P. 241 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	tdrgtmd 24001	A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)
Theorem	tdrgtps 24002	A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)
Theorem	istdrg2 24003	A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
Theorem	mulrcn 24004	The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝑇 = (+𝑓‘(mulGrp‘𝑅))    ⇒   ⊢ (𝑅 ∈ TopRing → 𝑇 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	invrcn2 24005	The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)))
Theorem	invrcn 24006	The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽))
Theorem	cnmpt1mulr 24007*	Continuity of ring multiplication; analogue of cnmpt12f 23491 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽))
Theorem	cnmpt2mulr 24008*	Continuity of ring multiplication; analogue of cnmpt22f 23500 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Theorem	dvrcn 24009	The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽))
Theorem	istlm 24010	The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Theorem	vscacn 24011	The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Theorem	tlmtmd 24012	A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Theorem	tlmtps 24013	A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
Theorem	tlmlmod 24014	A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Theorem	tlmtrg 24015	The scalar ring of a topological module is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopRing)
Theorem	tlmscatps 24016	The scalar ring of a topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
Theorem	istvc 24017	A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ TopVec ↔ (𝑊 ∈ TopMod ∧ 𝐹 ∈ TopDRing))
Theorem	tvctdrg 24018	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 24019*	Continuity of scalar multiplication; analogue of cnmpt12f 23491 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 24020*	Continuity of scalar multiplication; analogue of cnmpt22f 23500 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 24021	A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)
Theorem	tvctlm 24022	A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
Theorem	tvclmod 24023	A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Theorem	tvclvec 24024	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 24025	Extend class notation with the class function of uniform structures.
class UnifOn
Definition	df-ust 24026*	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 24027	The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn Fn V
Theorem	ustval 24028*	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 24029*	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 24030	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 24031	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 24032	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 24033	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 24034	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 24035	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 24036*	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 24037	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 24038	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 24039	A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Theorem	ustssco 24040	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 24041*	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 24042*	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 24043*	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 24044	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 24045	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 24046	The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ ¬ ∅ ∈ ∪ ran UnifOn
Theorem	ustund 24047	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 24048	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 24049	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 24050	Obsolete version of elfvunirn 6913 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 24051	Second direction for ustbas 24053. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
Theorem	ustuni 24052	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 24053	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 24054	Lemma for ustuqtop 24072. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Theorem	trust 24055	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 24056	Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
Definition	df-utop 24057*	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 24058*	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 24059*	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 24060	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 24061	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 24062	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 24063	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 24064	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 24065*	Lemma for ustuqtop 24072. (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 24066*	Lemma for ustuqtop 24072. (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 24067*	Lemma for ustuqtop 24072, similar to ssnei2 22941. (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 24068*	Lemma for ustuqtop 24072. (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 24069*	Lemma for ustuqtop 24072, similar to elnei 22936. (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 24070*	Lemma for ustuqtop 24072. (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 24071*	Lemma for ustuqtop 24072. (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 24072*	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 24073*	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 24074*	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 24075	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 24076	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 24077	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 24078	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 24079	Extend class notation with the Uniform Structure extractor function.
class UnifSt
Syntax	cusp 24080	Extend class notation with the class of uniform spaces.
class UnifSp
Syntax	ctus 24081	Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp
Definition	df-uss 24082	Define the uniform structure extractor function. Similarly with df-topn 17367 this differs from df-unif 17218 when a structure has been restricted using df-ress 17172; 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 24083	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 24084	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 24085	The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6873 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 24086	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 24087	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 24088	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 24089	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 24090	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 24091	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 24092	Lemma for tusbas 24094, tusunif 24095, and tustopn 24097. (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 24093	Obsolete proof of tuslem 24092 as of 28-Oct-2024. Lemma for tusbas 24094, tusunif 24095, and tustopn 24097. (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 24094	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 24095	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 24096	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 24097	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 24098	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 24099	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 24100	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)