[simp]

theorem AddOpposite.unop_eq_zero_iff {α : Type u_1} [Zero α] (a : αᵃᵒᵖ) : AddOpposite.unop a = 0 ↔ a = 0

@[simp]

theorem MulOpposite.op_eq_one_iff {α : Type u_1} [One α] (a : α) : MulOpposite.op a = 1 ↔ a = 1

@[simp]

theorem AddOpposite.op_eq_zero_iff {α : Type u_1} [Zero α] (a : α) : AddOpposite.op a = 0 ↔ a = 0

instance AddOpposite.instOne {α : Type u_1} [One α] : One αᵃᵒᵖ

Equations

• AddOpposite.instOne = { one := AddOpposite.op 1 }

@[simp]

theorem AddOpposite.op_one {α : Type u_1} [One α] : AddOpposite.op 1 = 1

@[simp]

theorem AddOpposite.unop_one {α : Type u_1} [One α] : AddOpposite.unop 1 = 1

@[simp]

theorem AddOpposite.op_eq_one_iff {α : Type u_1} [One α] {a : α} : AddOpposite.op a = 1 ↔ a = 1

@[simp]

theorem AddOpposite.unop_eq_one_iff {α : Type u_1} [One α] {a : αᵃᵒᵖ} : AddOpposite.unop a = 1 ↔ a = 1

instance AddOpposite.instMul {α : Type u_1} [Mul α] : Mul αᵃᵒᵖ

Equations

• AddOpposite.instMul = { mul := fun (a b : αᵃᵒᵖ) => AddOpposite.op (AddOpposite.unop a * AddOpposite.unop b) }

@[simp]

theorem AddOpposite.op_mul {α : Type u_1} [Mul α] (a : α) (b : α) : AddOpposite.op (a * b) = AddOpposite.op a * AddOpposite.op b

@[simp]

theorem AddOpposite.unop_mul {α : Type u_1} [Mul α] (a : αᵃᵒᵖ) (b : αᵃᵒᵖ) : AddOpposite.unop (a * b) = AddOpposite.unop a * AddOpposite.unop b

instance AddOpposite.instInv {α : Type u_1} [Inv α] : Inv αᵃᵒᵖ

Equations

• AddOpposite.instInv = { inv := fun (a : αᵃᵒᵖ) => AddOpposite.op (AddOpposite.unop a)⁻¹ }

instance AddOpposite.instInvolutiveInv {α : Type u_1} [InvolutiveInv α] : InvolutiveInv αᵃᵒᵖ

Equations

• AddOpposite.instInvolutiveInv = InvolutiveInv.mk ⋯

@[simp]

theorem AddOpposite.op_inv {α : Type u_1} [Inv α] (a : α) : AddOpposite.op a⁻¹ = (AddOpposite.op a)⁻¹

@[simp]

theorem AddOpposite.unop_inv {α : Type u_1} [Inv α] (a : αᵃᵒᵖ) : AddOpposite.unop a⁻¹ = (AddOpposite.unop a)⁻¹

instance AddOpposite.instDiv {α : Type u_1} [Div α] : Div αᵃᵒᵖ

Equations

• AddOpposite.instDiv = { div := fun (a b : αᵃᵒᵖ) => AddOpposite.op (AddOpposite.unop a / AddOpposite.unop b) }

@[simp]

theorem AddOpposite.op_div {α : Type u_1} [Div α] (a : α) (b : α) : AddOpposite.op (a / b) = AddOpposite.op a / AddOpposite.op b

@[simp]

theorem AddOpposite.unop_div {α : Type u_1} [Div α] (a : αᵃᵒᵖ) (b : αᵃᵒᵖ) : AddOpposite.unop (a / b) = AddOpposite.unop a / AddOpposite.unop b