Documentation

Mathlib.Algebra.Order.Group.OrderIso

Inverse and multiplication as order isomorphisms in ordered groups #

def OrderIso.neg (α : Type u) [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] :

x ↦ -x as an order-reversing equivalence.

Equations
def OrderIso.neg.proof_1 (α : Type u_1) [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] :
∀ {x x_1 : α}, -x_1 -x x x_1
Equations
@[simp]
theorem OrderIso.neg_symm_apply (α : Type u) [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] :
∀ (a : αᵒᵈ), (RelIso.toRelEmbedding (RelIso.symm (OrderIso.neg α))).toEmbedding a = -OrderDual.ofDual a
@[simp]
theorem OrderIso.inv_apply (α : Type u) [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] :
∀ (a : α), (RelIso.toRelEmbedding (OrderIso.inv α)).toEmbedding a = OrderDual.toDual a⁻¹
@[simp]
theorem OrderIso.neg_apply (α : Type u) [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] :
∀ (a : α), (RelIso.toRelEmbedding (OrderIso.neg α)).toEmbedding a = OrderDual.toDual (-a)
@[simp]
theorem OrderIso.inv_symm_apply (α : Type u) [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] :
∀ (a : αᵒᵈ), (RelIso.toRelEmbedding (RelIso.symm (OrderIso.inv α))).toEmbedding a = (OrderDual.ofDual a)⁻¹
def OrderIso.inv (α : Type u) [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] :

x ↦ x⁻¹ as an order-reversing equivalence.

Equations
theorem neg_le {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] {a : α} {b : α} :
-a b -b a
theorem inv_le' {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] {a : α} {b : α} :
theorem inv_le_of_inv_le' {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] {a : α} {b : α} :
a⁻¹ bb⁻¹ a

Alias of the forward direction of inv_le'.

theorem neg_le_of_neg_le {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] {a : α} {b : α} :
-a b-b a
theorem le_neg {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] {a : α} {b : α} :
a -b b -a
theorem le_inv' {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] {a : α} {b : α} :
theorem le_inv_of_le_inv {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] {a : α} {b : α} :
a b⁻¹b a⁻¹

Alias of the forward direction of le_inv'.

theorem le_neg_of_le_neg {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] {a : α} {b : α} :
a -bb -a
def OrderIso.addRight.proof_1 {α : Type u_1} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
∀ {x x_1 : α}, x + a x_1 + a x x_1
Equations
def OrderIso.addRight {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
α ≃o α

Equiv.addRight as an OrderIso. See also OrderEmbedding.addRight.

Equations
@[simp]
theorem OrderIso.addRight_apply {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) (x : α) :
(RelIso.toRelEmbedding (OrderIso.addRight a)).toEmbedding x = x + a
@[simp]
theorem OrderIso.addRight_toEquiv {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.mulRight_toEquiv {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.mulRight_apply {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) (x : α) :
(RelIso.toRelEmbedding (OrderIso.mulRight a)).toEmbedding x = x * a
def OrderIso.mulRight {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :
α ≃o α

Equiv.mulRight as an OrderIso. See also OrderEmbedding.mulRight.

Equations
@[simp]
theorem OrderIso.addRight_symm {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.mulRight_symm {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :
def OrderIso.addLeft {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
α ≃o α

Equiv.addLeft as an OrderIso. See also OrderEmbedding.addLeft.

Equations
def OrderIso.addLeft.proof_1 {α : Type u_1} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
∀ {x x_1 : α}, a + x a + x_1 x x_1
Equations
@[simp]
theorem OrderIso.addLeft_apply {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) (x : α) :
(RelIso.toRelEmbedding (OrderIso.addLeft a)).toEmbedding x = a + x
@[simp]
theorem OrderIso.mulLeft_toEquiv {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.addLeft_toEquiv {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.mulLeft_apply {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) (x : α) :
(RelIso.toRelEmbedding (OrderIso.mulLeft a)).toEmbedding x = a * x
def OrderIso.mulLeft {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :
α ≃o α

Equiv.mulLeft as an OrderIso. See also OrderEmbedding.mulLeft.

Equations
@[simp]
theorem OrderIso.addLeft_symm {α : Type u} [inst : AddGroup α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x x_1] (a : α) :
@[simp]
theorem OrderIso.mulLeft_symm {α : Type u} [inst : Group α] [inst : LE α] [inst : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x x_1] (a : α) :