Inverse and multiplication as order isomorphisms in ordered groups

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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] : 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' := (_ : ∀ {x x_1 : α}, -x_1 ≤ -x ↔ x ≤ x_1) }

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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

• (_ : -x ≤ -x ↔ x ≤ x) = (_ : -x ≤ -x ↔ x ≤ x)

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@[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

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@[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

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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)

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@[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

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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] : 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' := (_ : ∀ {x x_1 : α}, x_1⁻¹ ≤ x⁻¹ ↔ x ≤ x_1) }

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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

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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} : a⁻¹ ≤ b ↔ b⁻¹ ≤ a

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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 ≤ b → b⁻¹ ≤ a

Alias of the forward direction of inv_le'.

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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

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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

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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} : a ≤ b⁻¹ ↔ b ≤ a⁻¹

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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⁻¹ → b ≤ a⁻¹

Alias of the forward direction of le_inv'.

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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 ≤ -b → b ≤ -a

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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

• (_ : x + a ≤ x + a ↔ x ≤ x) = (_ : x + a ≤ x + a ↔ x ≤ x)

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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

• OrderIso.addRight a = { toEquiv := Equiv.addRight a, map_rel_iff' := (_ : ∀ {x x_1 : α}, x + a ≤ x_1 + a ↔ x ≤ x_1) }

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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

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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) : (OrderIso.addRight a).toEquiv = Equiv.addRight a

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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) : (OrderIso.mulRight a).toEquiv = Equiv.mulRight a

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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

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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

• OrderIso.mulRight a = { toEquiv := Equiv.mulRight a, map_rel_iff' := (_ : ∀ {x x_1 : α}, x * a ≤ x_1 * a ↔ x ≤ x_1) }

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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) : OrderIso.symm (OrderIso.addRight a) = OrderIso.addRight (-a)

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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) : OrderIso.symm (OrderIso.mulRight a) = OrderIso.mulRight a⁻¹

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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

• OrderIso.addLeft a = { toEquiv := Equiv.addLeft a, map_rel_iff' := (_ : ∀ {x x_1 : α}, a + x ≤ a + x_1 ↔ x ≤ x_1) }

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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

• (_ : a + x ≤ a + x ↔ x ≤ x) = (_ : a + x ≤ a + x ↔ x ≤ x)

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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

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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) : (OrderIso.mulLeft a).toEquiv = Equiv.mulLeft a

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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) : (OrderIso.addLeft a).toEquiv = Equiv.addLeft a

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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

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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

• OrderIso.mulLeft a = { toEquiv := Equiv.mulLeft a, map_rel_iff' := (_ : ∀ {x x_1 : α}, a * x ≤ a * x_1 ↔ x ≤ x_1) }

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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) : OrderIso.symm (OrderIso.addLeft a) = OrderIso.addLeft (-a)

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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) : OrderIso.symm (OrderIso.mulLeft a) = OrderIso.mulLeft a⁻¹