P. 243 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tvctdrg 24201	The scalar field of a topological vector space is a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopVec → 𝐹 ∈ TopDRing)
Theorem	cnmpt1vsca 24202*	Continuity of scalar multiplication; analogue of cnmpt12f 23674 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ TopMod)    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))
Theorem	cnmpt2vsca 24203*	Continuity of scalar multiplication; analogue of cnmpt22f 23683 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ TopMod)    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
Theorem	tlmtgp 24204	A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)
Theorem	tvctlm 24205	A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
Theorem	tvclmod 24206	A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Theorem	tvclvec 24207	A topological vector space is a vector space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LVec)
12.3  Uniform Structures and Spaces
12.3.1  Uniform structures
Syntax	cust 24208	Extend class notation with the class function of uniform structures.
class UnifOn
Definition	df-ust 24209*	Definition of a uniform structure. Definition 1 of [BourbakiTop1] p. II.1. A uniform structure is used to give a generalization of the idea of Cauchy's sequence. This definition is analogous to TopOn. Elements of an uniform structure are called entourages. (Contributed by FL, 29-May-2014.) (Revised by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
Theorem	ustfn 24210	The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn Fn V
Theorem	ustval 24211*	The class of all uniform structures for a base 𝑋. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
⊢ (𝑋 ∈ 𝑉 → (UnifOn‘𝑋) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
Theorem	isust 24212*	The predicate "𝑈 is a uniform structure with base 𝑋". (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
⊢ (𝑋 ∈ 𝑉 → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
Theorem	ustssxp 24213	Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
Theorem	ustssel 24214	A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) (Proof shortened by AV, 17-Sep-2021.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉 ⊆ 𝑊 → 𝑊 ∈ 𝑈))
Theorem	ustbasel 24215	The full set is always an entourage. Condition FIIb of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
Theorem	ustincl 24216	A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ∈ 𝑈) → (𝑉 ∩ 𝑊) ∈ 𝑈)
Theorem	ustdiag 24217	The diagonal set is included in any entourage, i.e. any point is 𝑉 -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
Theorem	ustinvel 24218	If 𝑉 is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ◡𝑉 ∈ 𝑈)
Theorem	ustexhalf 24219*	For each entourage 𝑉 there is an entourage 𝑤 that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉)
Theorem	ustrel 24220	The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
Theorem	ustfilxp 24221	A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))
Theorem	ustne0 24222	A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Theorem	ustssco 24223	In an uniform structure, any entourage 𝑉 is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → 𝑉 ⊆ (𝑉 ∘ 𝑉))
Theorem	ustexsym 24224*	In an uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑉))
Theorem	ustex2sym 24225*	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) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑉))
Theorem	ustex3sym 24226*	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) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))
Theorem	ustref 24227	Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑉𝐴)
Theorem	ust0 24228	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‘∅) = {{∅}}
Theorem	ustn0 24229	The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ ¬ ∅ ∈ ∪ ran UnifOn
Theorem	ustund 24230	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 → (𝐴 ∩ 𝐵) ≠ ∅)    ⇒   ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))
Theorem	ustelimasn 24231	Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))
Theorem	ustneism 24232	For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉 ∘ ◡𝑉). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
⊢ ((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴 ∈ 𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉 ∘ ◡𝑉))
Theorem	elrnustOLD 24233	Obsolete version of elfvunirn 6938 as of 12-Jan-2025. (Contributed by Thierry Arnoux, 16-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
Theorem	ustbas2 24234	Second direction for ustbas 24236. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
Theorem	ustuni 24235	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))
Theorem	ustbas 24236	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))
Theorem	ustimasn 24237	Lemma for ustuqtop 24255. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Theorem	trust 24238	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) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))
12.3.2  The topology induced by an uniform structure
Syntax	cutop 24239	Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
Definition	df-utop 24240*	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 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
Theorem	utopval 24241*	The topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
Theorem	elutop 24242*	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) ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))
Theorem	utoptop 24243	The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
Theorem	utopbas 24244	The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
Theorem	utoptopon 24245	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))
Theorem	restutop 24246	Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈 ↾t (𝐴 × 𝐴))))
Theorem	restutopopn 24247	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‘(𝑈 ↾t (𝐴 × 𝐴))))
Theorem	ustuqtoplem 24248*	Lemma for ustuqtop 24255. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑁‘𝑃) ↔ ∃𝑤 ∈ 𝑈 𝐴 = (𝑤 “ {𝑃})))
Theorem	ustuqtop0 24249*	Lemma for ustuqtop 24255. (Contributed by Thierry Arnoux, 11-Jan-2018.)
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
Theorem	ustuqtop1 24250*	Lemma for ustuqtop 24255, similar to ssnei2 23124. (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 24251*	Lemma for ustuqtop 24255. (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 24252*	Lemma for ustuqtop 24255, similar to elnei 23119. (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 24253*	Lemma for ustuqtop 24255. (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 24254*	Lemma for ustuqtop 24255. (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 24255*	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 24256*	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 24257*	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 24258	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 24259	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 24260	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 24261	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 24262	Extend class notation with the Uniform Structure extractor function.
class UnifSt
Syntax	cusp 24263	Extend class notation with the class of uniform spaces.
class UnifSp
Syntax	ctus 24264	Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp
Definition	df-uss 24265	Define the uniform structure extractor function. Similarly with df-topn 17468 this differs from df-unif 17320 when a structure has been restricted using df-ress 17275; 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 24266	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 24267	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 24268	The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6898 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 24269	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 24270	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 24271	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 24272	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 24273	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 24274	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 24275	Lemma for tusbas 24277, tusunif 24278, and tustopn 24280. (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 24276	Obsolete version of tuslem 24275 as of 28-Oct-2024. Lemma for tusbas 24277, tusunif 24278, and tustopn 24280. (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 24277	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 24278	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 24279	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 24280	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 24281	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 24282	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 24283	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 24284	Extend class notation with the uniform continuity operation.
class Cnu
Definition	df-ucn 24285*	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 24286*	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 24287*	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 24288*	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 24289*	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 24290*	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 24291*	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 24292	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 24293	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 24294	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 24295	Extend class notation with the set of Cauchy filter bases.
class CauFilu
Definition	df-cfilu 24296*	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 24297*	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 24298	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 24299*	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 24300*	Lemma for fmucnd 24301. (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)))