mathlib documentation

algebra.opposites

Multiplicative opposite and algebraic operations on it #

In this file we define mul_opposite 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 mul_opposite.op is the canonical map from α to αᵐᵒᵖ.

We also define add_opposite 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 add_opposite.op is the canonical map from α to αᵃᵒᵖ.

Notation #

αᵐᵒᵖ = mul_opposite α
αᵃᵒᵖ = add_opposite α

Tags #

multiplicative opposite, additive opposite

def mul_opposite (α : Type u) :

Type u

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

Equations

αᵐᵒᵖ = α

def add_opposite (α : Type u) :

Type u

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

Equations

αᵃᵒᵖ = α

def mul_opposite.op {α : Type u} :

α → αᵐᵒᵖ

The element of mul_opposite α that represents x : α.

Equations

mul_opposite.op = id

def add_opposite.op {α : Type u} :

α → αᵃᵒᵖ

The element of αᵃᵒᵖ that represents x : α.

Equations

add_opposite.op = id

def mul_opposite.unop {α : Type u} :

αᵐᵒᵖ → α

The element of α represented by x : αᵐᵒᵖ.

Equations

mul_opposite.unop = id

def add_opposite.unop {α : Type u} :

αᵃᵒᵖ → α

The element of α represented by x : αᵃᵒᵖ.

Equations

add_opposite.unop = id

@[simp]

theorem mul_opposite.unop_op {α : Type u} (x : α) :

mul_opposite.unop (mul_opposite.op x) = x

@[simp]

theorem add_opposite.unop_op {α : Type u} (x : α) :

add_opposite.unop (add_opposite.op x) = x

@[simp]

theorem mul_opposite.op_unop {α : Type u} (x : αᵐᵒᵖ) :

mul_opposite.op (mul_opposite.unop x) = x

@[simp]

theorem add_opposite.op_unop {α : Type u} (x : αᵃᵒᵖ) :

add_opposite.op (add_opposite.unop x) = x

@[simp]

theorem add_opposite.op_comp_unop {α : Type u} :

add_opposite.op ∘ add_opposite.unop = id

@[simp]

theorem mul_opposite.op_comp_unop {α : Type u} :

mul_opposite.op ∘ mul_opposite.unop = id

@[simp]

theorem mul_opposite.unop_comp_op {α : Type u} :

mul_opposite.unop ∘ mul_opposite.op = id

@[simp]

theorem add_opposite.unop_comp_op {α : Type u} :

add_opposite.unop ∘ add_opposite.op = id

@[protected, simp]

def mul_opposite.rec {α : Type u} {F : αᵐᵒᵖ → Sort v} (h : Π (X : α), F (mul_opposite.op X)) (X : αᵐᵒᵖ) :

F X

A recursor for mul_opposite. Use as induction x using mul_opposite.rec.

Equations

mul_opposite.rec h = λ (X : αᵐᵒᵖ), h (mul_opposite.unop X)

@[protected, simp]

def add_opposite.rec {α : Type u} {F : αᵃᵒᵖ → Sort v} (h : Π (X : α), F (add_opposite.op X)) (X : αᵃᵒᵖ) :

F X

A recursor for add_opposite. Use as induction x using add_opposite.rec.

Equations

add_opposite.rec h = λ (X : αᵃᵒᵖ), h (add_opposite.unop X)

@[simp]

theorem mul_opposite.op_equiv_apply {α : Type u} :

⇑mul_opposite.op_equiv = mul_opposite.op

def mul_opposite.op_equiv {α : Type u} :

α ≃ αᵐᵒᵖ

The canonical bijection between α and αᵐᵒᵖ.

Equations

mul_opposite.op_equiv = {to_fun := mul_opposite.op α, inv_fun := mul_opposite.unop α, left_inv := _, right_inv := _}

@[simp]

theorem add_opposite.op_equiv_apply {α : Type u} :

⇑add_opposite.op_equiv = add_opposite.op

@[simp]

theorem add_opposite.op_equiv_symm_apply {α : Type u} :

⇑(add_opposite.op_equiv.symm) = add_opposite.unop

@[simp]

theorem mul_opposite.op_equiv_symm_apply {α : Type u} :

⇑(mul_opposite.op_equiv.symm) = mul_opposite.unop

def add_opposite.op_equiv {α : Type u} :

α ≃ αᵃᵒᵖ

The canonical bijection between α and αᵃᵒᵖ.

Equations

add_opposite.op_equiv = {to_fun := add_opposite.op α, inv_fun := add_opposite.unop α, left_inv := _, right_inv := _}

theorem mul_opposite.op_bijective {α : Type u} :

function.bijective mul_opposite.op

theorem add_opposite.op_bijective {α : Type u} :

function.bijective add_opposite.op

theorem mul_opposite.unop_bijective {α : Type u} :

function.bijective mul_opposite.unop

theorem add_opposite.unop_bijective {α : Type u} :

function.bijective add_opposite.unop

theorem mul_opposite.op_injective {α : Type u} :

function.injective mul_opposite.op

theorem add_opposite.op_injective {α : Type u} :

function.injective add_opposite.op

theorem add_opposite.op_surjective {α : Type u} :

function.surjective add_opposite.op

theorem mul_opposite.op_surjective {α : Type u} :

function.surjective mul_opposite.op

theorem mul_opposite.unop_injective {α : Type u} :

function.injective mul_opposite.unop

theorem add_opposite.unop_injective {α : Type u} :

function.injective add_opposite.unop

theorem mul_opposite.unop_surjective {α : Type u} :

function.surjective mul_opposite.unop

theorem add_opposite.unop_surjective {α : Type u} :

function.surjective add_opposite.unop

@[simp]

theorem add_opposite.op_inj {α : Type u} {x y : α} :

add_opposite.op x = add_opposite.op y ↔ x = y

@[simp]

theorem mul_opposite.op_inj {α : Type u} {x y : α} :

mul_opposite.op x = mul_opposite.op y ↔ x = y

@[simp]

theorem add_opposite.unop_inj {α : Type u} {x y : αᵃᵒᵖ} :

add_opposite.unop x = add_opposite.unop y ↔ x = y

@[simp]

theorem mul_opposite.unop_inj {α : Type u} {x y : αᵐᵒᵖ} :

mul_opposite.unop x = mul_opposite.unop y ↔ x = y

@[protected, instance]

def mul_opposite.nontrivial (α : Type u) [nontrivial α] :

nontrivial αᵐᵒᵖ

@[protected, instance]

def add_opposite.nontrivial (α : Type u) [nontrivial α] :

nontrivial αᵃᵒᵖ

@[protected, instance]

def add_opposite.inhabited (α : Type u) [inhabited α] :

inhabited αᵃᵒᵖ

Equations

add_opposite.inhabited α = {default := add_opposite.op default}

@[protected, instance]

def mul_opposite.inhabited (α : Type u) [inhabited α] :

inhabited αᵐᵒᵖ

Equations

mul_opposite.inhabited α = {default := mul_opposite.op default}

@[protected, instance]

def mul_opposite.subsingleton (α : Type u) [subsingleton α] :

subsingleton αᵐᵒᵖ

@[protected, instance]

def add_opposite.subsingleton (α : Type u) [subsingleton α] :

subsingleton αᵃᵒᵖ

@[protected, instance]

def add_opposite.unique (α : Type u) [unique α] :

unique αᵃᵒᵖ

Equations

add_opposite.unique α = unique.mk' αᵃᵒᵖ

@[protected, instance]

def mul_opposite.unique (α : Type u) [unique α] :

unique αᵐᵒᵖ

Equations

mul_opposite.unique α = unique.mk' αᵐᵒᵖ

@[protected, instance]

def mul_opposite.is_empty (α : Type u) [is_empty α] :

is_empty αᵐᵒᵖ

@[protected, instance]

def add_opposite.is_empty (α : Type u) [is_empty α] :

is_empty αᵃᵒᵖ

@[protected, instance]

def mul_opposite.has_zero (α : Type u) [has_zero α] :

has_zero αᵐᵒᵖ

Equations

mul_opposite.has_zero α = {zero := mul_opposite.op 0}

@[protected, instance]

def mul_opposite.has_one (α : Type u) [has_one α] :

has_one αᵐᵒᵖ

Equations

mul_opposite.has_one α = {one := mul_opposite.op 1}

@[protected, instance]

def add_opposite.has_zero (α : Type u) [has_zero α] :

has_zero αᵃᵒᵖ

Equations

add_opposite.has_zero α = {zero := add_opposite.op 0}

@[protected, instance]

def mul_opposite.has_add (α : Type u) [has_add α] :

has_add αᵐᵒᵖ

Equations

mul_opposite.has_add α = {add := λ (x y : αᵐᵒᵖ), mul_opposite.op (mul_opposite.unop x + mul_opposite.unop y)}

@[protected, instance]

def mul_opposite.has_sub (α : Type u) [has_sub α] :

has_sub αᵐᵒᵖ

Equations

mul_opposite.has_sub α = {sub := λ (x y : αᵐᵒᵖ), mul_opposite.op (mul_opposite.unop x - mul_opposite.unop y)}

@[protected, instance]

def mul_opposite.has_neg (α : Type u) [has_neg α] :

has_neg αᵐᵒᵖ

Equations

mul_opposite.has_neg α = {neg := λ (x : αᵐᵒᵖ), mul_opposite.op (-mul_opposite.unop x)}

@[protected, instance]

def add_opposite.has_add (α : Type u) [has_add α] :

has_add αᵃᵒᵖ

Equations

add_opposite.has_add α = {add := λ (x y : αᵃᵒᵖ), add_opposite.op (add_opposite.unop y + add_opposite.unop x)}

@[protected, instance]

def mul_opposite.has_mul (α : Type u) [has_mul α] :

has_mul αᵐᵒᵖ

Equations

mul_opposite.has_mul α = {mul := λ (x y : αᵐᵒᵖ), mul_opposite.op ((mul_opposite.unop y) * mul_opposite.unop x)}

@[protected, instance]

def mul_opposite.has_inv (α : Type u) [has_inv α] :

has_inv αᵐᵒᵖ

Equations

mul_opposite.has_inv α = {inv := λ (x : αᵐᵒᵖ), mul_opposite.op (mul_opposite.unop x)⁻¹}

@[protected, instance]

def add_opposite.has_neg (α : Type u) [has_neg α] :

has_neg αᵃᵒᵖ

Equations

add_opposite.has_neg α = {neg := λ (x : αᵃᵒᵖ), add_opposite.op (-add_opposite.unop x)}

@[protected, instance]

def mul_opposite.has_scalar (α : Type u) (R : Type u_1) [has_scalar R α] :

has_scalar R αᵐᵒᵖ

Equations

mul_opposite.has_scalar α R = {smul := λ (c : R) (x : αᵐᵒᵖ), mul_opposite.op (c • mul_opposite.unop x)}

@[protected, instance]

def add_opposite.has_scalar (α : Type u) (R : Type u_1) [has_vadd R α] :

has_vadd R αᵃᵒᵖ

Equations

add_opposite.has_scalar α R = {vadd := λ (c : R) (x : αᵃᵒᵖ), add_opposite.op (c +ᵥ add_opposite.unop x)}

@[simp]

theorem mul_opposite.op_zero (α : Type u) [has_zero α] :

mul_opposite.op 0 = 0

@[simp]

theorem mul_opposite.unop_zero (α : Type u) [has_zero α] :

mul_opposite.unop 0 = 0

@[simp]

theorem add_opposite.op_zero (α : Type u) [has_zero α] :

add_opposite.op 0 = 0

@[simp]

theorem mul_opposite.op_one (α : Type u) [has_one α] :

mul_opposite.op 1 = 1

@[simp]

theorem mul_opposite.unop_one (α : Type u) [has_one α] :

mul_opposite.unop 1 = 1

@[simp]

theorem add_opposite.unop_zero (α : Type u) [has_zero α] :

add_opposite.unop 0 = 0

@[simp]

theorem mul_opposite.op_add {α : Type u} [has_add α] (x y : α) :

mul_opposite.op (x + y) = mul_opposite.op x + mul_opposite.op y

@[simp]

theorem mul_opposite.unop_add {α : Type u} [has_add α] (x y : αᵐᵒᵖ) :

mul_opposite.unop (x + y) = mul_opposite.unop x + mul_opposite.unop y

@[simp]

theorem mul_opposite.op_neg {α : Type u} [has_neg α] (x : α) :

mul_opposite.op (-x) = -mul_opposite.op x

@[simp]

theorem mul_opposite.unop_neg {α : Type u} [has_neg α] (x : αᵐᵒᵖ) :

mul_opposite.unop (-x) = -mul_opposite.unop x

@[simp]

theorem add_opposite.op_add {α : Type u} [has_add α] (x y : α) :

add_opposite.op (x + y) = add_opposite.op y + add_opposite.op x

@[simp]

theorem mul_opposite.op_mul {α : Type u} [has_mul α] (x y : α) :

mul_opposite.op (x * y) = (mul_opposite.op y) * mul_opposite.op x

@[simp]

theorem mul_opposite.unop_mul {α : Type u} [has_mul α] (x y : αᵐᵒᵖ) :

mul_opposite.unop (x * y) = (mul_opposite.unop y) * mul_opposite.unop x

@[simp]

theorem add_opposite.unop_add {α : Type u} [has_add α] (x y : αᵃᵒᵖ) :

add_opposite.unop (x + y) = add_opposite.unop y + add_opposite.unop x

@[simp]

theorem add_opposite.op_neg {α : Type u} [has_neg α] (x : α) :

add_opposite.op (-x) = -add_opposite.op x

@[simp]

theorem mul_opposite.op_inv {α : Type u} [has_inv α] (x : α) :

mul_opposite.op x⁻¹ = (mul_opposite.op x)⁻¹

@[simp]

theorem mul_opposite.unop_inv {α : Type u} [has_inv α] (x : αᵐᵒᵖ) :

mul_opposite.unop x⁻¹ = (mul_opposite.unop x)⁻¹

@[simp]

theorem add_opposite.unop_neg {α : Type u} [has_neg α] (x : αᵃᵒᵖ) :

add_opposite.unop (-x) = -add_opposite.unop x

@[simp]

theorem mul_opposite.op_sub {α : Type u} [has_sub α] (x y : α) :

mul_opposite.op (x - y) = mul_opposite.op x - mul_opposite.op y

@[simp]

theorem mul_opposite.unop_sub {α : Type u} [has_sub α] (x y : αᵐᵒᵖ) :

mul_opposite.unop (x - y) = mul_opposite.unop x - mul_opposite.unop y

@[simp]

theorem add_opposite.op_vadd {α : Type u} {R : Type u_1} [has_vadd R α] (c : R) (a : α) :

add_opposite.op (c +ᵥ a) = c +ᵥ add_opposite.op a

@[simp]

theorem mul_opposite.op_smul {α : Type u} {R : Type u_1} [has_scalar R α] (c : R) (a : α) :

mul_opposite.op (c • a) = c • mul_opposite.op a

@[simp]

theorem mul_opposite.unop_smul {α : Type u} {R : Type u_1} [has_scalar R α] (c : R) (a : αᵐᵒᵖ) :

mul_opposite.unop (c • a) = c • mul_opposite.unop a

@[simp]

theorem add_opposite.unop_vadd {α : Type u} {R : Type u_1} [has_vadd R α] (c : R) (a : αᵃᵒᵖ) :

add_opposite.unop (c +ᵥ a) = c +ᵥ add_opposite.unop a

@[simp]

theorem mul_opposite.unop_eq_zero_iff {α : Type u} [has_zero α] (a : αᵐᵒᵖ) :

mul_opposite.unop a = 0 ↔ a = 0

@[simp]

theorem mul_opposite.op_eq_zero_iff {α : Type u} [has_zero α] (a : α) :

mul_opposite.op a = 0 ↔ a = 0

theorem mul_opposite.unop_ne_zero_iff {α : Type u} [has_zero α] (a : αᵐᵒᵖ) :

mul_opposite.unop a ≠ 0 ↔ a ≠ 0

theorem mul_opposite.op_ne_zero_iff {α : Type u} [has_zero α] (a : α) :

mul_opposite.op a ≠ 0 ↔ a ≠ 0

@[simp]

theorem mul_opposite.unop_eq_one_iff {α : Type u} [has_one α] (a : αᵐᵒᵖ) :

mul_opposite.unop a = 1 ↔ a = 1

@[simp]

theorem add_opposite.unop_eq_zero_iff {α : Type u} [has_zero α] (a : αᵃᵒᵖ) :

add_opposite.unop a = 0 ↔ a = 0

@[simp]

theorem add_opposite.op_eq_zero_iff {α : Type u} [has_zero α] (a : α) :

add_opposite.op a = 0 ↔ a = 0

@[simp]

theorem mul_opposite.op_eq_one_iff {α : Type u} [has_one α] (a : α) :

mul_opposite.op a = 1 ↔ a = 1

@[protected, instance]

def add_opposite.has_one {α : Type u} [has_one α] :

has_one αᵃᵒᵖ

Equations

add_opposite.has_one = {one := add_opposite.op 1}

@[simp]

theorem add_opposite.op_one {α : Type u} [has_one α] :

add_opposite.op 1 = 1

@[simp]

theorem add_opposite.unop_one {α : Type u} [has_one α] :

add_opposite.unop 1 = 1

@[simp]

theorem add_opposite.op_eq_one_iff {α : Type u} [has_one α] {a : α} :

add_opposite.op a = 1 ↔ a = 1

@[simp]

theorem add_opposite.unop_eq_one_iff {α : Type u} [has_one α] {a : αᵃᵒᵖ} :

add_opposite.unop a = 1 ↔ a = 1

@[protected, instance]

def add_opposite.has_mul {α : Type u} [has_mul α] :

has_mul αᵃᵒᵖ

Equations

add_opposite.has_mul = {mul := λ (a b : αᵃᵒᵖ), add_opposite.op ((add_opposite.unop a) * add_opposite.unop b)}

@[simp]

theorem add_opposite.op_mul {α : Type u} [has_mul α] (a b : α) :

add_opposite.op (a * b) = (add_opposite.op a) * add_opposite.op b

@[simp]

theorem add_opposite.unop_mul {α : Type u} [has_mul α] (a b : αᵃᵒᵖ) :

add_opposite.unop (a * b) = (add_opposite.unop a) * add_opposite.unop b

@[protected, instance]

def add_opposite.has_inv {α : Type u} [has_inv α] :

has_inv αᵃᵒᵖ

Equations

add_opposite.has_inv = {inv := λ (a : αᵃᵒᵖ), add_opposite.op (add_opposite.unop a)⁻¹}

@[simp]

theorem add_opposite.op_inv {α : Type u} [has_inv α] (a : α) :

add_opposite.op a⁻¹ = (add_opposite.op a)⁻¹

@[simp]

theorem add_opposite.unop_inv {α : Type u} [has_inv α] (a : αᵃᵒᵖ) :

add_opposite.unop a⁻¹ = (add_opposite.unop a)⁻¹

@[protected, instance]

def add_opposite.has_div {α : Type u} [has_div α] :

has_div αᵃᵒᵖ

Equations

add_opposite.has_div = {div := λ (a b : αᵃᵒᵖ), add_opposite.op (add_opposite.unop a / add_opposite.unop b)}

@[simp]

theorem add_opposite.op_div {α : Type u} [has_div α] (a b : α) :

add_opposite.op (a / b) = add_opposite.op a / add_opposite.op b

@[simp]

theorem add_opposite.unop_div {α : Type u} [has_div α] (a b : α) :

add_opposite.unop (a / b) = add_opposite.unop a / add_opposite.unop b