File: Mathlib/Order/Category/DistLat.lean

full_name: DistLat
code: /-- The category of distributive lattices. -/
def DistLat 
goal: Type (u_1 + 1)

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

full_name: DistLat.instDistribLatticeα
code: instance (X : DistLat) : DistribLattice X 
goal: DistribLattice ↑X

full_name: DistLat.of
code: /-- Construct a bundled `DistLat` from a `DistribLattice` underlying type and typeclass. -/
def of (α : Type*) [DistribLattice α] : DistLat 
goal: DistLat

full_name: DistLat.coe_of
code: @[simp]
theorem coe_of (α : Type*) [DistribLattice α] : ↥(of α) = α 
goal: ↑(DistLat.of α) = α

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

full_name: DistLat.instParentProjectionLatticeDistribLatticeToLattice
code: instance : BundledHom.ParentProjection @DistribLattice.toLattice 
goal: CategoryTheory.BundledHom.ParentProjection @DistribLattice.toLattice

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

full_name: DistLat.hasForgetToLat
code: instance hasForgetToLat : HasForget₂ DistLat Lat 
goal: CategoryTheory.HasForget₂ DistLat Lat

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

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

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

full_name: DistLat.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : DistLat ⥤ DistLat 
goal: ∀ {X Y : DistLat} (a : LatticeHom ↑X ↑Y), DistLat.dual.map a = LatticeHom.dual a

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

full_name: DistLat.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : DistLat ⥤ DistLat 
goal: DistLat.dual.obj X = DistLat.of (↑X)ᵒᵈ

full_name: DistLat.dualEquiv_functor
code: /-- The equivalence between `DistLat` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : DistLat ≌ DistLat 
goal: DistLat.dualEquiv.functor = DistLat.dual

full_name: DistLat.dualEquiv_inverse
code: /-- The equivalence between `DistLat` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : DistLat ≌ DistLat 
goal: DistLat.dualEquiv.inverse = DistLat.dual

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

full_name: distLat_dual_comp_forget_to_Lat
code: theorem distLat_dual_comp_forget_to_Lat :
    DistLat.dual ⋙ forget₂ DistLat Lat = forget₂ DistLat Lat ⋙ Lat.dual 
goal: DistLat.dual ⋙ CategoryTheory.forget₂ DistLat Lat = CategoryTheory.forget₂ DistLat Lat ⋙ Lat.dual