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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremustex2sym 22001* In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than half 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑉))
 
Theoremustex3sym 22002* In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than a third of 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))
 
Theoremustref 22003 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑉𝐴)
 
Theoremust0 22004 The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
(UnifOn‘∅) = {{∅}}
 
Theoremustn0 22005 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
¬ ∅ ∈ ran UnifOn
 
Theoremustund 22006 If two intersecting sets 𝐴 and 𝐵 are both small in 𝑉, their union is small in (𝑉↑2). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)    &   (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)    &   (𝜑 → (𝐴𝐵) ≠ ∅)       (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))
 
Theoremustelimasn 22007 Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
 
Theoremustneism 22008 For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉𝑉). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))
 
Theoremelrnust 22009 First direction for ustbas 22012. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
 
Theoremustbas2 22010 Second direction for ustbas 22012. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
 
Theoremustuni 22011 The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
 
Theoremustbas 22012 Recover the base of an uniform structure 𝑈. ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
𝑋 = dom 𝑈       (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))
 
Theoremustimasn 22013 Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 5-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
 
Theoremtrust 22014 The trace of a uniform structure 𝑈 on a subset 𝐴 is a uniform structure on 𝐴. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
 
12.3.2  The topology induced by an uniform structure
 
Syntaxcutop 22015 Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
 
Definitiondf-utop 22016* Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
 
Theoremutopval 22017* The topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
 
Theoremelutop 22018* Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
 
Theoremutoptop 22019 The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
 
Theoremutopbas 22020 The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
 
Theoremutoptopon 22021 Topology induced by a uniform structure 𝑈 with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
 
Theoremrestutop 22022 Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))
 
Theoremrestutopopn 22023 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈t (𝐴 × 𝐴))))
 
Theoremustuqtoplem 22024* Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))
 
Theoremustuqtop0 22025* Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
 
Theoremustuqtop1 22026* Lemma for ustuqtop 22031, similar to ssnei2 20901. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
 
Theoremustuqtop2 22027* Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
 
Theoremustuqtop3 22028* Lemma for ustuqtop 22031, similar to elnei 20896. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
 
Theoremustuqtop4 22029* Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
 
Theoremustuqtop5 22030* Lemma for ustuqtop 22031. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
 
Theoremustuqtop 22031* For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) is the filter of the neighborhoods of 𝑝 for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
 
Theoremutopsnneiplem 22032* The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝐽 = (unifTop‘𝑈)    &   𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
 
Theoremutopsnneip 22033* The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
 
Theoremutopsnnei 22034 Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
 
Theoremutop2nei 22035 For any symmetrical entourage 𝑉 and any relation 𝑀, build a neighborhood of 𝑀. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
 
Theoremutop3cls 22036 Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝐽 = (unifTop‘𝑈)       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀𝑉)))
 
Theoremutopreg 22037 All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
 
12.3.3  Uniform Spaces
 
Syntaxcuss 22038 Extend class notation with the Uniform Structure extractor function.
class UnifSt
 
Syntaxcusp 22039 Extend class notation with the class of uniform spaces.
class UnifSp
 
Syntaxctus 22040 Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp
 
Definitiondf-uss 22041 Define the uniform structure extractor function. Similarly with df-topn 16065 this differs from df-unif 15946 when a structure has been restricted using df-ress 15846; in this case the UnifSet component will still have a uniform set over the larger set, and this function fixes this by restricting the uniform set as well. (Contributed by Thierry Arnoux, 1-Dec-2017.)
UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t ((Base‘𝑓) × (Base‘𝑓))))
 
Definitiondf-usp 22042 Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}
 
Definitiondf-tus 22043 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))
 
Theoremussval 22044 The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6172 to avoid requiring 𝑊 to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSet‘𝑊)       (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
 
Theoremussid 22045 In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSet‘𝑊)       ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))
 
Theoremisusp 22046 The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSt‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
 
Theoremressunif 22047 UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
𝐻 = (𝐺s 𝐴)    &   𝑈 = (UnifSet‘𝐺)       (𝐴𝑉𝑈 = (UnifSet‘𝐻))
 
Theoremressuss 22048 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
(𝐴𝑉 → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
 
Theoremressust 22049 The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
𝑋 = (Base‘𝑊)    &   𝑇 = (UnifSt‘(𝑊s 𝐴))       ((𝑊 ∈ UnifSp ∧ 𝐴𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
 
Theoremressusp 22050 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)
 
Theoremtusval 22051 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
 
Theoremtuslem 22052 Lemma for tusbas 22053, tusunif 22054, and tustopn 22056. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
 
Theoremtusbas 22053 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
 
Theoremtusunif 22054 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
 
Theoremtususs 22055 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSt‘𝐾))
 
Theoremtustopn 22056 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)    &   𝐽 = (unifTop‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = (TopOpen‘𝐾))
 
Theoremtususp 22057 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp)
 
Theoremtustps 22058 A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ TopSp)
 
Theoremuspreg 22059 If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.)
𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
 
12.3.4  Uniform continuity
 
Syntaxcucn 22060 Extend class notation with the uniform continuity operation.
class Cnu
 
Definitiondf-ucn 22061* Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function 𝑓 is uniformly continuous if, roughly speaking, it is possible to guarantee that (𝑓𝑥) and (𝑓𝑦) be as close to each other as we please by requiring only that 𝑥 and 𝑦 are sufficiently close to each other; unlike ordinary continuity, the maximum distance between (𝑓𝑥) and (𝑓𝑦) cannot depend on 𝑥 and 𝑦 themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
 
Theoremucnval 22062* The set of all uniformly continuous function from uniform space 𝑈 to uniform space 𝑉. (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})
 
Theoremisucn 22063* The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉." (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
 
Theoremisucn2 22064* The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉." , expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)
𝑈 = ((𝑋 × 𝑋)filGen𝑅)    &   𝑉 = ((𝑌 × 𝑌)filGen𝑆)    &   (𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝑅 ∈ (fBas‘(𝑋 × 𝑋)))    &   (𝜑𝑆 ∈ (fBas‘(𝑌 × 𝑌)))       (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
 
Theoremucnimalem 22065* Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
 
Theoremucnima 22066* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)
 
Theoremucnprima 22067* The preimage by a uniformly continuous function 𝐹 of an entourage 𝑊 of 𝑌 is an entourage of 𝑋. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       (𝜑 → (𝐺𝑊) ∈ 𝑈)
 
Theoremiducn 22068 The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))
 
Theoremcstucnd 22069 A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐴𝑌)       (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
 
Theoremucncn 22070 Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.)
𝐽 = (TopOpen‘𝑅)    &   𝐾 = (TopOpen‘𝑆)    &   (𝜑𝑅 ∈ UnifSp)    &   (𝜑𝑆 ∈ UnifSp)    &   (𝜑𝑅 ∈ TopSp)    &   (𝜑𝑆 ∈ TopSp)    &   (𝜑𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 
12.3.5  Cauchy filters in uniform spaces
 
Syntaxccfilu 22071 Extend class notation with the set of Cauchy filter bases.
class CauFilu
 
Definitiondf-cfilu 22072* Define the set of Cauchy filter bases on a uniform space. A Cauchy filter base is a filter base on the set such that for every entourage 𝑣, there is an element 𝑎 of the filter "small enough in 𝑣 " i.e. such that every pair {𝑥, 𝑦} of points in 𝑎 is related by 𝑣". Definition 2 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 16-Nov-2017.)
CauFilu = (𝑢 ran UnifOn ↦ {𝑓 ∈ (fBas‘dom 𝑢) ∣ ∀𝑣𝑢𝑎𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
 
Theoremiscfilu 22073* The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈." (Contributed by Thierry Arnoux, 18-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
 
Theoremcfilufbas 22074 A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))
 
Theoremcfiluexsm 22075* For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈) ∧ 𝑉𝑈) → ∃𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑉)
 
Theoremfmucndlem 22076* Lemma for fmucnd 22077. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹𝐴) × (𝐹𝐴)))
 
Theoremfmucnd 22077* The image of a Cauchy filter base by an uniformly continuous function is a Cauchy filter base. Deduction form. Proposition 3 of [BourbakiTop1] p. II.13. (Contributed by Thierry Arnoux, 18-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝐶 ∈ (CauFilu𝑈))    &   𝐷 = ran (𝑎𝐶 ↦ (𝐹𝑎))       (𝜑𝐷 ∈ (CauFilu𝑉))
 
Theoremcfilufg 22078 The filter generated by a Cauchy filter base is still a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu𝑈))
 
Theoremtrcfilu 22079 Condition for the trace of a Cauchy filter base to be a Cauchy filter base for the restricted uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu𝑈) ∧ ¬ ∅ ∈ (𝐹t 𝐴)) ∧ 𝐴𝑋) → (𝐹t 𝐴) ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴))))
 
Theoremcfiluweak 22080 A Cauchy filter base is also a Cauchy filter base on any coarser uniform structure. (Contributed by Thierry Arnoux, 24-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋𝐹 ∈ (CauFilu‘(𝑈t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu𝑈))
 
Theoremneipcfilu 22081 In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
𝑋 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝑈 = (UnifSt‘𝑊)       ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))
 
12.3.6  Complete uniform spaces
 
Syntaxccusp 22082 Extend class notation with the class of all complete uniform spaces.
class CUnifSp
 
Definitiondf-cusp 22083* Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
 
Theoremiscusp 22084* The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 3-Dec-2017.)
(𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
 
Theoremcuspusp 22085 A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
(𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
 
Theoremcuspcvg 22086 In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)
 
Theoremiscusp2 22087* The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSt‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
 
Theoremcnextucn 22088* Extension by continuity. Proposition 11 of [BourbakiTop1] p. II.20. Given a topology 𝐽 on 𝑋, a subset 𝐴 dense in 𝑋, this states a condition for 𝐹 from 𝐴 to a space 𝑌 Hausdorff and complete to be extensible by continuity. (Contributed by Thierry Arnoux, 4-Dec-2017.)
𝑋 = (Base‘𝑉)    &   𝑌 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑉)    &   𝐾 = (TopOpen‘𝑊)    &   𝑈 = (UnifSt‘𝑊)    &   (𝜑𝑉 ∈ TopSp)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CUnifSp)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹:𝐴𝑌)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    &   ((𝜑𝑥𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu𝑈))       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
 
Theoremucnextcn 22089 Extension by continuity. Theorem 2 of [BourbakiTop1] p. II.20. Given an uniform space on a set 𝑋, a subset 𝐴 dense in 𝑋, and a function 𝐹 uniformly continuous from 𝐴 to 𝑌, that function can be extended by continuity to the whole 𝑋, and its extension is uniformly continuous. (Contributed by Thierry Arnoux, 25-Jan-2018.)
𝑋 = (Base‘𝑉)    &   𝑌 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑉)    &   𝐾 = (TopOpen‘𝑊)    &   𝑆 = (UnifSt‘𝑉)    &   𝑇 = (UnifSt‘(𝑉s 𝐴))    &   𝑈 = (UnifSt‘𝑊)    &   (𝜑𝑉 ∈ TopSp)    &   (𝜑𝑉 ∈ UnifSp)    &   (𝜑𝑊 ∈ TopSp)    &   (𝜑𝑊 ∈ CUnifSp)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝑋)    &   (𝜑𝐹 ∈ (𝑇 Cnu𝑈))    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
 
12.4  Metric spaces
 
12.4.1  Pseudometric spaces
 
Theoremispsmet 22090* Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝑋𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦𝑋𝑧𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
 
Theorempsmetdmdm 22091 Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
 
Theorempsmetf 22092 The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
 
Theorempsmetcl 22093 Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
 
Theorempsmet0 22094 The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 0)
 
Theorempsmettri2 22095 Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
 
Theorempsmetsym 22096 The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
 
Theorempsmettri 22097 Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
 
Theorempsmetge0 22098 The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))
 
Theorempsmetxrge0 22099 The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶(0[,]+∞))
 
Theorempsmetres2 22100 Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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