File: Mathlib/Order/Category/Semilat.lean

full_name: SemilatSupCat.X
code: /-- The category of sup-semilattices with a bottom element. -/
structure SemilatSupCat : Type (u + 1) where
  /-- The underlying type of a sup-semilattice with a bottom element. -/
  protected X : Type u
  [isSemilatticeSup : SemilatticeSup X]
  [isOrderBot : OrderBot.{u} X]
goal: Type u

full_name: SemilatSupCat.isOrderBot
code: /-- The category of sup-semilattices with a bottom element. -/
structure SemilatSupCat : Type (u + 1) where
  /-- The underlying type of a sup-semilattice with a bottom element. -/
  protected X : Type u
  [isSemilatticeSup : SemilatticeSup X]
  [isOrderBot : OrderBot.{u} X]
goal: OrderBot self.X

full_name: SemilatSupCat.isSemilatticeSup
code: /-- The category of sup-semilattices with a bottom element. -/
structure SemilatSupCat : Type (u + 1) where
  /-- The underlying type of a sup-semilattice with a bottom element. -/
  protected X : Type u
  [isSemilatticeSup : SemilatticeSup X]
  [isOrderBot : OrderBot.{u} X]
goal: SemilatticeSup self.X

full_name: SemilatSupCat
code: /-- The category of sup-semilattices with a bottom element. -/
structure SemilatSupCat : Type (u + 1) where
  /-- The underlying type of a sup-semilattice with a bottom element. -/
  protected X : Type u
  [isSemilatticeSup : SemilatticeSup X]
  [isOrderBot : OrderBot.{u} X]
goal: Type (u + 1)

full_name: SemilatInfCat.X
code: /-- The category of inf-semilattices with a top element. -/
structure SemilatInfCat : Type (u + 1) where
  /-- The underlying type of an inf-semilattice with a top element. -/
  protected X : Type u
  [isSemilatticeInf : SemilatticeInf X]
  [isOrderTop : OrderTop.{u} X]
goal: Type u

full_name: SemilatInfCat.isSemilatticeInf
code: /-- The category of inf-semilattices with a top element. -/
structure SemilatInfCat : Type (u + 1) where
  /-- The underlying type of an inf-semilattice with a top element. -/
  protected X : Type u
  [isSemilatticeInf : SemilatticeInf X]
  [isOrderTop : OrderTop.{u} X]
goal: SemilatticeInf self.X

full_name: SemilatInfCat.isOrderTop
code: /-- The category of inf-semilattices with a top element. -/
structure SemilatInfCat : Type (u + 1) where
  /-- The underlying type of an inf-semilattice with a top element. -/
  protected X : Type u
  [isSemilatticeInf : SemilatticeInf X]
  [isOrderTop : OrderTop.{u} X]
goal: OrderTop self.X

full_name: SemilatInfCat
code: /-- The category of inf-semilattices with a top element. -/
structure SemilatInfCat : Type (u + 1) where
  /-- The underlying type of an inf-semilattice with a top element. -/
  protected X : Type u
  [isSemilatticeInf : SemilatticeInf X]
  [isOrderTop : OrderTop.{u} X]
goal: Type (u + 1)

full_name: SemilatSupCat.instCoeSortType
code: instance : CoeSort SemilatSupCat Type* 
goal: CoeSort SemilatSupCat (Type u_1)

full_name: SemilatSupCat.of
code: /-- Construct a bundled `SemilatSupCat` from a `SemilatticeSup`. -/
def of (α : Type*) [SemilatticeSup α] [OrderBot α] : SemilatSupCat 
goal: SemilatSupCat

full_name: SemilatSupCat.coe_of
code: @[simp]
theorem coe_of (α : Type*) [SemilatticeSup α] [OrderBot α] : ↥(of α) = α 
goal: (SemilatSupCat.of α).X = α

full_name: SemilatSupCat.instInhabited
code: instance : Inhabited SemilatSupCat 
goal: Inhabited SemilatSupCat

full_name: SemilatSupCat.instLargeCategory
code: instance : LargeCategory.{u} SemilatSupCat 
goal: CategoryTheory.LargeCategory SemilatSupCat

full_name: SemilatSupCat.instFunLike
code: instance instFunLike (X Y : SemilatSupCat) : FunLike (X ⟶ Y) X Y 
goal: FunLike (X ⟶ Y) X.X Y.X

full_name: SemilatSupCat.instConcreteCategory
code: instance : ConcreteCategory SemilatSupCat 
goal: CategoryTheory.ConcreteCategory SemilatSupCat

full_name: SemilatSupCat.hasForgetToPartOrd
code: instance hasForgetToPartOrd : HasForget₂ SemilatSupCat PartOrd 
goal: CategoryTheory.HasForget₂ SemilatSupCat PartOrd

full_name: SemilatSupCat.coe_forget_to_partOrd
code: @[simp]
theorem coe_forget_to_partOrd (X : SemilatSupCat) :
    ↥((forget₂ SemilatSupCat PartOrd).obj X) = ↥X 
goal: ↑((CategoryTheory.forget₂ SemilatSupCat PartOrd).obj X) = X.X

full_name: SemilatInfCat.instCoeSortType
code: instance : CoeSort SemilatInfCat Type* 
goal: CoeSort SemilatInfCat (Type u_1)

full_name: SemilatInfCat.of
code: /-- Construct a bundled `SemilatInfCat` from a `SemilatticeInf`. -/
def of (α : Type*) [SemilatticeInf α] [OrderTop α] : SemilatInfCat 
goal: SemilatInfCat

full_name: SemilatInfCat.coe_of
code: @[simp]
theorem coe_of (α : Type*) [SemilatticeInf α] [OrderTop α] : ↥(of α) = α 
goal: (SemilatInfCat.of α).X = α

full_name: SemilatInfCat.instInhabited
code: instance : Inhabited SemilatInfCat 
goal: Inhabited SemilatInfCat

full_name: SemilatInfCat.instLargeCategory
code: instance : LargeCategory.{u} SemilatInfCat 
goal: CategoryTheory.LargeCategory SemilatInfCat

full_name: SemilatInfCat.instFunLike
code: instance instFunLike (X Y : SemilatInfCat) : FunLike (X ⟶ Y) X Y 
goal: FunLike (X ⟶ Y) X.X Y.X

full_name: SemilatInfCat.instConcreteCategory
code: instance : ConcreteCategory SemilatInfCat 
goal: CategoryTheory.ConcreteCategory SemilatInfCat

full_name: SemilatInfCat.hasForgetToPartOrd
code: instance hasForgetToPartOrd : HasForget₂ SemilatInfCat PartOrd 
goal: CategoryTheory.HasForget₂ SemilatInfCat PartOrd

full_name: SemilatInfCat.coe_forget_to_partOrd
code: @[simp]
theorem coe_forget_to_partOrd (X : SemilatInfCat) :
    ↥((forget₂ SemilatInfCat PartOrd).obj X) = ↥X 
goal: ↑((CategoryTheory.forget₂ SemilatInfCat PartOrd).obj X) = X.X

full_name: SemilatSupCat.Iso.mk
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatSupCat.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

full_name: SemilatSupCat.Iso.mk_inv_toSupHom_toFun
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatSupCat.{u}} (e : α ≃o β) : α ≅ β 
goal: (SemilatSupCat.Iso.mk e).inv.toSupHom a = e.symm a

full_name: SemilatSupCat.Iso.mk_hom_toSupHom_toFun
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatSupCat.{u}} (e : α ≃o β) : α ≅ β 
goal: (SemilatSupCat.Iso.mk e).hom.toSupHom a = e a

full_name: SemilatSupCat.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatSupCat ⥤ SemilatInfCat 
goal: SemilatSupCat.dual.obj X = SemilatInfCat.of X.Xᵒᵈ

full_name: SemilatSupCat.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatSupCat ⥤ SemilatInfCat 
goal: SemilatSupCat.dual.map a = SupBotHom.dual a

full_name: SemilatSupCat.dual
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatSupCat ⥤ SemilatInfCat 
goal: SemilatSupCat ⥤ SemilatInfCat

full_name: SemilatInfCat.Iso.mk
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatInfCat.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

full_name: SemilatInfCat.Iso.mk_hom_toInfHom_toFun
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatInfCat.{u}} (e : α ≃o β) : α ≅ β 
goal: (SemilatInfCat.Iso.mk e).hom.toInfHom a = e a

full_name: SemilatInfCat.Iso.mk_inv_toInfHom_toFun
code: /-- Constructs an isomorphism of lattices from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : SemilatInfCat.{u}} (e : α ≃o β) : α ≅ β 
goal: (SemilatInfCat.Iso.mk e).inv.toInfHom a = e.symm a

full_name: SemilatInfCat.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatInfCat ⥤ SemilatSupCat 
goal: SemilatInfCat.dual.map a = InfTopHom.dual a

full_name: SemilatInfCat.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatInfCat ⥤ SemilatSupCat 
goal: SemilatInfCat.dual.obj X = SemilatSupCat.of X.Xᵒᵈ

full_name: SemilatInfCat.dual
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : SemilatInfCat ⥤ SemilatSupCat 
goal: SemilatInfCat ⥤ SemilatSupCat

full_name: SemilatSupCatEquivSemilatInfCat_inverse
code: /-- The equivalence between `SemilatSupCat` and `SemilatInfCat` induced by `OrderDual` both ways. -/
@[simps functor inverse]
def SemilatSupCatEquivSemilatInfCat : SemilatSupCat ≌ SemilatInfCat 
goal: SemilatSupCatEquivSemilatInfCat.inverse = SemilatInfCat.dual

full_name: SemilatSupCatEquivSemilatInfCat_functor
code: /-- The equivalence between `SemilatSupCat` and `SemilatInfCat` induced by `OrderDual` both ways. -/
@[simps functor inverse]
def SemilatSupCatEquivSemilatInfCat : SemilatSupCat ≌ SemilatInfCat 
goal: SemilatSupCatEquivSemilatInfCat.functor = SemilatSupCat.dual

full_name: SemilatSupCatEquivSemilatInfCat
code: /-- The equivalence between `SemilatSupCat` and `SemilatInfCat` induced by `OrderDual` both ways. -/
@[simps functor inverse]
def SemilatSupCatEquivSemilatInfCat : SemilatSupCat ≌ SemilatInfCat 
goal: SemilatSupCat ≌ SemilatInfCat

full_name: SemilatSupCat_dual_comp_forget_to_partOrd
code: theorem SemilatSupCat_dual_comp_forget_to_partOrd :
    SemilatSupCat.dual ⋙ forget₂ SemilatInfCat PartOrd =
      forget₂ SemilatSupCat PartOrd ⋙ PartOrd.dual 
goal: SemilatSupCat.dual ⋙ CategoryTheory.forget₂ SemilatInfCat PartOrd =
  CategoryTheory.forget₂ SemilatSupCat PartOrd ⋙ PartOrd.dual

full_name: SemilatInfCat_dual_comp_forget_to_partOrd
code: theorem SemilatInfCat_dual_comp_forget_to_partOrd :
    SemilatInfCat.dual ⋙ forget₂ SemilatSupCat PartOrd =
      forget₂ SemilatInfCat PartOrd ⋙ PartOrd.dual 
goal: SemilatInfCat.dual ⋙ CategoryTheory.forget₂ SemilatSupCat PartOrd =
  CategoryTheory.forget₂ SemilatInfCat PartOrd ⋙ PartOrd.dual