Order homomorphisms and sets

ink

theorem OrderIso.range_eq {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) : Set.range ↑(RelIso.toRelEmbedding e).toEmbedding = Set.univ

ink

@[simp]

theorem OrderIso.symm_image_image {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set α) : ↑(RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding '' (↑(RelIso.toRelEmbedding e).toEmbedding '' s) = s

ink

@[simp]

theorem OrderIso.image_symm_image {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set β) : ↑(RelIso.toRelEmbedding e).toEmbedding '' (↑(RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding '' s) = s

ink

theorem OrderIso.image_eq_preimage {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set α) : ↑(RelIso.toRelEmbedding e).toEmbedding '' s = ↑(RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding ⁻¹' s

ink

@[simp]

theorem OrderIso.preimage_symm_preimage {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set α) : ↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' (↑(RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding ⁻¹' s) = s

ink

@[simp]

theorem OrderIso.symm_preimage_preimage {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set β) : ↑(RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding ⁻¹' (↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' s) = s

ink

@[simp]

theorem OrderIso.image_preimage {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set β) : ↑(RelIso.toRelEmbedding e).toEmbedding '' (↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' s) = s

ink

@[simp]

theorem OrderIso.preimage_image {α : Type u_1} {β : Type u_2} [inst : LE α] [inst : LE β] (e : α ≃o β) (s : Set α) : ↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' (↑(RelIso.toRelEmbedding e).toEmbedding '' s) = s

ink

def OrderIso.setCongr {α : Type u_1} [inst : Preorder α] (s : Set α) (t : Set α) (h : s = t) : ↑s ≃o ↑t

Order isomorphism between two equal sets.

Equations

• One or more equations did not get rendered due to their size.

ink

def OrderIso.Set.univ {α : Type u_1} [inst : Preorder α] : ↑Set.univ ≃o α

Order isomorphism between univ : Set α and α.

Equations

• One or more equations did not get rendered due to their size.

ink

noncomputable def StrictMonoOn.orderIso {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (s : Set α) (hf : StrictMonoOn f s) : ↑s ≃o ↑(f '' s)

If a function f is strictly monotone on a set s, then it defines an order isomorphism between s and its image.

Equations

• StrictMonoOn.orderIso f s hf = { toEquiv := Set.BijOn.equiv f (_ : Set.BijOn f s (f '' s)), map_rel_iff' := (_ : ∀ {a b : ↑s}, f ↑a ≤ f ↑b ↔ ↑a ≤ ↑b) }

ink

@[simp]

theorem StrictMono.orderIso_apply {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) (a : α) : ↑(RelIso.toRelEmbedding (StrictMono.orderIso f h_mono)).toEmbedding a = { val := f a, property := (_ : ∃ y, f y = f a) }

ink

noncomputable def StrictMono.orderIso {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) : α ≃o ↑(Set.range f)

A strictly monotone function from a linear order is an order isomorphism between its domain and its range.

Equations

• StrictMono.orderIso f h_mono = { toEquiv := Equiv.ofInjective f (_ : Function.Injective f), map_rel_iff' := (_ : ∀ {a b : α}, f a ≤ f b ↔ a ≤ b) }

ink

noncomputable def StrictMono.orderIsoOfSurjective {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) (h_surj : Function.Surjective f) : α ≃o β

A strictly monotone surjective function from a linear order is an order isomorphism.

Equations

• One or more equations did not get rendered due to their size.

ink

@[simp]

theorem StrictMono.coe_orderIsoOfSurjective {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) (h_surj : Function.Surjective f) : ↑(RelIso.toRelEmbedding (StrictMono.orderIsoOfSurjective f h_mono h_surj)).toEmbedding = f

ink

@[simp]

theorem StrictMono.orderIsoOfSurjective_symm_apply_self {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) (h_surj : Function.Surjective f) (a : α) : ↑(RelIso.toRelEmbedding (OrderIso.symm (StrictMono.orderIsoOfSurjective f h_mono h_surj))).toEmbedding (f a) = a

ink

theorem StrictMono.orderIsoOfSurjective_self_symm_apply {α : Type u_2} {β : Type u_1} [inst : LinearOrder α] [inst : Preorder β] (f : α → β) (h_mono : StrictMono f) (h_surj : Function.Surjective f) (b : β) : f (↑(RelIso.toRelEmbedding (OrderIso.symm (StrictMono.orderIsoOfSurjective f h_mono h_surj))).toEmbedding b) = b

ink

@[simp]

theorem OrderIso.compl_symm_apply (α : Type u_1) [inst : BooleanAlgebra α] : ∀ (a : αᵒᵈ), ↑(RelIso.toRelEmbedding (RelIso.symm (OrderIso.compl α))).toEmbedding a = (compl ∘ ↑OrderDual.ofDual) a

ink

@[simp]

theorem OrderIso.compl_apply (α : Type u_1) [inst : BooleanAlgebra α] : ∀ (a : α), ↑(RelIso.toRelEmbedding (OrderIso.compl α)).toEmbedding a = (↑OrderDual.toDual ∘ compl) a

ink

def OrderIso.compl (α : Type u_1) [inst : BooleanAlgebra α] : α ≃o αᵒᵈ

Taking complements as an order isomorphism to the order dual.

Equations

• One or more equations did not get rendered due to their size.

ink

theorem compl_strictAnti (α : Type u_1) [inst : BooleanAlgebra α] : StrictAnti compl

ink

theorem compl_antitone (α : Type u_1) [inst : BooleanAlgebra α] : Antitone compl