Documentation

Mathlib.Data.Set.Intervals.OrderIso

Lemmas about images of intervals under order isomorphisms. #

@[simp]
theorem OrderIso.preimage_Iic {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Iic b = Set.Iic ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Ici {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Ici b = Set.Ici ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Iio {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Iio b = Set.Iio ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Ioi {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Ioi b = Set.Ioi ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Icc {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Icc a b = Set.Icc ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding a) ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Ico {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Ico a b = Set.Ico ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding a) ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Ioc {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Ioc a b = Set.Ioc ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding a) ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.preimage_Ioo {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : β) (b : β) :
(RelIso.toRelEmbedding e).toEmbedding ⁻¹' Set.Ioo a b = Set.Ioo ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding a) ((RelIso.toRelEmbedding (OrderIso.symm e)).toEmbedding b)
@[simp]
theorem OrderIso.image_Iic {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Iic a = Set.Iic ((RelIso.toRelEmbedding e).toEmbedding a)
@[simp]
theorem OrderIso.image_Ici {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Ici a = Set.Ici ((RelIso.toRelEmbedding e).toEmbedding a)
@[simp]
theorem OrderIso.image_Iio {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Iio a = Set.Iio ((RelIso.toRelEmbedding e).toEmbedding a)
@[simp]
theorem OrderIso.image_Ioi {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Ioi a = Set.Ioi ((RelIso.toRelEmbedding e).toEmbedding a)
@[simp]
theorem OrderIso.image_Ioo {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) (b : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Ioo a b = Set.Ioo ((RelIso.toRelEmbedding e).toEmbedding a) ((RelIso.toRelEmbedding e).toEmbedding b)
@[simp]
theorem OrderIso.image_Ioc {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) (b : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Ioc a b = Set.Ioc ((RelIso.toRelEmbedding e).toEmbedding a) ((RelIso.toRelEmbedding e).toEmbedding b)
@[simp]
theorem OrderIso.image_Ico {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) (b : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Ico a b = Set.Ico ((RelIso.toRelEmbedding e).toEmbedding a) ((RelIso.toRelEmbedding e).toEmbedding b)
@[simp]
theorem OrderIso.image_Icc {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) (a : α) (b : α) :
(RelIso.toRelEmbedding e).toEmbedding '' Set.Icc a b = Set.Icc ((RelIso.toRelEmbedding e).toEmbedding a) ((RelIso.toRelEmbedding e).toEmbedding b)
def OrderIso.IicTop {α : Type u_1} [inst : Preorder α] [inst : OrderTop α] :
↑(Set.Iic ) ≃o α

Order isomorphism between Iic (⊤ : α)⊤ : α) and α when α has a top element

Equations
  • One or more equations did not get rendered due to their size.
def OrderIso.IciBot {α : Type u_1} [inst : Preorder α] [inst : OrderBot α] :
↑(Set.Ici ) ≃o α

Order isomorphism between Ici (⊥ : α)⊥ : α) and α when α has a bottom element

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