Documentation

Mathlib.Algebra.Opposites

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 #

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 https://github.com/leanprover-community/mathlib4/pull/1036).

  • op' :: (
    • unop' : α

      The element of α represented by x : PreOpposite α.

  • )

Multiplicative opposite of a type.

Equations

Additive opposite of a type.

Equations
def MulOpposite.op {α : Type u_1} :
ααᵐᵒᵖ

The element of MulOpposite α that represents x : α.

Equations
  • MulOpposite.op = PreOpposite.op'
def AddOpposite.op {α : Type u_1} :
ααᵃᵒᵖ

The element of αᵃᵒᵖ that represents x : α.

Equations
  • AddOpposite.op = PreOpposite.op'
def MulOpposite.unop {α : Type u_1} :
αᵐᵒᵖα

The element of α represented by x : αᵐᵒᵖ.

Equations
  • MulOpposite.unop = PreOpposite.unop'
def AddOpposite.unop {α : Type u_1} :
αᵃᵒᵖα

The element of α represented by x : αᵃᵒᵖ.

Equations
  • AddOpposite.unop = PreOpposite.unop'
@[simp]
theorem MulOpposite.unop_op {α : Type u_1} (x : α) :
@[simp]
theorem AddOpposite.unop_op {α : Type u_1} (x : α) :
@[simp]
@[simp]
@[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 : αᵐᵒᵖSort u_3} (h : (X : α) → F (MulOpposite.op X)) (X : αᵐᵒᵖ) :
F X

A recursor for MulOpposite. Use as induction x.

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

A recursor for AddOpposite. Use as induction x.

Equations
def MulOpposite.opEquiv {α : Type u_1} :

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} :

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_apply {α : Type u_1} :
AddOpposite.opEquiv = AddOpposite.op
@[simp]
theorem MulOpposite.opEquiv_symm_apply {α : Type u_1} :
MulOpposite.opEquiv.symm = MulOpposite.unop
@[simp]
theorem AddOpposite.opEquiv_symm_apply {α : Type u_1} :
AddOpposite.opEquiv.symm = AddOpposite.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 y : α} :
@[simp]
theorem AddOpposite.op_inj {α : Type u_1} {x y : α} :
@[simp]
@[simp]
@[simp]
theorem MulOpposite.forall {α : Type u_1} {p : αᵐᵒᵖProp} :
(∀ (a : αᵐᵒᵖ), p a) ∀ (a : α), p (MulOpposite.op a)
@[simp]
theorem AddOpposite.forall {α : Type u_1} {p : αᵃᵒᵖProp} :
(∀ (a : αᵃᵒᵖ), p a) ∀ (a : α), p (AddOpposite.op a)
@[simp]
theorem MulOpposite.exists {α : Type u_1} {p : αᵐᵒᵖProp} :
(∃ (a : αᵐᵒᵖ), p a) ∃ (a : α), p (MulOpposite.op a)
@[simp]
theorem AddOpposite.exists {α : Type u_1} {p : αᵃᵒᵖProp} :
(∃ (a : αᵃᵒᵖ), p a) ∃ (a : α), p (AddOpposite.op a)
Equations
Equations
instance MulOpposite.instUnique {α : Type u_1} [Unique α] :
Equations
instance AddOpposite.instUnique {α : Type u_1} [Unique α] :
Equations
Equations
  • MulOpposite.instDecidableEq = .decidableEq
Equations
  • AddOpposite.instDecidableEq = .decidableEq
instance MulOpposite.instZero {α : Type u_1} [Zero α] :
Equations
instance MulOpposite.instOne {α : Type u_1} [One α] :
Equations
instance AddOpposite.instZero {α : Type u_1} [Zero α] :
Equations
instance MulOpposite.instAdd {α : Type u_1} [Add α] :
Equations
instance MulOpposite.instSub {α : Type u_1} [Sub α] :
Equations
instance MulOpposite.instNeg {α : Type u_1} [Neg α] :
Equations
Equations
instance MulOpposite.instMul {α : Type u_1} [Mul α] :
Equations
instance AddOpposite.instAdd {α : Type u_1} [Add α] :
Equations
instance MulOpposite.instInv {α : Type u_1} [Inv α] :
Equations
instance AddOpposite.instNeg {α : Type u_1} [Neg α] :
Equations
Equations
Equations
instance MulOpposite.instSMul {α : Type u_1} {β : Type u_2} [SMul α β] :
Equations
instance AddOpposite.instVAdd {α : Type u_1} {β : Type u_2} [VAdd α β] :
Equations
@[simp]
theorem MulOpposite.op_zero {α : Type u_1} [Zero α] :
@[simp]
theorem MulOpposite.unop_zero {α : Type u_1} [Zero α] :
@[simp]
theorem MulOpposite.op_one {α : Type u_1} [One α] :
@[simp]
theorem AddOpposite.op_zero {α : Type u_1} [Zero α] :
@[simp]
theorem MulOpposite.unop_one {α : Type u_1} [One α] :
@[simp]
theorem AddOpposite.unop_zero {α : Type u_1} [Zero α] :
@[simp]
theorem MulOpposite.op_add {α : Type u_1} [Add α] (x y : α) :
@[simp]
@[simp]
theorem MulOpposite.op_neg {α : Type u_1} [Neg α] (x : α) :
@[simp]
@[simp]
theorem MulOpposite.op_mul {α : Type u_1} [Mul α] (x y : α) :
@[simp]
theorem AddOpposite.op_add {α : Type u_1} [Add α] (x y : α) :
@[simp]
@[simp]
@[simp]
theorem MulOpposite.op_inv {α : Type u_1} [Inv α] (x : α) :
@[simp]
theorem AddOpposite.op_neg {α : Type u_1} [Neg α] (x : α) :
@[simp]
@[simp]
theorem MulOpposite.op_sub {α : Type u_1} [Sub α] (x y : α) :
@[simp]
@[simp]
theorem MulOpposite.op_smul {α : Type u_1} {β : Type u_2} [SMul α β] (a : α) (b : β) :
@[simp]
theorem AddOpposite.op_vadd {α : Type u_1} {β : Type u_2} [VAdd α β] (a : α) (b : β) :
@[simp]
theorem MulOpposite.unop_smul {α : Type u_1} {β : Type u_2} [SMul α β] (a : α) (b : βᵐᵒᵖ) :
@[simp]
theorem AddOpposite.unop_vadd {α : Type u_1} {β : Type u_2} [VAdd α β] (a : α) (b : βᵃᵒᵖ) :
@[simp]
theorem MulOpposite.unop_eq_zero_iff {α : Type u_1} [Zero α] (a : αᵐᵒᵖ) :
@[simp]
theorem MulOpposite.op_eq_zero_iff {α : Type u_1} [Zero α] (a : α) :
theorem MulOpposite.op_ne_zero_iff {α : Type u_1} [Zero α] (a : α) :
@[simp]
theorem MulOpposite.unop_eq_one_iff {α : Type u_1} [One α] (a : αᵐᵒᵖ) :
@[simp]
theorem AddOpposite.unop_eq_zero_iff {α : Type u_1} [Zero α] (a : αᵃᵒᵖ) :
@[simp]
theorem MulOpposite.op_eq_one_iff {α : Type u_1} [One α] (a : α) :
@[simp]
theorem AddOpposite.op_eq_zero_iff {α : Type u_1} [Zero α] (a : α) :
instance AddOpposite.instOne {α : Type u_1} [One α] :
Equations
@[simp]
theorem AddOpposite.op_one {α : Type u_1} [One α] :
@[simp]
theorem AddOpposite.unop_one {α : Type u_1} [One α] :
@[simp]
theorem AddOpposite.op_eq_one_iff {α : Type u_1} [One α] {a : α} :
@[simp]
theorem AddOpposite.unop_eq_one_iff {α : Type u_1} [One α] {a : αᵃᵒᵖ} :
instance AddOpposite.instMul {α : Type u_1} [Mul α] :
Equations
@[simp]
theorem AddOpposite.op_mul {α : Type u_1} [Mul α] (a b : α) :
@[simp]
instance AddOpposite.instInv {α : Type u_1} [Inv α] :
Equations
Equations
@[simp]
theorem AddOpposite.op_inv {α : Type u_1} [Inv α] (a : α) :
instance AddOpposite.instDiv {α : Type u_1} [Div α] :
Equations
@[simp]
theorem AddOpposite.op_div {α : Type u_1} [Div α] (a b : α) :
@[simp]