Inverse and multiplication as order isomorphisms in ordered groups

def OrderIso.inv (α : Type u) [Group α] [LE α] [MulLeftMono α] [MulRightMono α] : α ≃o αᵒᵈ

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

Equations

• OrderIso.inv α = { toEquiv := Equiv.trans (Equiv.inv α) OrderDual.toDual, map_rel_iff' := ⋯ }

def OrderIso.neg (α : Type u) [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] : α ≃o αᵒᵈ

x ↦ -x as an order-reversing equivalence.

Equations

• OrderIso.neg α = { toEquiv := Equiv.trans (Equiv.neg α) OrderDual.toDual, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.neg_symm_apply (α : Type u) [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a✝ : αᵒᵈ) : (RelIso.symm (neg α)) a✝ = -OrderDual.ofDual a✝

@[simp]

theorem OrderIso.inv_apply (α : Type u) [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a✝ : α) : (inv α) a✝ = OrderDual.toDual a✝⁻¹

@[simp]

theorem OrderIso.inv_symm_apply (α : Type u) [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a✝ : αᵒᵈ) : (RelIso.symm (inv α)) a✝ = (OrderDual.ofDual a✝)⁻¹

@[simp]

theorem OrderIso.neg_apply (α : Type u) [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a✝ : α) : (neg α) a✝ = OrderDual.toDual (-a✝)

theorem inv_le' {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] {a b : α} : a⁻¹ ≤ b ↔ b⁻¹ ≤ a

theorem neg_le {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] {a b : α} : -a ≤ b ↔ -b ≤ a

theorem inv_le_of_inv_le' {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] {a b : α} : a⁻¹ ≤ b → b⁻¹ ≤ a

Alias of the forward direction of inv_le'.

theorem neg_le_of_neg_le {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] {a b : α} : -a ≤ b → -b ≤ a

theorem le_inv' {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] {a b : α} : a ≤ b⁻¹ ↔ b ≤ a⁻¹

theorem le_neg {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] {a b : α} : a ≤ -b ↔ b ≤ -a

def OrderIso.divLeft {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a : α) : α ≃o αᵒᵈ

x ↦ a / x as an order-reversing equivalence.

Equations

• OrderIso.divLeft a = { toEquiv := (Equiv.divLeft a).trans OrderDual.toDual, map_rel_iff' := ⋯ }

def OrderIso.subLeft {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a : α) : α ≃o αᵒᵈ

x ↦ a - x as an order-reversing equivalence.

Equations

• OrderIso.subLeft a = { toEquiv := (Equiv.subLeft a).trans OrderDual.toDual, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.divLeft_apply {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a a✝ : α) : (divLeft a) a✝ = OrderDual.toDual (a / a✝)

@[simp]

theorem OrderIso.subLeft_apply {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a a✝ : α) : (subLeft a) a✝ = OrderDual.toDual (a - a✝)

@[simp]

theorem OrderIso.subLeft_symm_apply {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] (a : α) (a✝ : αᵒᵈ) : (RelIso.symm (subLeft a)) a✝ = -OrderDual.ofDual a✝ + a

@[simp]

theorem OrderIso.divLeft_symm_apply {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] (a : α) (a✝ : αᵒᵈ) : (RelIso.symm (divLeft a)) a✝ = (OrderDual.ofDual a✝)⁻¹ * a

theorem le_inv_of_le_inv {α : Type u} [Group α] [LE α] [MulLeftMono α] [MulRightMono α] {a b : α} : a ≤ b⁻¹ → b ≤ a⁻¹

Alias of the forward direction of le_inv'.

theorem le_neg_of_le_neg {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] [AddRightMono α] {a b : α} : a ≤ -b → b ≤ -a

def OrderIso.mulRight {α : Type u} [Group α] [LE α] [MulRightMono α] (a : α) : α ≃o α

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

Equations

• OrderIso.mulRight a = { toEquiv := Equiv.mulRight a, map_rel_iff' := ⋯ }

def OrderIso.addRight {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a : α) : α ≃o α

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

Equations

• OrderIso.addRight a = { toEquiv := Equiv.addRight a, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.mulRight_apply {α : Type u} [Group α] [LE α] [MulRightMono α] (a x : α) : (mulRight a) x = x * a

@[simp]

theorem OrderIso.addRight_toEquiv {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a : α) : (addRight a).toEquiv = Equiv.addRight a

@[simp]

theorem OrderIso.mulRight_toEquiv {α : Type u} [Group α] [LE α] [MulRightMono α] (a : α) : (mulRight a).toEquiv = Equiv.mulRight a

@[simp]

theorem OrderIso.addRight_apply {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a x : α) : (addRight a) x = x + a

@[simp]

theorem OrderIso.mulRight_symm {α : Type u} [Group α] [LE α] [MulRightMono α] (a : α) : (mulRight a).symm = mulRight a⁻¹

@[simp]

theorem OrderIso.addRight_symm {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a : α) : (addRight a).symm = addRight (-a)

def OrderIso.divRight {α : Type u} [Group α] [LE α] [MulRightMono α] (a : α) : α ≃o α

x ↦ x / a as an order isomorphism.

Equations

• OrderIso.divRight a = { toEquiv := Equiv.divRight a, map_rel_iff' := ⋯ }

def OrderIso.subRight {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a : α) : α ≃o α

x ↦ x - a as an order isomorphism.

Equations

• OrderIso.subRight a = { toEquiv := Equiv.subRight a, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.divRight_apply {α : Type u} [Group α] [LE α] [MulRightMono α] (a b : α) : (divRight a) b = b / a

@[simp]

theorem OrderIso.subRight_symm_apply {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a b : α) : (RelIso.symm (subRight a)) b = b + a

@[simp]

theorem OrderIso.subRight_apply {α : Type u} [AddGroup α] [LE α] [AddRightMono α] (a b : α) : (subRight a) b = b - a

@[simp]

theorem OrderIso.divRight_symm_apply {α : Type u} [Group α] [LE α] [MulRightMono α] (a b : α) : (RelIso.symm (divRight a)) b = b * a

def OrderIso.mulLeft {α : Type u} [Group α] [LE α] [MulLeftMono α] (a : α) : α ≃o α

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

Equations

• OrderIso.mulLeft a = { toEquiv := Equiv.mulLeft a, map_rel_iff' := ⋯ }

def OrderIso.addLeft {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] (a : α) : α ≃o α

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

Equations

• OrderIso.addLeft a = { toEquiv := Equiv.addLeft a, map_rel_iff' := ⋯ }

@[simp]

theorem OrderIso.mulLeft_apply {α : Type u} [Group α] [LE α] [MulLeftMono α] (a x : α) : (mulLeft a) x = a * x

@[simp]

theorem OrderIso.mulLeft_toEquiv {α : Type u} [Group α] [LE α] [MulLeftMono α] (a : α) : (mulLeft a).toEquiv = Equiv.mulLeft a

@[simp]

theorem OrderIso.addLeft_toEquiv {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] (a : α) : (addLeft a).toEquiv = Equiv.addLeft a

@[simp]

theorem OrderIso.addLeft_apply {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] (a x : α) : (addLeft a) x = a + x

@[simp]

theorem OrderIso.mulLeft_symm {α : Type u} [Group α] [LE α] [MulLeftMono α] (a : α) : (mulLeft a).symm = mulLeft a⁻¹

@[simp]

theorem OrderIso.addLeft_symm {α : Type u} [AddGroup α] [LE α] [AddLeftMono α] (a : α) : (addLeft a).symm = addLeft (-a)