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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtmdcn2 22301* Write out the definition of continuity of +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ TopMnd ∧ 𝑈𝐽) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢𝐽𝑣𝐽 (𝑋𝑢𝑌𝑣 ∧ ∀𝑥𝑢𝑦𝑣 (𝑥 + 𝑦) ∈ 𝑈))

Theoremtgpsubcn 22302 In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
𝐽 = (TopOpen‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ TopGrp → ∈ ((𝐽 ×t 𝐽) Cn 𝐽))

Theoremistgp2 22303 A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    = (-g𝐺)       (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))

Theoremtmdmulg 22304* In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))

Theoremtgpmulg 22305* In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))

Theoremtgpmulg2 22306 In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 23027 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &    · = (.g𝐺)       (𝐺 ∈ TopGrp → · ∈ ((𝒫 ℤ ×t 𝐽) Cn 𝐽))

Theoremtmdgsum 22307* In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
𝐽 = (TopOpen‘𝐺)    &   𝐵 = (Base‘𝐺)       ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))

Theoremtmdgsum2 22308* For any neighborhood 𝑈 of 𝑛𝑋, there is a neighborhood 𝑢 of 𝑋 such that any sum of 𝑛 elements in 𝑢 sums to an element of 𝑈. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋𝐵)    &   (𝜑 → ((♯‘𝐴) · 𝑋) ∈ 𝑈)       (𝜑 → ∃𝑢𝐽 (𝑋𝑢 ∧ ∀𝑓 ∈ (𝑢𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))

Theoremoppgtmd 22309 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd)

Theoremoppgtgp 22310 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp)

Theoremdistgp 22311 Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Theoremindistgp 22312 Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐵 = (Base‘𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐽 = {∅, 𝐵}) → 𝐺 ∈ TopGrp)

Theoremsymgtgp 22313 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ TopGrp)

Theoremtmdlactcn 22314* The left group action of element 𝐴 in a topological monoid 𝐺 is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
𝐹 = (𝑥𝑋 ↦ (𝐴 + 𝑥))    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐽 = (TopOpen‘𝐺)       ((𝐺 ∈ TopMnd ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))

Theoremtgplacthmeo 22315* 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 22316 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 22317 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 22318 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 22320, 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 22319 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 22320 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 22321 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 22322 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 22323* 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 22324* The identity component, the connected component containing the identity element, is a closed (conncompcld 21646) 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 22325* 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 22326 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 22327 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 22328 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 22329 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 22330 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 22331* 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 22332* Lemma for qustgp 22333. (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 22333 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 22334 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 22335 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 22336 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 22337 Extend class notation to include infinite group sums in a topological group.
class tsums

Definitiondf-tsms 22338* 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 14825) and Σg (df-gsum 16489), 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 22339* 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 22340 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 22341* 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 22342* 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 22343 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 17702 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 22344* 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 22345* 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 22346 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 22347* 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 22348 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 22349 One half of tgptsmscls 22361, 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 22350 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 22351 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 22352 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 22353* 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 22354 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 22355 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 22356 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 22357 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 22358 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 22359 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 22360 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 22361 A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 22321, 0nsg 18023. (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 22362 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 22363 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 22364* Lemma for tsmsxp 22366. (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 22365* Lemma for tsmsxp 22366. (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 22366* Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18761 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 22364 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 22367 The class of all topological division rings.
class TopRing

Syntaxctdrg 22368 The class of all topological division rings.
class TopDRing

Syntaxctlm 22369 The class of all topological modules.
class TopMod

Syntaxctvc 22370 The class of all topological vector spaces.
class TopVec

Definitiondf-trg 22371 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 22372 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 22373 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 22374 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 22375 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 22376 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 22377 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 22378 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 22379 A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Theoremtrgtmd2 22380 A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Theoremtrgtps 22381 A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Theoremtrgring 22382 A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Theoremtrggrp 22383 A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Theoremtdrgtrg 22384 A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Theoremtdrgdrng 22385 A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Theoremtdrgring 22386 A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ Ring)

Theoremtdrgtmd 22387 A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)

Theoremtdrgtps 22388 A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)

Theoremistdrg2 22389 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 22390 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 22391 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 22392 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 22393* Continuity of ring multiplication; analogue of cnmpt12f 21878 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 22394* Continuity of ring multiplication; analogue of cnmpt22f 21887 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 22395 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 22396 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 22397 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 22398 A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)

Theoremtlmtps 22399 A topological module is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ TopSp)

Theoremtlmlmod 22400 A topological module is a left module. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑊 ∈ TopMod → 𝑊 ∈ LMod)

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