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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtgptsmscld 22001 The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))

Theoremtsmssplit 22002 Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums (𝐹𝐶)))    &   (𝜑𝑌 ∈ (𝐺 tsums (𝐹𝐷)))    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums 𝐹))

Theoremtsmsxplem1 22003* Lemma for tsmsxp 22005. (Contributed by Mario Carneiro, 21-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))    &   𝐽 = (TopOpen‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐿𝐽)    &   (𝜑0𝐿)    &   (𝜑𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   (𝜑 → dom 𝐷𝐾)    &   (𝜑𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))       (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))

Theoremtsmsxplem2 22004* Lemma for tsmsxp 22005. (Contributed by Mario Carneiro, 21-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))    &   𝐽 = (TopOpen‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐿𝐽)    &   (𝜑0𝐿)    &   (𝜑𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   (𝜑 → ∀𝑐𝑆𝑑𝑇 (𝑐 + 𝑑) ∈ 𝑈)    &   (𝜑𝑁 ∈ (𝒫 𝐶 ∩ Fin))    &   (𝜑𝐷 ⊆ (𝐾 × 𝑁))    &   (𝜑 → ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿)    &   (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆)    &   (𝜑 → ∀𝑔 ∈ (𝐿𝑚 𝐾)(𝐺 Σg 𝑔) ∈ 𝑇)       (𝜑 → (𝐺 Σg (𝐻𝐾)) ∈ 𝑈)

Theoremtsmsxp 22005* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18421 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22003 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))       (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))

12.2.8  Topological rings, fields, vector spaces

Syntaxctrg 22006 The class of all topological division rings.
class TopRing

Syntaxctdrg 22007 The class of all topological division rings.
class TopDRing

Syntaxctlm 22008 The class of all topological modules.
class TopMod

Syntaxctvc 22009 The class of all topological vector spaces.
class TopVec

Definitiondf-trg 22010 Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}

Definitiondf-tdrg 22011 Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}

Definitiondf-tlm 22012 Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣ ((Scalar‘𝑤) ∈ TopRing ∧ ( ·sf𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}

Definitiondf-tvc 22013 Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈ TopDRing}

Theoremistrg 22014 Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))

Theoremtrgtmd 22015 The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)       (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)

Theoremistdrg 22016 Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))

Theoremtdrgunit 22017 The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ TopDRing → (𝑀s 𝑈) ∈ TopGrp)

Theoremtrgtgp 22018 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Theoremtrgtmd2 22019 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Theoremtrgtps 22020 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Theoremtrgring 22021 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Theoremtrggrp 22022 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Theoremtdrgtrg 22023 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Theoremtdrgdrng 22024 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Theoremtdrgring 22025 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ Ring)

Theoremtdrgtmd 22026 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)

Theoremtdrgtps 22027 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)

Theoremistdrg2 22028 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 = (0g𝑅)       (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s (𝐵 ∖ { 0 })) ∈ TopGrp))

Theoremmulrcn 22029 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))

Theoreminvrcn2 22030 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 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽t 𝑈) Cn (𝐽t 𝑈)))

Theoreminvrcn 22031 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 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽t 𝑈) Cn 𝐽))

Theoremcnmpt1mulr 22032* Continuity of ring multiplication; analogue of cnmpt12f 21517 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐽 = (TopOpen‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ TopRing)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽))

Theoremcnmpt2mulr 22033* Continuity of ring multiplication; analogue of cnmpt22f 21526 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐽 = (TopOpen‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ TopRing)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))

Theoremdvrcn 22034 The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐽 = (TopOpen‘𝑅)    &    / = (/r𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽t 𝑈)) Cn 𝐽))

Theoremistlm 22035 The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
· = ( ·sf𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (TopOpen‘𝐹)       (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))

Theoremvscacn 22036 The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
· = ( ·sf𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (TopOpen‘𝐹)       (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Theoremtlmtmd 22037 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Theoremtlmtps 22038 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Theoremtlmlmod 22039 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Theoremtlmtrg 22040 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 22041 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 22042 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 22043 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 22044* Continuity of scalar multiplication; analogue of cnmpt12f 21517 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 22045* Continuity of scalar multiplication; analogue of cnmpt22f 21526 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 22046 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)

Theoremtvctlm 22047 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Theoremtvclmod 22048 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Theoremtvclvec 22049 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 22050 Extend class notation with the class function of uniform structures.
class UnifOn

Definitiondf-ust 22051* 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 22052 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn Fn V

Theoremustval 22053* 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 22054* 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 22055 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 22056 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 22057 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 22058 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 22059 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 22060 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 22061* 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 22062 The elements of uniform structures, called entourages, are relations. (Contributed by Thierry Arnoux, 15-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)

Theoremustfilxp 22063 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 22064 A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)

Theoremustssco 22065 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 22066* 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 22067* 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 22068* 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 22069 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 22070 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 22071 The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)
¬ ∅ ∈ ran UnifOn

Theoremustund 22072 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 22073 Any point 𝐴 is near enough to itself. (Contributed by Thierry Arnoux, 18-Nov-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝐴𝑋) → 𝐴 ∈ (𝑉 “ {𝐴}))

Theoremustneism 22074 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 22075 First direction for ustbas 22078. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)

Theoremustbas2 22076 Second direction for ustbas 22078. (Contributed by Thierry Arnoux, 16-Nov-2017.)
(𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)

Theoremustuni 22077 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 22078 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 22079 Lemma for ustuqtop 22097. (Contributed by Thierry Arnoux, 5-Dec-2017.)
((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑃𝑋) → (𝑉 “ {𝑃}) ⊆ 𝑋)

Theoremtrust 22080 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 22081 Extend class notation with the function inducing a topology from a uniform structure.
class unifTop

Definitiondf-utop 22082* 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 22083* 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 22084* 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 22085 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 22086 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 22087 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 22088 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 22089 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 22090* Lemma for ustuqtop 22097. (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 22091* Lemma for ustuqtop 22097. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)

Theoremustuqtop1 22092* Lemma for ustuqtop 22097, similar to ssnei2 20968. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))

Theoremustuqtop2 22093* Lemma for ustuqtop 22097. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))

Theoremustuqtop3 22094* Lemma for ustuqtop 22097, similar to elnei 20963. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)

Theoremustuqtop4 22095* Lemma for ustuqtop 22097. (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 22096* Lemma for ustuqtop 22097. (Contributed by Thierry Arnoux, 11-Jan-2018.)
𝑁 = (𝑝𝑋 ↦ ran (𝑣𝑈 ↦ (𝑣 “ {𝑝})))       ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))

Theoremustuqtop 22097* 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 22098* 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 22099* 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 22100 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}))

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