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 #

Tags #

multiplicative opposite, additive opposite

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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.

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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.

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def mul_opposite.op {α : Type u} :
α → αᵐᵒᵖ

The element of mul_opposite α that represents x : α.

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def add_opposite.op {α : Type u} :
α → αᵃᵒᵖ

The element of αᵃᵒᵖ that represents x : α.

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def mul_opposite.unop {α : Type u} :
αᵐᵒᵖ → α

The element of α represented by x : αᵐᵒᵖ.

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def add_opposite.unop {α : Type u} :
αᵃᵒᵖ → α

The element of α represented by x : αᵃᵒᵖ.

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@[simp]
theorem mul_opposite.unop_op {α : Type u} (x : α) :
mul_opposite.unop (mul_opposite.op x) = x
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@[simp]
theorem add_opposite.unop_op {α : Type u} (x : α) :
add_opposite.unop (add_opposite.op x) = x
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@[simp]
theorem mul_opposite.op_unop {α : Type u} (x : αᵐᵒᵖ) :
mul_opposite.op (mul_opposite.unop x) = x
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@[simp]
theorem add_opposite.op_unop {α : Type u} (x : αᵃᵒᵖ) :
add_opposite.op (add_opposite.unop x) = x
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@[simp]
theorem add_opposite.op_comp_unop {α : Type u} :
add_opposite.op ∘ add_opposite.unop = id
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@[simp]
theorem mul_opposite.op_comp_unop {α : Type u} :
mul_opposite.op ∘ mul_opposite.unop = id
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@[simp]
theorem mul_opposite.unop_comp_op {α : Type u} :
mul_opposite.unop ∘ mul_opposite.op = id
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@[simp]
theorem add_opposite.unop_comp_op {α : Type u} :
add_opposite.unop ∘ add_opposite.op = id
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@[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.

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@[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.

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@[simp]
theorem mul_opposite.op_equiv_apply {α : Type u} :
⇑mul_opposite.op_equiv = mul_opposite.op
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def mul_opposite.op_equiv {α : Type u} :
α ≃ αᵐᵒᵖ

The canonical bijection between α and αᵐᵒᵖ.

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@[simp]
theorem add_opposite.op_equiv_apply {α : Type u} :
⇑add_opposite.op_equiv = add_opposite.op
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@[simp]
theorem add_opposite.op_equiv_symm_apply {α : Type u} :
⇑(add_opposite.op_equiv.symm) = add_opposite.unop
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@[simp]
theorem mul_opposite.op_equiv_symm_apply {α : Type u} :
⇑(mul_opposite.op_equiv.symm) = mul_opposite.unop
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def add_opposite.op_equiv {α : Type u} :
α ≃ αᵃᵒᵖ

The canonical bijection between α and αᵃᵒᵖ.

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theorem mul_opposite.op_bijective {α : Type u} :
function.bijective mul_opposite.op
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theorem add_opposite.op_bijective {α : Type u} :
function.bijective add_opposite.op
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theorem mul_opposite.unop_bijective {α : Type u} :
function.bijective mul_opposite.unop
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theorem add_opposite.unop_bijective {α : Type u} :
function.bijective add_opposite.unop
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theorem mul_opposite.op_injective {α : Type u} :
function.injective mul_opposite.op
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theorem add_opposite.op_injective {α : Type u} :
function.injective add_opposite.op
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theorem add_opposite.op_surjective {α : Type u} :
function.surjective add_opposite.op
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theorem mul_opposite.op_surjective {α : Type u} :
function.surjective mul_opposite.op
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theorem mul_opposite.unop_injective {α : Type u} :
function.injective mul_opposite.unop
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theorem add_opposite.unop_injective {α : Type u} :
function.injective add_opposite.unop
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theorem mul_opposite.unop_surjective {α : Type u} :
function.surjective mul_opposite.unop
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theorem add_opposite.unop_surjective {α : Type u} :
function.surjective add_opposite.unop
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@[simp]
theorem add_opposite.op_inj {α : Type u} {x y : α} :
add_opposite.op x = add_opposite.op y ↔ x = y
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@[simp]
theorem mul_opposite.op_inj {α : Type u} {x y : α} :
mul_opposite.op x = mul_opposite.op y ↔ x = y
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@[simp]
theorem add_opposite.unop_inj {α : Type u} {x y : αᵃᵒᵖ} :
add_opposite.unop x = add_opposite.unop y ↔ x = y
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@[simp]
theorem mul_opposite.unop_inj {α : Type u} {x y : αᵐᵒᵖ} :
mul_opposite.unop x = mul_opposite.unop y ↔ x = y
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@[protected, instance]
def mul_opposite.nontrivial (α : Type u) [nontrivial α] :
nontrivial αᵐᵒᵖ
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@[protected, instance]
def add_opposite.nontrivial (α : Type u) [nontrivial α] :
nontrivial αᵃᵒᵖ
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@[protected, instance]
def add_opposite.inhabited (α : Type u) [inhabited α] :
inhabited αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.inhabited (α : Type u) [inhabited α] :
inhabited αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.subsingleton (α : Type u) [subsingleton α] :
subsingleton αᵐᵒᵖ
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@[protected, instance]
def add_opposite.subsingleton (α : Type u) [subsingleton α] :
subsingleton αᵃᵒᵖ
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@[protected, instance]
def add_opposite.unique (α : Type u) [unique α] :
unique αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.unique (α : Type u) [unique α] :
unique αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.is_empty (α : Type u) [is_empty α] :
is_empty αᵐᵒᵖ
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@[protected, instance]
def add_opposite.is_empty (α : Type u) [is_empty α] :
is_empty αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.has_zero (α : Type u) [has_zero α] :
has_zero αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.has_one (α : Type u) [has_one α] :
has_one αᵐᵒᵖ
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@[protected, instance]
def add_opposite.has_zero (α : Type u) [has_zero α] :
has_zero αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.has_add (α : Type u) [has_add α] :
has_add αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.has_sub (α : Type u) [has_sub α] :
has_sub αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.has_neg (α : Type u) [has_neg α] :
has_neg αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.has_involutive_neg (α : Type u) [has_involutive_neg α] :
has_involutive_neg αᵐᵒᵖ
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@[protected, instance]
def add_opposite.has_add (α : Type u) [has_add α] :
has_add αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.has_mul (α : Type u) [has_mul α] :
has_mul αᵐᵒᵖ
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@[protected, instance]
def mul_opposite.has_inv (α : Type u) [has_inv α] :
has_inv αᵐᵒᵖ
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@[protected, instance]
def add_opposite.has_neg (α : Type u) [has_neg α] :
has_neg αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.has_involutive_inv (α : Type u) [has_involutive_inv α] :
has_involutive_inv αᵐᵒᵖ
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@[protected, instance]
def add_opposite.has_involutive_neg (α : Type u) [has_involutive_neg α] :
has_involutive_neg αᵃᵒᵖ
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@[protected, instance]
def mul_opposite.has_scalar (α : Type u) (R : Type u_1) [has_scalar R α] :
has_scalar R αᵐᵒᵖ
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@[protected, instance]
def add_opposite.has_scalar (α : Type u) (R : Type u_1) [has_vadd R α] :
has_vadd R αᵃᵒᵖ
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@[simp]
theorem mul_opposite.op_zero (α : Type u) [has_zero α] :
mul_opposite.op 0 = 0
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@[simp]
theorem mul_opposite.unop_zero (α : Type u) [has_zero α] :
mul_opposite.unop 0 = 0
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@[simp]
theorem add_opposite.op_zero (α : Type u) [has_zero α] :
add_opposite.op 0 = 0
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@[simp]
theorem mul_opposite.op_one (α : Type u) [has_one α] :
mul_opposite.op 1 = 1
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@[simp]
theorem mul_opposite.unop_one (α : Type u) [has_one α] :
mul_opposite.unop 1 = 1
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@[simp]
theorem add_opposite.unop_zero (α : Type u) [has_zero α] :
add_opposite.unop 0 = 0
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@[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
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@[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
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@[simp]
theorem mul_opposite.op_neg {α : Type u} [has_neg α] (x : α) :
mul_opposite.op (-x) = -mul_opposite.op x
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@[simp]
theorem mul_opposite.unop_neg {α : Type u} [has_neg α] (x : αᵐᵒᵖ) :
mul_opposite.unop (-x) = -mul_opposite.unop x
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@[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
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@[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
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@[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
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@[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
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@[simp]
theorem add_opposite.op_neg {α : Type u} [has_neg α] (x : α) :
add_opposite.op (-x) = -add_opposite.op x
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@[simp]
theorem mul_opposite.op_inv {α : Type u} [has_inv α] (x : α) :
mul_opposite.op x⁻¹ = (mul_opposite.op x)⁻¹
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@[simp]
theorem mul_opposite.unop_inv {α : Type u} [has_inv α] (x : αᵐᵒᵖ) :
mul_opposite.unop x⁻¹ = (mul_opposite.unop x)⁻¹
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@[simp]
theorem add_opposite.unop_neg {α : Type u} [has_neg α] (x : αᵃᵒᵖ) :
add_opposite.unop (-x) = -add_opposite.unop x
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[simp]
theorem mul_opposite.unop_eq_zero_iff {α : Type u} [has_zero α] (a : αᵐᵒᵖ) :
mul_opposite.unop a = 0 ↔ a = 0
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@[simp]
theorem mul_opposite.op_eq_zero_iff {α : Type u} [has_zero α] (a : α) :
mul_opposite.op a = 0 ↔ a = 0
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theorem mul_opposite.unop_ne_zero_iff {α : Type u} [has_zero α] (a : αᵐᵒᵖ) :
mul_opposite.unop a ≠ 0 ↔ a ≠ 0
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theorem mul_opposite.op_ne_zero_iff {α : Type u} [has_zero α] (a : α) :
mul_opposite.op a ≠ 0 ↔ a ≠ 0
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@[simp]
theorem mul_opposite.unop_eq_one_iff {α : Type u} [has_one α] (a : αᵐᵒᵖ) :
mul_opposite.unop a = 1 ↔ a = 1
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@[simp]
theorem add_opposite.unop_eq_zero_iff {α : Type u} [has_zero α] (a : αᵃᵒᵖ) :
add_opposite.unop a = 0 ↔ a = 0
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@[simp]
theorem add_opposite.op_eq_zero_iff {α : Type u} [has_zero α] (a : α) :
add_opposite.op a = 0 ↔ a = 0
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@[simp]
theorem mul_opposite.op_eq_one_iff {α : Type u} [has_one α] (a : α) :
mul_opposite.op a = 1 ↔ a = 1
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@[protected, instance]
def add_opposite.has_one {α : Type u} [has_one α] :
has_one αᵃᵒᵖ
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@[simp]
theorem add_opposite.op_one {α : Type u} [has_one α] :
add_opposite.op 1 = 1
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@[simp]
theorem add_opposite.unop_one {α : Type u} [has_one α] :
add_opposite.unop 1 = 1
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@[simp]
theorem add_opposite.op_eq_one_iff {α : Type u} [has_one α] {a : α} :
add_opposite.op a = 1 ↔ a = 1
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@[simp]
theorem add_opposite.unop_eq_one_iff {α : Type u} [has_one α] {a : αᵃᵒᵖ} :
add_opposite.unop a = 1 ↔ a = 1
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@[protected, instance]
def add_opposite.has_mul {α : Type u} [has_mul α] :
has_mul αᵃᵒᵖ
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@[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
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@[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
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@[protected, instance]
def add_opposite.has_inv {α : Type u} [has_inv α] :
has_inv αᵃᵒᵖ
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@[protected, instance]
def add_opposite.has_involutive_inv {α : Type u} [has_involutive_inv α] :
has_involutive_inv αᵃᵒᵖ
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@[simp]
theorem add_opposite.op_inv {α : Type u} [has_inv α] (a : α) :
add_opposite.op a⁻¹ = (add_opposite.op a)⁻¹
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@[simp]
theorem add_opposite.unop_inv {α : Type u} [has_inv α] (a : αᵃᵒᵖ) :
add_opposite.unop a⁻¹ = (add_opposite.unop a)⁻¹
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@[protected, instance]
def add_opposite.has_div {α : Type u} [has_div α] :
has_div αᵃᵒᵖ
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@[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
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@[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