File: Mathlib/Topology/Order/Hom/Basic.lean

full_name: ContinuousOrderHom
code: /-- The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms. -/
structure ContinuousOrderHom (α β : Type*) [Preorder α] [Preorder β] [TopologicalSpace α]
  [TopologicalSpace β] extends OrderHom α β where
  continuous_toFun : Continuous toFun
goal: Type (max u_6 u_7)

full_name: ContinuousOrderHom.continuous_toFun
code: /-- The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms. -/
structure ContinuousOrderHom (α β : Type*) [Preorder α] [Preorder β] [TopologicalSpace α]
  [TopologicalSpace β] extends OrderHom α β where
  continuous_toFun : Continuous toFun
goal: Continuous self.toFun

full_name: _aux_Mathlib_Topology_Order_Hom_Basic___unexpand_ContinuousOrderHom_1
code: infixr:25 " →Co " => ContinuousOrderHom
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Topology_Order_Hom_Basic___macroRules_term_→Co__1»
code: infixr:25 " →Co " => ContinuousOrderHom
goal: Lean.Macro

full_name: ContinuousOrderHomClass
code: /-- `ContinuousOrderHomClass F α β` states that `F` is a type of continuous monotone maps.

You should extend this class when you extend `ContinuousOrderHom`. -/
class ContinuousOrderHomClass (F : Type*) (α β : outParam Type*) [Preorder α] [Preorder β]
    [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] extends
    ContinuousMapClass F α β : Prop where
  map_monotone (f : F) : Monotone f
goal: Prop

full_name: ContinuousOrderHomClass.map_monotone
code: /-- `ContinuousOrderHomClass F α β` states that `F` is a type of continuous monotone maps.

You should extend this class when you extend `ContinuousOrderHom`. -/
class ContinuousOrderHomClass (F : Type*) (α β : outParam Type*) [Preorder α] [Preorder β]
    [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] extends
    ContinuousMapClass F α β : Prop where
  map_monotone (f : F) : Monotone f
goal: Monotone ⇑f

full_name: ContinuousOrderHomClass.toOrderHomClass
code: instance (priority := 100) toOrderHomClass  :
    OrderHomClass F α β 
goal: OrderHomClass F α β

full_name: ContinuousOrderHomClass.toContinuousOrderHom
code: /-- Turn an element of a type `F` satisfying `ContinuousOrderHomClass F α β` into an actual
`ContinuousOrderHom`. This is declared as the default coercion from `F` to `α →Co β`. -/
@[coe]
def toContinuousOrderHom (f : F) : α →Co β 
goal: α →Co β

full_name: ContinuousOrderHomClass.instCoeTCContinuousOrderHom
code: instance : CoeTC F (α →Co β) 
goal: CoeTC F (α →Co β)

full_name: ContinuousOrderHom.toContinuousMap
code: /-- Reinterpret a `ContinuousOrderHom` as a `ContinuousMap`. -/
def toContinuousMap (f : α →Co β) : C(α, β) 
goal: C(α, β)

full_name: ContinuousOrderHom.instFunLike
code: instance instFunLike : FunLike (α →Co β) α β 
goal: FunLike (α →Co β) α β

full_name: ContinuousOrderHom.instContinuousOrderHomClass
code: instance : ContinuousOrderHomClass (α →Co β) α β 
goal: ContinuousOrderHomClass (α →Co β) α β

full_name: ContinuousOrderHom.coe_toOrderHom
code: @[simp] theorem coe_toOrderHom (f : α →Co β) : ⇑f.toOrderHom = f 
goal: ⇑f.toOrderHom = ⇑f

full_name: ContinuousOrderHom.toFun_eq_coe
code: theorem toFun_eq_coe {f : α →Co β} : f.toFun = (f : α → β) 
goal: f.toFun = ⇑f

full_name: ContinuousOrderHom.ext
code: @[ext]
theorem ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g 
goal: f = g

full_name: ContinuousOrderHom.copy
code: /-- Copy of a `ContinuousOrderHom` with a new `ContinuousMap` equal to the old one. Useful to fix
definitional equalities. -/
protected def copy (f : α →Co β) (f' : α → β) (h : f' = f) : α →Co β 
goal: α →Co β

full_name: ContinuousOrderHom.coe_copy
code: @[simp]
theorem coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
goal: ⇑(f.copy f' h) = f'

full_name: ContinuousOrderHom.copy_eq
code: theorem copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
goal: f.copy f' h = f

full_name: ContinuousOrderHom.id
code: /-- `id` as a `ContinuousOrderHom`. -/
protected def id : α →Co α 
goal: α →Co α

full_name: ContinuousOrderHom.instInhabited
code: instance : Inhabited (α →Co α) 
goal: Inhabited (α →Co α)

full_name: ContinuousOrderHom.coe_id
code: @[simp]
theorem coe_id : ⇑(ContinuousOrderHom.id α) = id 
goal: ⇑(ContinuousOrderHom.id α) = id

full_name: ContinuousOrderHom.id_apply
code: @[simp]
theorem id_apply (a : α) : ContinuousOrderHom.id α a = a 
goal: (ContinuousOrderHom.id α) a = a

full_name: ContinuousOrderHom.comp
code: /-- Composition of `ContinuousOrderHom`s as a `ContinuousOrderHom`. -/
def comp (f : β →Co γ) (g : α →Co β) : ContinuousOrderHom α γ 
goal: α →Co γ

full_name: ContinuousOrderHom.coe_comp
code: @[simp]
theorem coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g 
goal: ⇑(f.comp g) = ⇑f ∘ ⇑g

full_name: ContinuousOrderHom.comp_apply
code: @[simp]
theorem comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) 
goal: (f.comp g) a = f (g a)

full_name: ContinuousOrderHom.comp_assoc
code: @[simp]
theorem comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) :
    (f.comp g).comp h = f.comp (g.comp h) 
goal: (f.comp g).comp h = f.comp (g.comp h)

full_name: ContinuousOrderHom.comp_id
code: @[simp]
theorem comp_id (f : α →Co β) : f.comp (ContinuousOrderHom.id α) = f 
goal: f.comp (ContinuousOrderHom.id α) = f

full_name: ContinuousOrderHom.id_comp
code: @[simp]
theorem id_comp (f : α →Co β) : (ContinuousOrderHom.id β).comp f = f 
goal: (ContinuousOrderHom.id β).comp f = f

full_name: ContinuousOrderHom.cancel_right
code: @[simp]
theorem cancel_right {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂

full_name: ContinuousOrderHom.cancel_left
code: @[simp]
theorem cancel_left {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂

full_name: ContinuousOrderHom.instPreorder
code: instance : Preorder (α →Co β) 
goal: Preorder (α →Co β)

full_name: ContinuousOrderHom.instPartialOrder
code: instance [PartialOrder β] : PartialOrder (α →Co β) 
goal: PartialOrder (α →Co β)