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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtgplacthmeo 21901* The left group action of element 𝐴 in a topological group 𝐺 is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ (𝐴 + 𝑥))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))

Theoremsubmtmd 21902 A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)

Theoremsubgtgp 21903 A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Theoremsubgntr 21904 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21906, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ ((int‘𝐽)‘𝑆)) → 𝑆𝐽)

Theoremopnsubg 21905 An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆𝐽) → 𝑆 ∈ (Clsd‘𝐽))

Theoremclssubg 21906 The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (SubGrp‘𝐺))

Theoremclsnsg 21907 The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (NrmSGrp‘𝐺)) → ((cls‘𝐽)‘𝑆) ∈ (NrmSGrp‘𝐺))

Theoremcldsubg 21908 A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝑅 = (𝐺 ~QG 𝑆)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑋 / 𝑅) ∈ Fin) → (𝑆 ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))

Theoremtgpconncompeqg 21909* The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}    &    = (𝐺 ~QG 𝑆)       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})

Theoremtgpconncomp 21910* The identity component, the connected component containing the identity element, is a closed (conncompcld 21231) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))

Theoremtgpconncompss 21911* The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}       ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇𝐽) → 𝑆𝑇)

Theoremghmcnp 21912 A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝐾 = (TopOpen‘𝐻)       ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))

Theoremsnclseqg 21913 The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &    0 = (0g𝐺)    &    = (𝐺 ~QG 𝑆)    &   𝑆 = ((cls‘𝐽)‘{ 0 })       ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((cls‘𝐽)‘{𝐴}))

Theoremtgphaus 21914 A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
0 = (0g𝐺)    &   𝐽 = (TopOpen‘𝐺)       (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))

Theoremtgpt1 21915 Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))

Theoremtgpt0 21916 Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐽 = (TopOpen‘𝐺)       (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))

Theoremqustgpopn 21917* A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   𝑋 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝐾 = (TopOpen‘𝐻)    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))       ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆𝐽) → (𝐹𝑆) ∈ 𝐾)

Theoremqustgplem 21918* Lemma for qustgp 21919. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   𝑋 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝐾 = (TopOpen‘𝐻)    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))    &    = (𝑧𝑋, 𝑤𝑋 ↦ [(𝑧(-g𝐺)𝑤)](𝐺 ~QG 𝑌))       ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Theoremqustgp 21919 The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))       ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Theoremqustgphaus 21920 The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   𝐽 = (TopOpen‘𝐺)    &   𝐾 = (TopOpen‘𝐻)       ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐾 ∈ Haus)

Theoremprdstmdd 21921 The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶TopMnd)       (𝜑𝑌 ∈ TopMnd)

Theoremprdstgpd 21922 The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶TopGrp)       (𝜑𝑌 ∈ TopGrp)

12.2.7  Infinite group sum on topological groups

Syntaxctsu 21923 Extend class notation to include infinite group sums in a topological group.
class tsums

Definitiondf-tsms 21924* Define the set of limit points of an infinite group sum for the topological group 𝐺. If 𝐺 is Hausdorff, then there will be at most one element in this set and (𝑊 tsums 𝐹) selects this unique element if it exists. (𝑊 tsums 𝐹) ≈ 1𝑜 is a way to say that the sum exists and is unique. Note that unlike Σ (df-sum 14411) and Σg (df-gsum 16097), this does not return the sum itself, but rather the set of all such sums, which is usually either empty or a singleton. (Contributed by Mario Carneiro, 2-Sep-2015.)
tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ (𝒫 dom 𝑓 ∩ Fin) / 𝑠(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦)))))

Theoremtsmsfbas 21925* The collection of all sets of the form 𝐹(𝑧) = {𝑦𝑆𝑧𝑦}, which can be read as the set of all finite subsets of 𝐴 which contain 𝑧 as a subset, for each finite subset 𝑧 of 𝐴, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑆 = (𝒫 𝐴 ∩ Fin)    &   𝐹 = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})    &   𝐿 = ran 𝐹    &   (𝜑𝐴𝑊)       (𝜑𝐿 ∈ (fBas‘𝑆))

Theoremtsmslem1 21926 The finite partial sums of a function 𝐹 are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝑆 = (𝒫 𝐴 ∩ Fin)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑𝐹:𝐴𝐵)       ((𝜑𝑋𝑆) → (𝐺 Σg (𝐹𝑋)) ∈ 𝐵)

Theoremtsmsval2 21927* Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = (𝒫 𝐴 ∩ Fin)    &   𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})    &   (𝜑𝐺𝑉)    &   (𝜑𝐹𝑊)    &   (𝜑 → dom 𝐹 = 𝐴)       (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))

Theoremtsmsval 21928* Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = (𝒫 𝐴 ∩ Fin)    &   𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝑊)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))

Theoremtsmspropd 21929 The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17310 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
(𝜑𝐹𝑉)    &   (𝜑𝐺𝑊)    &   (𝜑𝐻𝑋)    &   (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   (𝜑 → (+g𝐺) = (+g𝐻))    &   (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻))       (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Theoremeltsms 21930* The property of being a sum of the sequence 𝐹 in the topological commutative monoid 𝐺. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = (𝒫 𝐴 ∩ Fin)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ (𝐶𝐵 ∧ ∀𝑢𝐽 (𝐶𝑢 → ∃𝑧𝑆𝑦𝑆 (𝑧𝑦 → (𝐺 Σg (𝐹𝑦)) ∈ 𝑢)))))

Theoremtsmsi 21931* The property of being a sum of the sequence 𝐹 in the topological commutative monoid 𝐺. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝑆 = (𝒫 𝐴 ∩ Fin)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶 ∈ (𝐺 tsums 𝐹))    &   (𝜑𝑈𝐽)    &   (𝜑𝐶𝑈)       (𝜑 → ∃𝑧𝑆𝑦𝑆 (𝑧𝑦 → (𝐺 Σg (𝐹𝑦)) ∈ 𝑈))

Theoremtsmscl 21932 A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)

Theoremhaustsms 21933* In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐽 ∈ Haus)       (𝜑 → ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹))

Theoremhaustsms2 21934 In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐽 ∈ Haus)       (𝜑 → (𝑋 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) = {𝑋}))

Theoremtsmscls 21935 One half of tgptsmscls 21947, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))       (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))

Theoremtsmsgsum 21936 The convergent points of a finite topological group sum are the closure of the finite group sum operation. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by AV, 24-Jul-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   𝐽 = (TopOpen‘𝐺)       (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{(𝐺 Σg 𝐹)}))

Theoremtsmsid 21937 If a sum is finite, the usual sum is always a limit point of the topological sum (although it may not be the only limit point). (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 tsums 𝐹))

Theoremhaustsmsid 21938 In a Hausdorff topological group, a finite sum sums to exactly the usual number with no extraneous limit points. By setting the topology to the discrete topology (which is Hausdorff), this theorem can be used to turn any tsums theorem into a Σg theorem, so that the infinite group sum operation can be viewed as a generalization of the finite group sum. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐽 ∈ Haus)       (𝜑 → (𝐺 tsums 𝐹) = {(𝐺 Σg 𝐹)})

Theoremtsms0 21939* The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)       (𝜑0 ∈ (𝐺 tsums (𝑥𝐴0 )))

Theoremtsmssubm 21940 Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝑆 ∈ (SubMnd‘𝐺))    &   (𝜑𝐹:𝐴𝑆)    &   𝐻 = (𝐺s 𝑆)       (𝜑 → (𝐻 tsums 𝐹) = ((𝐺 tsums 𝐹) ∩ 𝑆))

Theoremtsmsres 21941 Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)       (𝜑 → (𝐺 tsums (𝐹𝑊)) = (𝐺 tsums 𝐹))

Theoremtsmsf1o 21942 Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐻:𝐶1-1-onto𝐴)       (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹𝐻)))

Theoremtsmsmhm 21943 Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   𝐾 = (TopOpen‘𝐻)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopSp)    &   (𝜑𝐻 ∈ CMnd)    &   (𝜑𝐻 ∈ TopSp)    &   (𝜑𝐶 ∈ (𝐺 MndHom 𝐻))    &   (𝜑𝐶 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))       (𝜑 → (𝐶𝑋) ∈ (𝐻 tsums (𝐶𝐹)))

Theoremtsmsadd 21944 The sum of two infinite group sums. (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)    &   (𝜑𝐻:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))    &   (𝜑𝑌 ∈ (𝐺 tsums 𝐻))       (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums (𝐹𝑓 + 𝐻)))

Theoremtsmsinv 21945 Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐼 = (invg𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))       (𝜑 → (𝐼𝑋) ∈ (𝐺 tsums (𝐼𝐹)))

Theoremtsmssub 21946 The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐻:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))    &   (𝜑𝑌 ∈ (𝐺 tsums 𝐻))       (𝜑 → (𝑋 𝑌) ∈ (𝐺 tsums (𝐹𝑓 𝐻)))

Theoremtgptsmscls 21947 A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 21907, 0nsg 17633. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopGrp)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝑋 ∈ (𝐺 tsums 𝐹))       (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Theoremtgptsmscld 21948 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 21949 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 21950* Lemma for tsmsxp 21952. (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 21951* Lemma for tsmsxp 21952. (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 21952* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18369 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21950 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 21953 The class of all topological division rings.
class TopRing

Syntaxctdrg 21954 The class of all topological division rings.
class TopDRing

Syntaxctlm 21955 The class of all topological modules.
class TopMod

Syntaxctvc 21956 The class of all topological vector spaces.
class TopVec

Definitiondf-trg 21957 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 21958 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 21959 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 21960 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 21961 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 21962 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 21963 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 21964 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 21965 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Theoremtrgtmd2 21966 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Theoremtrgtps 21967 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Theoremtrgring 21968 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Theoremtrggrp 21969 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Theoremtdrgtrg 21970 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Theoremtdrgdrng 21971 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Theoremtdrgring 21972 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ Ring)

Theoremtdrgtmd 21973 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)

Theoremtdrgtps 21974 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)

Theoremistdrg2 21975 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 21976 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 21977 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 21978 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 21979* Continuity of ring multiplication; analogue of cnmpt12f 21463 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 21980* Continuity of ring multiplication; analogue of cnmpt22f 21472 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 21981 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 21982 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 21983 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 21984 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Theoremtlmtps 21985 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Theoremtlmlmod 21986 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ LMod)

Theoremtlmtrg 21987 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 21988 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 21989 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 21990 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 21991* Continuity of scalar multiplication; analogue of cnmpt12f 21463 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 21992* Continuity of scalar multiplication; analogue of cnmpt22f 21472 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 21993 A topological vector space is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopGrp)

Theoremtvctlm 21994 A topological vector space is a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ TopMod)

Theoremtvclmod 21995 A topological vector space is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopVec → 𝑊 ∈ LMod)

Theoremtvclvec 21996 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 21997 Extend class notation with the class function of uniform structures.
class UnifOn

Definitiondf-ust 21998* 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 21999 The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
UnifOn Fn V

Theoremustval 22000* 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)))})

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