P. 234 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23301-23400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ustuqtop1 23301*	Lemma for ustuqtop 23306, similar to ssnei2 22175. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
Theorem	ustuqtop2 23302*	Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
Theorem	ustuqtop3 23303*	Lemma for ustuqtop 23306, similar to elnei 22170. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
Theorem	ustuqtop4 23304*	Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
Theorem	ustuqtop5 23305*	Lemma for ustuqtop 23306. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
Theorem	ustuqtop 23306*	For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣 ∈ 𝑈 ↦ (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 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
Theorem	utopsnneiplem 23307*	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 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}    &   ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
Theorem	utopsnneip 23308*	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) ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
Theorem	utopsnnei 23309	Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
⊢ 𝐽 = (unifTop‘𝑈)    ⇒   ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	utop2nei 23310	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) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀 ∘ 𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
Theorem	utop3cls 23311	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)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))
Theorem	utopreg 23312	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
Syntax	cuss 23313	Extend class notation with the Uniform Structure extractor function.
class UnifSt
Syntax	cusp 23314	Extend class notation with the class of uniform spaces.
class UnifSp
Syntax	ctus 23315	Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp
Definition	df-uss 23316	Define the uniform structure extractor function. Similarly with df-topn 17051 this differs from df-unif 16911 when a structure has been restricted using df-ress 16868; 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))))
Definition	df-usp 23317	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)))}
Definition	df-tus 23318	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)⟩))
Theorem	ussval 23319	The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6748 to avoid requiring 𝑊 to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑈 = (UnifSet‘𝑊)    ⇒   ⊢ (𝑈 ↾t (𝐵 × 𝐵)) = (UnifSt‘𝑊)
Theorem	ussid 23320	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) = ∪ 𝑈 → 𝑈 = (UnifSt‘𝑊))
Theorem	isusp 23321	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)))
Theorem	ressuss 23322	Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
⊢ (𝐴 ∈ 𝑉 → (UnifSt‘(𝑊 ↾s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))
Theorem	ressust 23323	The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 𝑇 = (UnifSt‘(𝑊 ↾s 𝐴))    ⇒   ⊢ ((𝑊 ∈ UnifSp ∧ 𝐴 ⊆ 𝑋) → 𝑇 ∈ (UnifOn‘𝐴))
Theorem	ressusp 23324	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 ∧ 𝐴 ∈ 𝐽) → (𝑊 ↾s 𝐴) ∈ UnifSp)
Theorem	tusval 23325	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)⟩))
Theorem	tuslem 23326	Lemma for tusbas 23328, tusunif 23329, and tustopn 23331. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof shortened by AV, 28-Oct-2024.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
Theorem	tuslemOLD 23327	Obsolete proof of tuslem 23326 as of 28-Oct-2024. Lemma for tusbas 23328, tusunif 23329, and tustopn 23331. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
Theorem	tusbas 23328	The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
Theorem	tusunif 23329	The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
Theorem	tususs 23330	The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSt‘𝐾))
Theorem	tustopn 23331	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))
Theorem	tususp 23332	A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp)
Theorem	tustps 23333	A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ TopSp)
Theorem	uspreg 23334	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
Syntax	cucn 23335	Extend class notation with the uniform continuity operation.
class Cnu
Definition	df-ucn 23336*	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 (𝑓‘𝑥) and (𝑓‘𝑦) 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 (𝑓‘𝑥) and (𝑓‘𝑦) 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 ∪ 𝑣 ↑m dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪ 𝑢∀𝑦 ∈ dom ∪ 𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
Theorem	ucnval 23337*	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 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))})
Theorem	isucn 23338*	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:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑉 ∃𝑟 ∈ 𝑈 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
Theorem	isucn2 23339*	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)    &   ⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝑅 ∈ (fBas‘(𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝑆 ∈ (fBas‘(𝑌 × 𝑌)))    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑠 ∈ 𝑆 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑟𝑦 → (𝐹‘𝑥)𝑠(𝐹‘𝑦)))))
Theorem	ucnimalem 23340*	Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    ⇒   ⊢ 𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st ‘𝑝)), (𝐹‘(2nd ‘𝑝))⟩)
Theorem	ucnima 23341*	An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ 𝑈 (𝐺 “ 𝑟) ⊆ 𝑊)
Theorem	ucnprima 23342*	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.)
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))    &   ⊢ (𝜑 → 𝑊 ∈ 𝑉)    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩)    ⇒   ⊢ (𝜑 → (◡𝐺 “ 𝑊) ∈ 𝑈)
Theorem	iducn 23343	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))
Theorem	cstucnd 23344	A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))
Theorem	ucncn 23345	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)    &   ⊢ (𝜑 → 𝑅 ∈ UnifSp)    &   ⊢ (𝜑 → 𝑆 ∈ UnifSp)    &   ⊢ (𝜑 → 𝑅 ∈ TopSp)    &   ⊢ (𝜑 → 𝑆 ∈ TopSp)    &   ⊢ (𝜑 → 𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
12.3.5  Cauchy filters in uniform spaces
Syntax	ccfilu 23346	Extend class notation with the set of Cauchy filter bases.
class CauFilu
Definition	df-cfilu 23347*	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) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣})
Theorem	iscfilu 23348*	The predicate "𝐹 is a Cauchy filter base on uniform space 𝑈". (Contributed by Thierry Arnoux, 18-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
Theorem	cfilufbas 23349	A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → 𝐹 ∈ (fBas‘𝑋))
Theorem	cfiluexsm 23350*	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 ∈ (CauFilu‘𝑈) ∧ 𝑉 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑉)
Theorem	fmucndlem 23351*	Lemma for fmucnd 23352. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ⊆ 𝑋) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ ⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹 “ 𝐴) × (𝐹 “ 𝐴)))
Theorem	fmucnd 23352*	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.)
⊢ (𝜑 → 𝑈 ∈ (UnifOn‘𝑋))    &   ⊢ (𝜑 → 𝑉 ∈ (UnifOn‘𝑌))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑈 Cnu𝑉))    &   ⊢ (𝜑 → 𝐶 ∈ (CauFilu‘𝑈))    &   ⊢ 𝐷 = ran (𝑎 ∈ 𝐶 ↦ (𝐹 “ 𝑎))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (CauFilu‘𝑉))
Theorem	cfilufg 23353	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 ∈ (CauFilu‘𝑈)) → (𝑋filGen𝐹) ∈ (CauFilu‘𝑈))
Theorem	trcfilu 23354	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 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))))
Theorem	cfiluweak 23355	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) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐹 ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) → 𝐹 ∈ (CauFilu‘𝑈))
Theorem	neipcfilu 23356	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 ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈))
12.3.6  Complete uniform spaces
Syntax	ccusp 23357	Extend class notation with the class of all complete uniform spaces.
class CUnifSp
Definition	df-cusp 23358*	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) ≠ ∅)}
Theorem	iscusp 23359*	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) ≠ ∅)))
Theorem	cuspusp 23360	A complete uniform space is an uniform space. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ (𝑊 ∈ CUnifSp → 𝑊 ∈ UnifSp)
Theorem	cuspcvg 23361	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) ≠ ∅)
Theorem	iscusp2 23362*	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 ∈ (CauFilu‘𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
Theorem	cnextucn 23363*	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)    &   ⊢ (𝜑 → 𝑉 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ CUnifSp)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑌)    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑌 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∈ (CauFilu‘𝑈))    ⇒   ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
Theorem	ucnextcn 23364	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‘(𝑉 ↾s 𝐴))    &   ⊢ 𝑈 = (UnifSt‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ TopSp)    &   ⊢ (𝜑 → 𝑉 ∈ UnifSp)    &   ⊢ (𝜑 → 𝑊 ∈ TopSp)    &   ⊢ (𝜑 → 𝑊 ∈ CUnifSp)    &   ⊢ (𝜑 → 𝐾 ∈ Haus)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑇 Cnu𝑈))    &   ⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝑋)    ⇒   ⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
12.4  Metric spaces
12.4.1  Pseudometric spaces
Theorem	ispsmet 23365*	Express the predicate "𝐷 is a pseudometric." (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝑋 ∈ 𝑉 → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ((𝑥𝐷𝑥) = 0 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Theorem	psmetdmdm 23366	Recover the base set from a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
Theorem	psmetf 23367	The distance function of a pseudometric as a function. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
Theorem	psmetcl 23368	Closure of the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
Theorem	psmet0 23369	The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)
Theorem	psmettri2 23370	Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
Theorem	psmetsym 23371	The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
Theorem	psmettri 23372	Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
Theorem	psmetge0 23373	The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
Theorem	psmetxrge0 23374	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[,]+∞))
Theorem	psmetres2 23375	Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Theorem	psmetlecl 23376	Real closure of an extended metric value that is upper bounded by a real. (Contributed by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ ℝ ∧ (𝐴𝐷𝐵) ≤ 𝐶)) → (𝐴𝐷𝐵) ∈ ℝ)
Theorem	distspace 23377	A set 𝑋 together with a (distance) function 𝐷 which is a pseudometric is a distance space (according to E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006), i.e. a (base) set 𝑋 equipped with a distance 𝐷, which is a mapping of two elements of the base set to the (extended) reals and which is nonnegative, symmetric and equal to 0 if the two elements are equal. (Contributed by AV, 15-Oct-2021.) (Revised by AV, 5-Jul-2022.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ (𝐴𝐷𝐴) = 0) ∧ (0 ≤ (𝐴𝐷𝐵) ∧ (𝐴𝐷𝐵) = (𝐵𝐷𝐴))))
12.4.2  Basic metric space properties
Syntax	cxms 23378	Extend class notation with the class of extended metric spaces.
class ∞MetSp
Syntax	cms 23379	Extend class notation with the class of metric spaces.
class MetSp
Syntax	ctms 23380	Extend class notation with the function mapping a metric to the metric space it defines.
class toMetSp
Definition	df-xms 23381	Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
Definition	df-ms 23382	Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
⊢ MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
Definition	df-tms 23383	Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
Theorem	ismet 23384*	Express the predicate "𝐷 is a metric." (Contributed by NM, 25-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦))))))
Theorem	isxmet 23385*	Express the predicate "𝐷 is an extended metric." (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝑋 ∈ 𝐴 → (𝐷 ∈ (∞Met‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦))))))
Theorem	ismeti 23386*	Properties that determine a metric. (Contributed by NM, 17-Nov-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐷:(𝑋 × 𝑋)⟶ℝ    &   ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))    &   ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))    ⇒   ⊢ 𝐷 ∈ (Met‘𝑋)
Theorem	isxmetd 23387*	Properties that determine an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 9-Apr-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +𝑒 (𝑧𝐷𝑦)))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
Theorem	isxmet2d 23388*	It is safe to only require the triangle inequality when the values are real (so that we can use the standard addition over the reals), but in this case the nonnegativity constraint cannot be deduced and must be provided separately. (Counterexample: 𝐷(𝑥, 𝑦) = if(𝑥 = 𝑦, 0, -∞) satisfies all hypotheses except nonnegativity.) (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷:(𝑋 × 𝑋)⟶ℝ*)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 0 ≤ (𝑥𝐷𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥𝐷𝑦) ≤ 0 ↔ 𝑥 = 𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) ∧ ((𝑧𝐷𝑥) ∈ ℝ ∧ (𝑧𝐷𝑦) ∈ ℝ)) → (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))    ⇒   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
Theorem	metflem 23389*	Lemma for metf 23391 and others. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝑋 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) + (𝑧𝐷𝑦)))))
Theorem	xmetf 23390	Mapping of the distance function of an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
Theorem	metf 23391	Mapping of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
Theorem	xmetcl 23392	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
Theorem	metcl 23393	Closure of the distance function of a metric space. Part of Property M1 of [Kreyszig] p. 3. (Contributed by NM, 30-Aug-2006.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ)
Theorem	ismet2 23394	An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷:(𝑋 × 𝑋)⟶ℝ))
Theorem	metxmet 23395	A metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
Theorem	xmetdmdm 23396	Recover the base set from an extended metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = dom dom 𝐷)
Theorem	metdmdm 23397	Recover the base set from a metric. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷)
Theorem	xmetunirn 23398	Two ways to express an extended metric on an unspecified base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ (𝐷 ∈ ∪ ran ∞Met ↔ 𝐷 ∈ (∞Met‘dom dom 𝐷))
Theorem	xmeteq0 23399	The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
Theorem	meteq0 23400	The value of a metric is zero iff its arguments are equal. Property M2 of [Kreyszig] p. 4. (Contributed by NM, 30-Aug-2006.)
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))