P. 225 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulrcn 22401	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 22402	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 22403	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 22404*	Continuity of ring multiplication; analogue of cnmpt12f 21889 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 22405*	Continuity of ring multiplication; analogue of cnmpt22f 21898 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 22406	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 22407	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 22408	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 22409	A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)
Theorem	tlmtps 22410	A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
Theorem	tlmlmod 22411	A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ LMod)
Theorem	tlmtrg 22412	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 22413	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 22414	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 22415	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 22416*	Continuity of scalar multiplication; analogue of cnmpt12f 21889 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 22417*	Continuity of scalar multiplication; analogue of cnmpt22f 21898 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 22418	A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)
Theorem	tvctlm 22419	A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ TopMod)
Theorem	tvclmod 22420	A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopVec → 𝑊 ∈ LMod)
Theorem	tvclvec 22421	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 22422	Extend class notation with the class function of uniform structures.
class UnifOn
Definition	df-ust 22423*	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 22424	The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
⊢ UnifOn Fn V
Theorem	ustval 22425*	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 22426*	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 22427	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 22428	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 22429	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 22430	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 22431	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 22432	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 22433*	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 22434	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 22435	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 22436	A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Theorem	ustssco 22437	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 22438*	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 22439*	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 22440*	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 22441	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 22442	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 22443	The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
⊢ ¬ ∅ ∈ ∪ ran UnifOn
Theorem	ustund 22444	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 22445	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 22446	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	elrnust 22447	First direction for ustbas 22450. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
Theorem	ustbas2 22448	Second direction for ustbas 22450. (Contributed by Thierry Arnoux, 16-Nov-2017.)
⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
Theorem	ustuni 22449	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 22450	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 22451	Lemma for ustuqtop 22469. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑃 ∈ 𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)
Theorem	trust 22452	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 22453	Extend class notation with the function inducing a topology from a uniform structure.
class unifTop
Definition	df-utop 22454*	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 22455*	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 22456*	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 22457	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 22458	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 22459	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 22460	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 22461	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 22462*	Lemma for ustuqtop 22469. (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 22463*	Lemma for ustuqtop 22469. (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 22464*	Lemma for ustuqtop 22469, similar to ssnei2 21339. (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 22465*	Lemma for ustuqtop 22469. (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 22466*	Lemma for ustuqtop 22469, similar to elnei 21334. (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 22467*	Lemma for ustuqtop 22469. (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 22468*	Lemma for ustuqtop 22469. (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 22469*	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 22470*	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 22471*	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 22472	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 22473	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 22474	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 22475	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 22476	Extend class notation with the Uniform Structure extractor function.
class UnifSt
Syntax	cusp 22477	Extend class notation with the class of uniform spaces.
class UnifSp
Syntax	ctus 22478	Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp
Definition	df-uss 22479	Define the uniform structure extractor function. Similarly with df-topn 16481 this differs from df-unif 16372 when a structure has been restricted using df-ress 16274; 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 22480	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 22481	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 22482	The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6441 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 22483	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 22484	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	ressunif 22485	UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑈 = (UnifSet‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝑈 = (UnifSet‘𝐻))
Theorem	ressuss 22486	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 22487	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 22488	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 22489	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 22490	Lemma for tusbas 22491, tusunif 22492, and tustopn 22494. (Contributed by Thierry Arnoux, 5-Dec-2017.)
⊢ 𝐾 = (toUnifSp‘𝑈)    ⇒   ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
Theorem	tusbas 22491	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 22492	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 22493	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 22494	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 22495	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 22496	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 22497	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 22498	Extend class notation with the uniform continuity operation.
class Cnu
Definition	df-ucn 22499*	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 22500*	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))})