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tendsto_untopA [Nonempty ι] {a : WithTop ι} (ha : a ≠ ⊤) :
Tendsto untopA (𝓝 a) (𝓝 a.untopA) | tendsto_untopD _ ha | lemma | WithTop.tendsto_untopA | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuousOn_untopA [Nonempty ι] : ContinuousOn untopA { a : WithTop ι | a ≠ ⊤ } | continuousOn_untopD _ | lemma | WithTop.continuousOn_untopA | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"ContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_untop (a : {a : WithTop ι | a ≠ ⊤}) :
Tendsto (fun x ↦ untop x.1 x.2) (𝓝 a) (𝓝 (untop a.1 a.2)) | by
have : Nonempty ι := ⟨untop a.1 a.2⟩
simp only [← untopA_eq_untop, ne_eq, coe_setOf, mem_setOf_eq]
exact (tendsto_untopA a.2).comp <| tendsto_subtype_rng.mp tendsto_id | lemma | WithTop.tendsto_untop | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
continuous_untop : Continuous (fun x : {a : WithTop ι | a ≠ ⊤} ↦ untop x.1 x.2) | continuous_iff_continuousAt.mpr tendsto_untop | lemma | WithTop.continuous_untop | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Continuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
neTopHomeomorph : { a : WithTop ι | a ≠ ⊤ } ≃ₜ ι | where
toEquiv := Equiv.withTopSubtypeNe
continuous_toFun := continuous_untop
continuous_invFun := continuous_coe.subtype_mk _ | def | WithTop.neTopHomeomorph | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [] | Homeomorphism between the non-top elements of `WithTop ι` and `ι`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
sumHomeomorph [OrderTop ι] : WithTop ι ≃ₜ ι ⊕ Unit | where
toFun x := if h : x = ⊤ then Sum.inr () else Sum.inl x.untopA
invFun x := match x with
| Sum.inl i => (i : WithTop ι)
| Sum.inr () => ⊤
left_inv x := by cases x <;> simp
right_inv x := by cases x <;> simp
continuous_toFun := by
have h_fr : frontier ({⊤} : Set (WithTop ι)) = ∅ := by
sim... | def | WithTop.sumHomeomorph | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Continuous.comp_continuousOn",
"IsClosed.closure_eq",
"OrderTop",
"Set",
"Set.Finite.isClosed",
"Set.finite_singleton",
"continuous_if'",
"frontier",
"interior",
"interior_eq_iff_isOpen",
"isOpen_Ioi"
] | If `ι` has a top element, then `WithTop ι` is homeomorphic to `ι ⊕ Unit`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
tendsto_nhds_top_iff {α : Type*} {f : Filter α} (x : α → WithTop ι) :
Tendsto x f (𝓝 ⊤) ↔ ∀ (i : ι), ∀ᶠ (a : α) in f, i < x a | by
obtain (h | h) := isEmpty_or_nonempty ι
· simpa using .of_forall fun _ ↦ Subsingleton.elim ..
refine nhds_top_basis.tendsto_right_iff.trans ?_
rw [← Set.forall_mem_range (p := (∀ᶠ a in f, · < x a)), WithTop.range_coe]
simp | lemma | WithTop.tendsto_nhds_top_iff | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"Filter",
"Set.forall_mem_range",
"WithTop.range_coe",
"isEmpty_or_nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_coe_atTop [NoMaxOrder ι] :
Tendsto ((↑) : ι → WithTop ι) atTop (𝓝 ⊤) | by
obtain (h | h) := isEmpty_or_nonempty ι
· simpa using Subsingleton.elim ..
rw [tendsto_nhds_top_iff]
intro i
filter_upwards [atTop_basis_Ioi.mem_of_mem (i := i) trivial]
simp | lemma | WithTop.tendsto_coe_atTop | Topology.Order | Mathlib/Topology/Order/WithTop.lean | [] | [
"NoMaxOrder",
"isEmpty_or_nonempty"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
AlexDisc extends TopCat where
[is_alexandrovDiscrete : AlexandrovDiscrete carrier] | structure | AlexDisc | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexandrovDiscrete",
"TopCat"
] | The category of Alexandrov-discrete spaces. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
category : Category AlexDisc | inferInstanceAs <| Category (InducedCategory _ toTopCat) | instance | AlexDisc.category | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
concreteCategory : ConcreteCategory AlexDisc (C(·, ·)) | inferInstanceAs <| ConcreteCategory (InducedCategory _ toTopCat) _ | instance | AlexDisc.concreteCategory | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instHasForgetToTop : HasForget₂ AlexDisc TopCat | inferInstanceAs <| HasForget₂ (InducedCategory _ toTopCat) _ | instance | AlexDisc.instHasForgetToTop | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"TopCat"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
forgetToTop_full : (forget₂ AlexDisc TopCat).Full | where
map_surjective f := ⟨InducedCategory.homMk f, rfl⟩ | instance | AlexDisc.forgetToTop_full | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"TopCat"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
forgetToTop_faithful : (forget₂ AlexDisc TopCat).Faithful | where
map_injective {X Y f g} h := by
ext x
exact ConcreteCategory.congr_hom h x | instance | AlexDisc.forgetToTop_faithful | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"TopCat"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
of (X : Type*) [TopologicalSpace X] [AlexandrovDiscrete X] : AlexDisc | where
toTopCat := TopCat.of X | abbrev | AlexDisc.of | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"AlexandrovDiscrete",
"TopologicalSpace"
] | Construct a bundled `AlexDisc` from the underlying topological space. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] : ↥(of α) = α | rfl | lemma | AlexDisc.coe_of | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexandrovDiscrete",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
forgetToTop_of (α : Type*) [TopologicalSpace α] [AlexandrovDiscrete α] :
(forget₂ AlexDisc TopCat).obj (of α) = TopCat.of α | rfl | lemma | AlexDisc.forgetToTop_of | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"AlexandrovDiscrete",
"TopCat",
"TopologicalSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_forgetToTop (X : AlexDisc) : ↥((forget₂ _ TopCat).obj X) = X | rfl | lemma | AlexDisc.coe_forgetToTop | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"TopCat"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Iso.mk {α β : AlexDisc} (e : α ≃ₜ β) : α ≅ β | where
hom := ConcreteCategory.ofHom (e : ContinuousMap α β)
inv := ConcreteCategory.ofHom (e.symm : ContinuousMap β α)
hom_inv_id := by ext; apply e.symm_apply_apply
inv_hom_id := by ext; apply e.apply_symm_apply | def | AlexDisc.Iso.mk | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"ContinuousMap"
] | Constructs an equivalence between preorders from an order isomorphism between them. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
alexDiscEquivPreord : AlexDisc ≌ Preord | where
functor := forget₂ _ _ ⋙ topToPreord
inverse.obj X := AlexDisc.of (WithUpperSet X)
inverse.map f := ConcreteCategory.ofHom (WithUpperSet.map f.hom)
unitIso := NatIso.ofComponents fun X ↦ AlexDisc.Iso.mk <| by
dsimp; exact homeoWithUpperSetTopologyorderIso X
counitIso := NatIso.ofComponents fun X ↦ P... | def | alexDiscEquivPreord | Topology.Order.Category | Mathlib/Topology/Order/Category/AlexDisc.lean | [] | [
"AlexDisc",
"AlexDisc.Iso.mk",
"AlexDisc.of",
"Preord",
"Preord.Iso.mk",
"homeoWithUpperSetTopologyorderIso",
"orderIsoSpecializationWithUpperSetTopology",
"symm",
"topToPreord"
] | Sends a topological space to its specialisation order. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
PT | FrameHom L Prop | abbrev | Locale.PT | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"FrameHom"
] | The type of points of a complete lattice `L`, where a *point* of a complete lattice is,
by definition, a frame homomorphism from `L` to `Prop`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
openOfElementHom : FrameHom L (Set (PT L)) | where
toFun u := {x | x u}
map_inf' a b := by simp [Set.setOf_and]
map_top' := by simp
map_sSup' S := by ext; simp [Prop.exists_iff] | def | Locale.openOfElementHom | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"FrameHom",
"Prop.exists_iff",
"Set",
"Set.setOf_and"
] | The frame homomorphism from a complete lattice `L` to the complete lattice of sets of
points of `L`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instTopologicalSpace : TopologicalSpace (PT L) | where
IsOpen s := ∃ u, {x | x u} = s
isOpen_univ := ⟨⊤, by simp⟩
isOpen_inter := by rintro s t ⟨u, rfl⟩ ⟨v, rfl⟩; use u ⊓ v; simp_rw [map_inf]; rfl
isOpen_sUnion S hS := by
choose f hf using hS
use ⨆ t, ⨆ ht, f t ht
simp_rw [map_iSup, iSup_Prop_eq, setOf_exists, hf, sUnion_eq_biUnion] | instance | Locale.PT.instTopologicalSpace | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"IsOpen",
"TopologicalSpace",
"iSup_Prop_eq",
"isOpen_sUnion",
"isOpen_univ",
"map_iSup"
] | The topology on the set of points of the complete lattice `L`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
isOpen_iff (U : Set (PT L)) : IsOpen U ↔ ∃ u : L, {x | x u} = U | Iff.rfl | lemma | Locale.PT.isOpen_iff | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"IsOpen",
"Set"
] | Characterization of when a subset of the space of points is open. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
pt : Locale ⥤ TopCat | where
obj L := .of (PT L.unop)
map f := TopCat.ofHom ⟨fun p ↦ p.comp f.unop.hom,
continuous_def.2 <| by rintro s ⟨u, rfl⟩; use f.unop u; rfl⟩ | def | Locale.pt | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"Locale",
"TopCat",
"TopCat.ofHom"
] | The covariant functor `pt` from the category of locales to the category of
topological spaces, which sends a locale `L` to the topological space `PT L` of homomorphisms
from `L` to `Prop` and a locale homomorphism `f` to a continuous function between the spaces
of points. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
localePointOfSpacePoint (x : X) : PT (Opens X) | where
toFun := (x ∈ ·)
map_inf' _ _ := rfl
map_top' := rfl
map_sSup' S := by simp [Prop.exists_iff] | def | Locale.localePointOfSpacePoint | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"Prop.exists_iff"
] | The unit of the adjunction between locales and topological spaces, which associates with
a point `x` of the space `X` a point of the locale of opens of `X`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
counitAppCont : FrameHom L (Opens <| PT L) | where
toFun u := ⟨openOfElementHom L u, u, rfl⟩
map_inf' a b := by simp
map_top' := by simp
map_sSup' S := by ext; simp | def | Locale.counitAppCont | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"FrameHom"
] | The counit is a frame homomorphism. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
adjunctionTopToLocalePT : topToLocale ⊣ pt | where
unit := { app := fun X ↦ TopCat.ofHom ⟨localePointOfSpacePoint X, continuous_def.2 <|
by rintro _ ⟨u, rfl⟩; simpa using! u.2⟩ }
counit := { app := fun L ↦ ⟨Frm.ofHom (counitAppCont L)⟩ } | def | Locale.adjunctionTopToLocalePT | Topology.Order.Category | Mathlib/Topology/Order/Category/FrameAdjunction.lean | [] | [
"TopCat.ofHom",
"topToLocale"
] | The forgetful functor `topToLocale` is left adjoint to the functor `pt`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
ContinuousOrderHom (α β : Type*) [Preorder α] [Preorder β] [TopologicalSpace α]
[TopologicalSpace β] extends OrderHom α β where
continuous_toFun : Continuous toFun | structure | ContinuousOrderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"Continuous",
"OrderHom",
"Preorder",
"TopologicalSpace"
] | The type of continuous monotone maps from `α` to `β`, aka Priestley homomorphisms. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ContinuousOrderHomClass (F : Type*) (α β : outParam Type*) [Preorder α] [Preorder β]
[TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] : Prop
extends ContinuousMapClass F α β where
map_monotone (f : F) : Monotone f | class | ContinuousOrderHomClass | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousMapClass",
"FunLike",
"Monotone",
"Preorder",
"TopologicalSpace"
] | `ContinuousOrderHomClass F α β` states that `F` is a type of continuous monotone maps.
You should extend this class when you extend `ContinuousOrderHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toContinuousOrderHom (f : F) : α →Co β | { toFun := f
monotone' := ContinuousOrderHomClass.map_monotone f
continuous_toFun := map_continuous f } | def | ContinuousOrderHomClass.toContinuousOrderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
toContinuousMap (f : α →Co β) : C(α, β) | { f with } | def | ContinuousOrderHom.toContinuousMap | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"toContinuousMap"
] | Reinterpret a `ContinuousOrderHom` as a `ContinuousMap`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
instFunLike : FunLike (α →Co β) α β | where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | ContinuousOrderHom.instFunLike | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"FunLike"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_toOrderHom (f : α →Co β) : ⇑f.toOrderHom = f | rfl | theorem | ContinuousOrderHom.coe_toOrderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun_eq_coe {f : α →Co β} : f.toFun = (f : α → β) | rfl | theorem | ContinuousOrderHom.toFun_eq_coe | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {f g : α →Co β} (h : ∀ a, f a = g a) : f = g | DFunLike.ext f g h | theorem | ContinuousOrderHom.ext | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"DFunLike.ext"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy (f : α →Co β) (f' : α → β) (h : f' = f) : α →Co β | ⟨f.toOrderHom.copy f' h, h.symm.subst f.continuous_toFun⟩ | def | ContinuousOrderHom.copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | Copy of a `ContinuousOrderHom` with a new `ContinuousMap` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_copy (f : α →Co β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | theorem | ContinuousOrderHom.coe_copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy_eq (f : α →Co β) (f' : α → β) (h : f' = f) : f.copy f' h = f | DFunLike.ext' h | theorem | ContinuousOrderHom.copy_eq | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"DFunLike.ext'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id : α →Co α | ⟨OrderHom.id, continuous_id⟩ | def | ContinuousOrderHom.id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | `id` as a `ContinuousOrderHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_id : ⇑(ContinuousOrderHom.id α) = id | rfl | theorem | ContinuousOrderHom.coe_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousOrderHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_apply (a : α) : ContinuousOrderHom.id α a = a | rfl | theorem | ContinuousOrderHom.id_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousOrderHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp (f : β →Co γ) (g : α →Co β) : ContinuousOrderHom α γ | ⟨f.toOrderHom.comp g.toOrderHom, f.continuous_toFun.comp g.continuous_toFun⟩ | def | ContinuousOrderHom.comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousOrderHom"
] | Composition of `ContinuousOrderHom`s as a `ContinuousOrderHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g | rfl | theorem | ContinuousOrderHom.coe_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) | rfl | theorem | ContinuousOrderHom.comp_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | theorem | ContinuousOrderHom.comp_assoc | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_id (f : α →Co β) : f.comp (ContinuousOrderHom.id α) = f | ext fun _ => rfl | theorem | ContinuousOrderHom.comp_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousOrderHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_comp (f : α →Co β) : (ContinuousOrderHom.id β).comp f = f | ext fun _ => rfl | theorem | ContinuousOrderHom.id_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"ContinuousOrderHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_right {g₁ g₂ : β →Co γ} {f : α →Co β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, fun h => congr_arg₂ _ h rfl⟩ | theorem | ContinuousOrderHom.cancel_right | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_left {g : β →Co γ} {f₁ f₂ : α →Co β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | ContinuousOrderHom.cancel_left | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Basic.lean | [] | [
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoEpimorphism (α β : Type*) [Preorder α] [Preorder β] extends α →o β where
exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b | structure | PseudoEpimorphism | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"Preorder"
] | The type of pseudo-epimorphisms, aka p-morphisms, aka bounded maps, from `α` to `β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EsakiaHom (α β : Type*) [TopologicalSpace α] [Preorder α] [TopologicalSpace β]
[Preorder β] extends α →Co β where
exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : toFun a ≤ b → ∃ c, a ≤ c ∧ toFun c = b | structure | EsakiaHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"Preorder",
"TopologicalSpace"
] | The type of Esakia morphisms, aka continuous pseudo-epimorphisms, from `α` to `β`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
PseudoEpimorphismClass (F : Type*) (α β : outParam Type*)
[Preorder α] [Preorder β] [FunLike F α β] : Prop
extends OrderHomClass F α β where
exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b | class | PseudoEpimorphismClass | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"FunLike",
"OrderHomClass",
"Preorder"
] | `PseudoEpimorphismClass F α β` states that `F` is a type of `⊔`-preserving morphisms.
You should extend this class when you extend `PseudoEpimorphism`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
EsakiaHomClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α] [Preorder α]
[TopologicalSpace β] [Preorder β] [FunLike F α β] : Prop
extends ContinuousOrderHomClass F α β where
exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b | class | EsakiaHomClass | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"ContinuousOrderHomClass",
"FunLike",
"Preorder",
"TopologicalSpace"
] | `EsakiaHomClass F α β` states that `F` is a type of lattice morphisms.
You should extend this class when you extend `EsakiaHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instFunLike : FunLike (PseudoEpimorphism α β) α β | where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨_, _⟩, _⟩ := f
obtain ⟨⟨_, _⟩, _⟩ := g
congr | instance | PseudoEpimorphism.instFunLike | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"FunLike",
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toOrderHom_eq_coe (f : PseudoEpimorphism α β) : ⇑f.toOrderHom = f | rfl | theorem | PseudoEpimorphism.toOrderHom_eq_coe | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun_eq_coe {f : PseudoEpimorphism α β} : f.toFun = (f : α → β) | rfl | theorem | PseudoEpimorphism.toFun_eq_coe | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {f g : PseudoEpimorphism α β} (h : ∀ a, f a = g a) : f = g | DFunLike.ext f g h | theorem | PseudoEpimorphism.ext | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"DFunLike.ext",
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : PseudoEpimorphism α β | ⟨f.toOrderHom.copy f' h, by simpa only [h.symm, toFun_eq_coe] using! f.exists_map_eq_of_map_le'⟩ | def | PseudoEpimorphism.copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | Copy of a `PseudoEpimorphism` with a new `toFun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_copy (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | theorem | PseudoEpimorphism.coe_copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy_eq (f : PseudoEpimorphism α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | DFunLike.ext' h | theorem | PseudoEpimorphism.copy_eq | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"DFunLike.ext'",
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id : PseudoEpimorphism α α | ⟨OrderHom.id, fun _ b h => ⟨b, h, rfl⟩⟩ | def | PseudoEpimorphism.id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | `id` as a `PseudoEpimorphism`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_id : ⇑(PseudoEpimorphism.id α) = id | rfl | theorem | PseudoEpimorphism.coe_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_id_orderHom : (PseudoEpimorphism.id α : α →o α) = OrderHom.id | rfl | theorem | PseudoEpimorphism.coe_id_orderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"OrderHom.id",
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_apply (a : α) : PseudoEpimorphism.id α a = a | rfl | theorem | PseudoEpimorphism.id_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) : PseudoEpimorphism α γ | ⟨g.toOrderHom.comp f.toOrderHom, fun a b h₀ => by
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁
exact ⟨b, h₂, rfl⟩⟩ | def | PseudoEpimorphism.comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | Composition of `PseudoEpimorphism`s as a `PseudoEpimorphism`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_comp (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
(g.comp f : α → γ) = g ∘ f | rfl | theorem | PseudoEpimorphism.coe_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comp_orderHom (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) :
(g.comp f : α →o γ) = (g : β →o γ).comp f | rfl | theorem | PseudoEpimorphism.coe_comp_orderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_apply (g : PseudoEpimorphism β γ) (f : PseudoEpimorphism α β) (a : α) :
(g.comp f) a = g (f a) | rfl | theorem | PseudoEpimorphism.comp_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_assoc (h : PseudoEpimorphism γ δ) (g : PseudoEpimorphism β γ)
(f : PseudoEpimorphism α β) : (h.comp g).comp f = h.comp (g.comp f) | rfl | theorem | PseudoEpimorphism.comp_assoc | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_id (f : PseudoEpimorphism α β) : f.comp (PseudoEpimorphism.id α) = f | ext fun _ => rfl | theorem | PseudoEpimorphism.comp_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism",
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_comp (f : PseudoEpimorphism α β) : (PseudoEpimorphism.id β).comp f = f | ext fun _ => rfl | theorem | PseudoEpimorphism.id_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism",
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_right {g₁ g₂ : PseudoEpimorphism β γ} {f : PseudoEpimorphism α β}
(hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩ | theorem | PseudoEpimorphism.cancel_right | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_left {g : PseudoEpimorphism β γ} {f₁ f₂ : PseudoEpimorphism α β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | PseudoEpimorphism.cancel_left | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"PseudoEpimorphism",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toPseudoEpimorphism (f : EsakiaHom α β) : PseudoEpimorphism α β | { f with } | def | EsakiaHom.toPseudoEpimorphism | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
instFunLike : FunLike (EsakiaHom α β) α β | where
coe f := f.toFun
coe_injective' f g h := by
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g
congr | instance | EsakiaHom.instFunLike | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"FunLike"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toContinuousOrderHom_coe {f : EsakiaHom α β} :
f.toContinuousOrderHom = (f : α → β) | rfl | theorem | EsakiaHom.toContinuousOrderHom_coe | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
toFun_eq_coe {f : EsakiaHom α β} : f.toFun = (f : α → β) | rfl | theorem | EsakiaHom.toFun_eq_coe | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ext {f g : EsakiaHom α β} (h : ∀ a, f a = g a) : f = g | DFunLike.ext f g h | theorem | EsakiaHom.ext | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"DFunLike.ext",
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : EsakiaHom α β | ⟨f.toContinuousOrderHom.copy f' h, by
simpa only [h.symm, toFun_eq_coe] using! f.exists_map_eq_of_map_le'⟩ | def | EsakiaHom.copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | Copy of an `EsakiaHom` with a new `toFun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_copy (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | theorem | EsakiaHom.coe_copy | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
copy_eq (f : EsakiaHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | DFunLike.ext' h | theorem | EsakiaHom.copy_eq | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"DFunLike.ext'",
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id : EsakiaHom α α | ⟨ContinuousOrderHom.id α, fun _ b h => ⟨b, h, rfl⟩⟩ | def | EsakiaHom.id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | `id` as an `EsakiaHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_id : ⇑(EsakiaHom.id α) = id | rfl | theorem | EsakiaHom.coe_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_id_pseudoEpimorphism :
(EsakiaHom.id α : PseudoEpimorphism α α) = PseudoEpimorphism.id α | rfl | theorem | EsakiaHom.coe_id_pseudoEpimorphism | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom.id",
"PseudoEpimorphism",
"PseudoEpimorphism.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_apply (a : α) : EsakiaHom.id α a = a | rfl | theorem | EsakiaHom.id_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_id_continuousOrderHom : (EsakiaHom.id α : α →Co α) = ContinuousOrderHom.id α | rfl | theorem | EsakiaHom.coe_id_continuousOrderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"ContinuousOrderHom.id",
"EsakiaHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : EsakiaHom α γ | ⟨g.toContinuousOrderHom.comp f.toContinuousOrderHom, fun a b h₀ => by
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁
exact ⟨b, h₂, rfl⟩⟩ | def | EsakiaHom.comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | Composition of `EsakiaHom`s as an `EsakiaHom`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
coe_comp_continuousOrderHom (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(g.comp f : α →Co γ) = (g : β →Co γ).comp f | rfl | theorem | EsakiaHom.coe_comp_continuousOrderHom | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comp_pseudoEpimorphism (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(g.comp f : PseudoEpimorphism α γ) = (g : PseudoEpimorphism β γ).comp f | rfl | theorem | EsakiaHom.coe_comp_pseudoEpimorphism | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"PseudoEpimorphism"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
coe_comp (g : EsakiaHom β γ) (f : EsakiaHom α β) : (g.comp f : α → γ) = g ∘ f | rfl | theorem | EsakiaHom.coe_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_apply (g : EsakiaHom β γ) (f : EsakiaHom α β) (a : α) : (g.comp f) a = g (f a) | rfl | theorem | EsakiaHom.comp_apply | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_assoc (h : EsakiaHom γ δ) (g : EsakiaHom β γ) (f : EsakiaHom α β) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | theorem | EsakiaHom.comp_assoc | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_id (f : EsakiaHom α β) : f.comp (EsakiaHom.id α) = f | ext fun _ => rfl | theorem | EsakiaHom.comp_id | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"EsakiaHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
id_comp (f : EsakiaHom α β) : (EsakiaHom.id β).comp f = f | ext fun _ => rfl | theorem | EsakiaHom.id_comp | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"EsakiaHom.id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_right {g₁ g₂ : EsakiaHom β γ} {f : EsakiaHom α β} (hf : Surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨fun h => ext <| hf.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (comp · f)⟩ | theorem | EsakiaHom.cancel_right | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
cancel_left {g : EsakiaHom β γ} {f₁ f₂ : EsakiaHom α β} (hg : Injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | EsakiaHom.cancel_left | Topology.Order.Hom | Mathlib/Topology/Order/Hom/Esakia.lean | [] | [
"EsakiaHom",
"f₁",
"f₂"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
LowerSemicontinuousOn.exists_isMinOn {s : Set α} (ne_s : s.Nonempty)
(hs : IsCompact s) (hf : LowerSemicontinuousOn f s) :
∃ a ∈ s, IsMinOn f s a | by
simp only [isMinOn_iff]
have _ : Nonempty α := Exists.nonempty ne_s
have _ : Nonempty s := Nonempty.to_subtype ne_s
let φ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)
let ℱ : Filter α := ⨅ a : s, φ (f a)
have : ℱ.NeBot := by
apply iInf_neBot_of_directed _ _
· change Directed GE.ge (fun x ↦ (φ ∘... | theorem | LowerSemicontinuousOn.exists_isMinOn | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Directed",
"Directed.mono_comp",
"Filter",
"IsCompact",
"IsMinOn",
"LowerSemicontinuousOn",
"Set",
"Std.Total.directed",
"isMinOn_iff",
"le_of_not_gt",
"le_refl",
"not_le_of_gt"
] | A lower semicontinuous function attains its lower bound on a nonempty compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
LowerSemicontinuousOn.bddBelow_of_isCompact [Nonempty β] {s : Set α} (hs : IsCompact s)
(hf : LowerSemicontinuousOn f s) : BddBelow (f '' s) | by
cases s.eq_empty_or_nonempty with
| inl h =>
simp only [h, Set.image_empty]
exact bddBelow_empty
| inr h =>
obtain ⟨a, _, has⟩ := LowerSemicontinuousOn.exists_isMinOn h hs hf
exact has.bddBelow | theorem | LowerSemicontinuousOn.bddBelow_of_isCompact | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"IsCompact",
"LowerSemicontinuousOn",
"LowerSemicontinuousOn.exists_isMinOn",
"Set",
"Set.image_empty"
] | A lower semicontinuous function is bounded below on a compact set. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
LowerSemicontinuous (indicator s fun _x => y) | by
intro x z hz
by_cases h : x ∈ s <;> simp [h] at hz
· filter_upwards [hs.mem_nhds h]
simp +contextual [hz]
· refine Filter.Eventually.of_forall fun x' => ?_
by_cases h' : x' ∈ s <;> simp [h', hz.trans_le hy, hz] | theorem | IsOpen.lowerSemicontinuous_indicator | Topology.Semicontinuity | Mathlib/Topology/Semicontinuity/Basic.lean | [] | [
"Filter.Eventually.of_forall",
"IsOpen",
"LowerSemicontinuous",
"by_cases"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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