mathlib documentation

algebra.opposites

Algebraic operations on αᵒᵖ #

This file records several basic facts about the opposite of an algebraic structure, e.g. the opposite of a ring is a ring (with multiplication x * y = yx). Use is made of the identity functions op : α → αᵒᵖ and unop : αᵒᵖ → α.

@[instance]
def opposite.has_add (α : Type u) [has_add α] :
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@[instance]
def opposite.has_sub (α : Type u) [has_sub α] :
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@[instance]
def opposite.has_zero (α : Type u) [has_zero α] :
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@[instance]
def opposite.nontrivial (α : Type u) [nontrivial α] :
@[simp]
theorem opposite.unop_eq_zero_iff (α : Type u) [has_zero α] (a : αᵒᵖ) :
@[simp]
theorem opposite.op_eq_zero_iff (α : Type u) [has_zero α] (a : α) :
opposite.op a = 0 a = 0
@[instance]
Equations
@[instance]
def opposite.has_neg (α : Type u) [has_neg α] :
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@[instance]
def opposite.has_mul (α : Type u) [has_mul α] :
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@[instance]
def opposite.semigroup (α : Type u) [semigroup α] :
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@[instance]
def opposite.has_one (α : Type u) [has_one α] :
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@[simp]
theorem opposite.unop_eq_one_iff (α : Type u) [has_one α] (a : αᵒᵖ) :
@[simp]
theorem opposite.op_eq_one_iff (α : Type u) [has_one α] (a : α) :
opposite.op a = 1 a = 1
@[instance]
Equations
@[instance]
def opposite.comm_monoid (α : Type u) [comm_monoid α] :
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@[instance]
def opposite.has_inv (α : Type u) [has_inv α] :
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@[instance]
def opposite.group (α : Type u) [group α] :
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@[instance]
def opposite.distrib (α : Type u) [distrib α] :
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@[instance]
@[instance]
def opposite.has_scalar (α : Type u) (R : Type u_1) [has_scalar R α] :
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@[instance]
def opposite.mul_action (α : Type u) (R : Type u_1) [monoid R] [mul_action R α] :
Equations
@[simp]
theorem opposite.op_zero (α : Type u) [has_zero α] :
@[simp]
theorem opposite.unop_zero (α : Type u) [has_zero α] :
@[simp]
theorem opposite.op_one (α : Type u) [has_one α] :
@[simp]
theorem opposite.unop_one (α : Type u) [has_one α] :
@[simp]
theorem opposite.op_add {α : Type u} [has_add α] (x y : α) :
@[simp]
theorem opposite.unop_add {α : Type u} [has_add α] (x y : αᵒᵖ) :
@[simp]
theorem opposite.op_neg {α : Type u} [has_neg α] (x : α) :
@[simp]
theorem opposite.unop_neg {α : Type u} [has_neg α] (x : αᵒᵖ) :
@[simp]
theorem opposite.op_mul {α : Type u} [has_mul α] (x y : α) :
@[simp]
theorem opposite.unop_mul {α : Type u} [has_mul α] (x y : αᵒᵖ) :
@[simp]
theorem opposite.op_inv {α : Type u} [has_inv α] (x : α) :
@[simp]
theorem opposite.unop_inv {α : Type u} [has_inv α] (x : αᵒᵖ) :
@[simp]
theorem opposite.op_sub {α : Type u} [add_group α] (x y : α) :
@[simp]
theorem opposite.unop_sub {α : Type u} [add_group α] (x y : αᵒᵖ) :
@[simp]
theorem opposite.op_smul {α : Type u} {R : Type u_1} [has_scalar R α] (c : R) (a : α) :
@[simp]
theorem opposite.unop_smul {α : Type u} {R : Type u_1} [has_scalar R α] (c : R) (a : αᵒᵖ) :
theorem opposite.semiconj_by.op {α : Type u} [has_mul α] {a x y : α} (h : semiconj_by a x y) :
theorem opposite.semiconj_by.unop {α : Type u} [has_mul α] {a x y : αᵒᵖ} (h : semiconj_by a x y) :
@[simp]
theorem opposite.semiconj_by_op {α : Type u} [has_mul α] {a x y : α} :
@[simp]
theorem opposite.semiconj_by_unop {α : Type u} [has_mul α] {a x y : αᵒᵖ} :
theorem opposite.commute.op {α : Type u} [has_mul α] {x y : α} (h : commute x y) :
theorem opposite.commute.unop {α : Type u} [has_mul α] {x y : αᵒᵖ} (h : commute x y) :
@[simp]
theorem opposite.commute_op {α : Type u} [has_mul α] {x y : α} :
@[simp]
theorem opposite.commute_unop {α : Type u} [has_mul α] {x y : αᵒᵖ} :
def opposite.op_add_equiv {α : Type u} [has_add α] :

The function op is an additive equivalence.

Equations
def ring_hom.to_opposite {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R →+* S) (hf : ∀ (x y : R), commute (f x) (f y)) :

A ring homomorphism f : R →+* S such that f x commutes with f y for all x, y defines a ring homomorphism to Sᵒᵖ.

Equations
@[simp]
theorem ring_hom.coe_to_opposite {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R →+* S) (hf : ∀ (x y : R), commute (f x) (f y)) :