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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtlmtps 21801 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Theoremtlmlmod 21802 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Theoremtlmtrg 21803 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)

Theoremtlmscatps 21804 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)

Theoremistvc 21805 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))

Theoremtvctdrg 21806 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)

Theoremcnmpt1vsca 21807* Continuity of scalar multiplication; analogue of cnmpt12f 21279 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘𝐹)    &   (𝜑𝑊 ∈ TopMod)    &   (𝜑𝐿 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐿 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐿 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Theoremcnmpt2vsca 21808* Continuity of scalar multiplication; analogue of cnmpt22f 21288 which cannot be used directly because ·𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘𝐹)    &   (𝜑𝑊 ∈ TopMod)    &   (𝜑𝐿 ∈ (TopOn‘𝑋))    &   (𝜑𝑀 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))

Theoremtlmtgp 21809 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)

Theoremtvctlm 21810 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Theoremtvclmod 21811 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Theoremtvclvec 21812 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

Syntaxcust 21813 Extend class notation with the class function of uniform structures.
class UnifOn

Definitiondf-ust 21814* 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 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})

Theoremustfn 21815 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn Fn V

Theoremustval 21816* The class of all uniform structures for a base 𝑋. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
(𝑋𝑉 → (UnifOn‘𝑋) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})

Theoremisust 21817* The predicate "𝑈 is a uniform structure with base 𝑋." (Contributed by Thierry Arnoux, 15-Nov-2017.) (Revised by AV, 17-Sep-2021.)
(𝑋𝑉 → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))

Theoremustssxp 21818 Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))

Theoremustssel 21819 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) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉𝑊𝑊𝑈))

Theoremustbasel 21820 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)

Theoremustincl 21821 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) ∧ 𝑉𝑈𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)

Theoremustdiag 21822 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) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)

Theoremustinvel 21823 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) ∧ 𝑉𝑈) → 𝑉𝑈)

Theoremustexhalf 21824* 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) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)

Theoremustrel 21825 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)

Theoremustfilxp 21826 A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Theoremustne0 21827 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)

Theoremustssco 21828 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) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑉𝑉))

Theoremustexsym 21829* In an uniform structure, for any entourage 𝑉, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑉))

Theoremustex2sym 21830* In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than half 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤𝑤) ⊆ 𝑉))

Theoremustex3sym 21831* In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than a third of 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑉))

Theoremustref 21832 Any element of the base set is "near" itself, i.e. entourages are reflexive. (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴𝑉𝐴)

Theoremust0 21833 The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
(UnifOn‘∅) = {{∅}}

Theoremustn0 21834 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
¬ ∅ ∈ ran UnifOn

Theoremustund 21835 If two intersecting sets 𝐴 and 𝐵 are both small in 𝑉, their union is small in (𝑉↑2). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)    &   (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)    &   (𝜑 → (𝐴𝐵) ≠ ∅)       (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))

Theoremustelimasn 21836 Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))

Theoremustneism 21837 For a point 𝐴 in 𝑋, (𝑉 “ {𝐴}) is small enough in (𝑉𝑉). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
((𝑉 ⊆ (𝑋 × 𝑋) ∧ 𝐴𝑋) → ((𝑉 “ {𝐴}) × (𝑉 “ {𝐴})) ⊆ (𝑉𝑉))

Theoremelrnust 21838 First direction for ustbas 21841. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)

Theoremustbas2 21839 Second direction for ustbas 21841. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)

Theoremustuni 21840 The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))

Theoremustbas 21841 Recover the base of an uniform structure 𝑈. ran UnifOn is to UnifOn what Top is to TopOn. (Contributed by Thierry Arnoux, 16-Nov-2017.)
𝑋 = dom 𝑈       (𝑈 ran UnifOn ↔ 𝑈 ∈ (UnifOn‘𝑋))

Theoremustimasn 21842 Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 5-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)

Theoremtrust 21843 The trace of a uniform structure 𝑈 on a subset 𝐴 is a uniform structure on 𝐴. Definition 3 of [BourbakiTop1] p. II.9. (Contributed by Thierry Arnoux, 2-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → (𝑈t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴))

12.3.2  The topology induced by an uniform structure

Syntaxcutop 21844 Extend class notation with the function inducing a topology from a uniform structure.
class unifTop

Definitiondf-utop 21845* Definition of a topology induced by a uniform structure. Definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})

Theoremutopval 21846* The topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})

Theoremelutop 21847* Open sets in the topology induced by an uniform structure 𝑈 on 𝑋 (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (𝐴 ∈ (unifTop‘𝑈) ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝐴)))

Theoremutoptop 21848 The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)

Theoremutopbas 21849 The base of the topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))

Theoremutoptopon 21850 Topology induced by a uniform structure 𝑈 with its base set. (Contributed by Thierry Arnoux, 5-Jan-2018.)
(𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))

Theoremrestutop 21851 Restriction of a topology induced by an uniform structure. (Contributed by Thierry Arnoux, 12-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴𝑋) → ((unifTop‘𝑈) ↾t 𝐴) ⊆ (unifTop‘(𝑈t (𝐴 × 𝐴))))

Theoremrestutopopn 21852 The restriction of the topology induced by an uniform structure to an open set. (Contributed by Thierry Arnoux, 16-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ∈ (unifTop‘𝑈)) → ((unifTop‘𝑈) ↾t 𝐴) = (unifTop‘(𝑈t (𝐴 × 𝐴))))

Theoremustuqtoplem 21853* Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) ∧ 𝐴𝑉) → (𝐴 ∈ (𝑁𝑃) ↔ ∃𝑤𝑈 𝐴 = (𝑤 “ {𝑃})))

Theoremustuqtop0 21854* Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

Theoremustuqtop1 21855* Lemma for ustuqtop 21860, similar to ssnei2 20730. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))

Theoremustuqtop2 21856* Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))

Theoremustuqtop3 21857* Lemma for ustuqtop 21860, similar to elnei 20725. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)

Theoremustuqtop4 21858* Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))

Theoremustuqtop5 21859* Lemma for ustuqtop 21860. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))

Theoremustuqtop 21860* For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})) is the filter of the neighborhoods of 𝑝 for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝𝑋 (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))

Theoremutopsnneiplem 21861* The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝐽 = (unifTop‘𝑈)    &   𝐾 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}    &   𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))

Theoremutopsnneip 21862* The neighborhoods of a point 𝑃 for the topology induced by an uniform space 𝑈. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))

Theoremutopsnnei 21863 Images of singletons by entourages 𝑉 are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))

Theoremutop2nei 21864 For any symmetrical entourage 𝑉 and any relation 𝑀, build a neighborhood of 𝑀. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))

Theoremutop3cls 21865 Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝐽 = (unifTop‘𝑈)       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀𝑉)))

Theoremutopreg 21866 All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)
𝐽 = (unifTop‘𝑈)       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

12.3.3  Uniform Spaces

Syntaxcuss 21867 Extend class notation with the Uniform Structure extractor function.
class UnifSt

Syntaxcusp 21868 Extend class notation with the class of uniform spaces.
class UnifSp

Syntaxctus 21869 Extend class notation with the function mapping a uniform structure to a uniform space.
class toUnifSp

Definitiondf-uss 21870 Define the uniform structure extractor function. Similarly with df-topn 15907 this differs from df-unif 15792 when a structure has been restricted using df-ress 15702; in this case the UnifSet component will still have a uniform set over the larger set, and this function fixes this by restricting the uniform set as well. (Contributed by Thierry Arnoux, 1-Dec-2017.)
UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t ((Base‘𝑓) × (Base‘𝑓))))

Definitiondf-usp 21871 Definition of a uniform space, i.e. a base set with an uniform structure and its induced topology. Derived from definition 3 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Nov-2017.)
UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) = (unifTop‘(UnifSt‘𝑓)))}

Definitiondf-tus 21872 Define the function mapping a uniform structure to a uniform space. (Contributed by Thierry Arnoux, 17-Nov-2017.)
toUnifSp = (𝑢 ran UnifOn ↦ ({⟨(Base‘ndx), dom 𝑢⟩, ⟨(UnifSet‘ndx), 𝑢⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑢)⟩))

Theoremussval 21873 The uniform structure on uniform space 𝑊. This proof uses a trick with fvprc 6097 to avoid requiring 𝑊 to be a set. (Contributed by Thierry Arnoux, 3-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSet‘𝑊)       (𝑈t (𝐵 × 𝐵)) = (UnifSt‘𝑊)

Theoremussid 21874 In case the base of the UnifSt element of the uniform space is the base of its element structure, then UnifSt does not restrict it further. (Contributed by Thierry Arnoux, 4-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSet‘𝑊)       ((𝐵 × 𝐵) = 𝑈𝑈 = (UnifSt‘𝑊))

Theoremisusp 21875 The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝑈 = (UnifSt‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Theoremressunif 21876 UnifSet is unaffected by restriction. (Contributed by Thierry Arnoux, 7-Dec-2017.)
𝐻 = (𝐺s 𝐴)    &   𝑈 = (UnifSet‘𝐺)       (𝐴𝑉𝑈 = (UnifSet‘𝐻))

Theoremressuss 21877 Value of the uniform structure of a restricted space. (Contributed by Thierry Arnoux, 12-Dec-2017.)
(𝐴𝑉 → (UnifSt‘(𝑊s 𝐴)) = ((UnifSt‘𝑊) ↾t (𝐴 × 𝐴)))

Theoremressust 21878 The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018.)
𝑋 = (Base‘𝑊)    &   𝑇 = (UnifSt‘(𝑊s 𝐴))       ((𝑊 ∈ UnifSp ∧ 𝐴𝑋) → 𝑇 ∈ (UnifOn‘𝐴))

Theoremressusp 21879 The restriction of a uniform topological space to an open set is a uniform space. (Contributed by Thierry Arnoux, 16-Dec-2017.)
𝐵 = (Base‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴𝐽) → (𝑊s 𝐴) ∈ UnifSp)

Theoremtusval 21880 The value of the uniform space mapping function. (Contributed by Thierry Arnoux, 5-Dec-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))

Theoremtuslem 21881 Lemma for tusbas 21882, tusunif 21883, and tustopn 21885. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Theoremtusbas 21882 The base set of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))

Theoremtusunif 21883 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))

Theoremtususs 21884 The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 15-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSt‘𝐾))

Theoremtustopn 21885 The topology induced by a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)    &   𝐽 = (unifTop‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 = (TopOpen‘𝐾))

Theoremtususp 21886 A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp)

Theoremtustps 21887 A constructed uniform space is a topological space. (Contributed by Thierry Arnoux, 25-Jan-2018.)
𝐾 = (toUnifSp‘𝑈)       (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ TopSp)

Theoremuspreg 21888 If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.)
𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ UnifSp ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

12.3.4  Uniform continuity

Syntaxcucn 21889 Extend class notation with the uniform continuity operation.
class Cnu

Definitiondf-ucn 21890* Define a function on two uniform structures which value is the set of uniformly continuous functions from the first uniform structure to the second. A function 𝑓 is uniformly continuous if, roughly speaking, it is possible to guarantee that (𝑓𝑥) and (𝑓𝑦) be as close to each other as we please by requiring only that 𝑥 and 𝑦 are sufficiently close to each other; unlike ordinary continuity, the maximum distance between (𝑓𝑥) and (𝑓𝑦) cannot depend on 𝑥 and 𝑦 themselves. This formulation is the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Cnu = (𝑢 ran UnifOn, 𝑣 ran UnifOn ↦ {𝑓 ∈ (dom 𝑣𝑚 dom 𝑢) ∣ ∀𝑠𝑣𝑟𝑢𝑥 ∈ dom 𝑢𝑦 ∈ dom 𝑢(𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})

Theoremucnval 21891* The set of all uniformly continuous function from uniform space 𝑈 to uniform space 𝑉. (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝑈 Cnu𝑉) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝑓𝑥)𝑠(𝑓𝑦))})

Theoremisucn 21892* The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉." (Contributed by Thierry Arnoux, 16-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑉𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))

Theoremisucn2 21893* The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉." , expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)
𝑈 = ((𝑋 × 𝑋)filGen𝑅)    &   𝑉 = ((𝑌 × 𝑌)filGen𝑆)    &   (𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝑅 ∈ (fBas‘(𝑋 × 𝑋)))    &   (𝜑𝑆 ∈ (fBas‘(𝑌 × 𝑌)))       (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))

Theoremucnimalem 21894* Reformulate the 𝐺 function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       𝐺 = (𝑝 ∈ (𝑋 × 𝑋) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)

Theoremucnima 21895* An equivalent statement of the definition of uniformly continuous function. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       (𝜑 → ∃𝑟𝑈 (𝐺𝑟) ⊆ 𝑊)

Theoremucnprima 21896* The preimage by a uniformly continuous function 𝐹 of an entourage 𝑊 of 𝑌 is an entourage of 𝑋. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐹 ∈ (𝑈 Cnu𝑉))    &   (𝜑𝑊𝑉)    &   𝐺 = (𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)       (𝜑 → (𝐺𝑊) ∈ 𝑈)

Theoremiducn 21897 The identity is uniformly continuous from a uniform structure to itself. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝑈 Cnu𝑈))

Theoremcstucnd 21898 A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝜑𝑈 ∈ (UnifOn‘𝑋))    &   (𝜑𝑉 ∈ (UnifOn‘𝑌))    &   (𝜑𝐴𝑌)       (𝜑 → (𝑋 × {𝐴}) ∈ (𝑈 Cnu𝑉))

Theoremucncn 21899 Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.)
𝐽 = (TopOpen‘𝑅)    &   𝐾 = (TopOpen‘𝑆)    &   (𝜑𝑅 ∈ UnifSp)    &   (𝜑𝑆 ∈ UnifSp)    &   (𝜑𝑅 ∈ TopSp)    &   (𝜑𝑆 ∈ TopSp)    &   (𝜑𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))

12.3.5  Cauchy filters in uniform spaces

Syntaxccfilu 21900 Extend class notation with the set of Cauchy filter bases.
class CauFilu

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