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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremalexsubALTlem4 21901* Lemma for alexsubALT 21902. If any cover taken from a subbase has a finite subcover, any cover taken from the corresponding base has a finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
𝑋 = 𝐽       (𝐽 = (topGen‘(fi‘𝑥)) → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) → ∀𝑎 ∈ 𝒫 (fi‘𝑥)(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
 
TheoremalexsubALT 21902* The Alexander Subbase Theorem: a space is compact iff it has a subbase such that any cover taken from the subbase has a finite subcover. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 11-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = 𝐽       (𝐽 ∈ Comp ↔ ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
 
Theoremptcmplem1 21903* Lemma for ptcmp 21909. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))       (𝜑 → (𝑋 = (ran 𝑆 ∪ {𝑋}) ∧ (∏t𝐹) = (topGen‘(fi‘(ran 𝑆 ∪ {𝑋})))))
 
Theoremptcmplem2 21904* Lemma for ptcmp 21909. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)       (𝜑 𝑘 ∈ {𝑛𝐴 ∣ ¬ (𝐹𝑛) ≈ 1𝑜} (𝐹𝑘) ∈ dom card)
 
Theoremptcmplem3 21905* Lemma for ptcmp 21909. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)    &   𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}       (𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑓𝑘) ∈ ( (𝐹𝑘) ∖ 𝐾)))
 
Theoremptcmplem4 21906* Lemma for ptcmp 21909. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))    &   (𝜑𝑈 ⊆ ran 𝑆)    &   (𝜑𝑋 = 𝑈)    &   (𝜑 → ¬ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)𝑋 = 𝑧)    &   𝐾 = {𝑢 ∈ (𝐹𝑘) ∣ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢) ∈ 𝑈}        ¬ 𝜑
 
Theoremptcmplem5 21907* Lemma for ptcmp 21909. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝑆 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))    &   𝑋 = X𝑛𝐴 (𝐹𝑛)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Comp)    &   (𝜑𝑋 ∈ (UFL ∩ dom card))       (𝜑 → (∏t𝐹) ∈ Comp)
 
Theoremptcmpg 21908 Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if 𝒫 𝒫 𝑋 is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 21909). (Contributed by Mario Carneiro, 27-Aug-2015.)
𝐽 = (∏t𝐹)    &   𝑋 = 𝐽       ((𝐴𝑉𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)
 
Theoremptcmp 21909 Tychonoff's theorem: The product of compact spaces is compact. The proof uses the Axiom of Choice. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐴𝑉𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)
 
12.2.5  Extension by continuity
 
Syntaxccnext 21910 Extend class notation with the continuous extension operation.
class CnExt
 
Definitiondf-cnext 21911* Define the continuous extension of a given function. (Contributed by Thierry Arnoux, 1-Dec-2017.)
CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ ( 𝑘pm 𝑗) ↦ 𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
 
Theoremcnextval 21912* The function applying continuous extension to a given function 𝑓. (Contributed by Thierry Arnoux, 1-Dec-2017.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽CnExt𝐾) = (𝑓 ∈ ( 𝐾pm 𝐽) ↦ 𝑥 ∈ ((cls‘𝐽)‘dom 𝑓)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t dom 𝑓))‘𝑓))))
 
Theoremcnextfval 21913* The continuous extension of a given function 𝐹. (Contributed by Thierry Arnoux, 1-Dec-2017.)
𝑋 = 𝐽    &   𝐵 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝑋)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
 
Theoremcnextrel 21914 In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
𝐶 = 𝐽    &   𝐵 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
 
Theoremcnextfun 21915 If the target space is Hausdorff, a continuous extension is a function. (Contributed by Thierry Arnoux, 20-Dec-2017.)
𝐶 = 𝐽    &   𝐵 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹))
 
Theoremcnextfvval 21916* The value of the continuous extension of a given function 𝐹 at a point 𝑋. (Contributed by Thierry Arnoux, 21-Dec-2017.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)       ((𝜑𝑋𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
 
Theoremcnextf 21917* Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶𝐵)
 
Theoremcnextcn 21918* Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology 𝐽 on 𝐶, a subset 𝐴 dense in 𝐶, this states a condition for 𝐹 from 𝐴 to a regular space 𝐾 to be extensible by continuity. (Contributed by Thierry Arnoux, 1-Jan-2018.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   (𝜑𝐾 ∈ Reg)       (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾))
 
Theoremcnextfres1 21919* 𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 17-Jan-2018.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶)    &   ((𝜑𝑥𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅)    &   (𝜑𝐾 ∈ Reg)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))       (𝜑 → (((𝐽CnExt𝐾)‘𝐹) ↾ 𝐴) = 𝐹)
 
Theoremcnextfres 21920 𝐹 and its extension by continuity agree on the domain of 𝐹. (Contributed by Thierry Arnoux, 29-Aug-2020.)
𝐶 = 𝐽    &   𝐵 = 𝐾    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐾 ∈ Haus)    &   (𝜑𝐴𝐶)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑𝑋𝐴)       (𝜑 → (((𝐽CnExt𝐾)‘𝐹)‘𝑋) = (𝐹𝑋))
 
12.2.6  Topological groups
 
Syntaxctmd 21921 Extend class notation with the class of all topological monoids.
class TopMnd
 
Syntaxctgp 21922 Extend class notation with the class of all topological groups.
class TopGrp
 
Definitiondf-tmd 21923* Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpen‘𝑓) / 𝑗](+𝑓𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)}
 
Definitiondf-tgp 21924* Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣ [(TopOpen‘𝑓) / 𝑗](invg𝑓) ∈ (𝑗 Cn 𝑗)}
 
Theoremistmd 21925 The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐹 = (+𝑓𝐺)    &   𝐽 = (TopOpen‘𝐺)       (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
 
Theoremtmdmnd 21926 A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
 
Theoremtmdtps 21927 A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
 
Theoremistgp 21928 The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
 
Theoremtgpgrp 21929 A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
 
Theoremtgptmd 21930 A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
 
Theoremtgptps 21931 A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
(𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
 
Theoremtmdtopon 21932 The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
 
Theoremtgptopon 21933 The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
 
Theoremtmdcn 21934 In a topological monoid, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ TopMnd → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
 
Theoremtgpcn 21935 In a topological group, the operation 𝐹 representing the functionalization of the operator slot +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐹 = (+𝑓𝐺)       (𝐺 ∈ TopGrp → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
 
Theoremtgpinv 21936 In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
𝐽 = (TopOpen‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
 
Theoremgrpinvhmeo 21937 The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽Homeo𝐽))
 
Theoremcnmpt1plusg 21938* Continuity of the group sum; analogue of cnmpt12f 21517 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ TopMnd)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
 
Theoremcnmpt2plusg 21939* Continuity of the group sum; analogue of cnmpt22f 21526 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
𝐽 = (TopOpen‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ TopMnd)    &   (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))
 
Theoremtmdcn2 21940* 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 21941 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 21942 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 21943* 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 21944* 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 21945 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 22666 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 21946* 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 21947* 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 21948 The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd)
 
Theoremoppgtgp 21949 The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp)
 
Theoremdistgp 21950 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 21951 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 21952 The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ TopGrp)
 
Theoremtmdlactcn 21953* 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 21954* 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 21955 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 21956 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 21957 A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21959, 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 21958 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 21959 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 21960 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 21961 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 21962* 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 21963* The identity component, the connected component containing the identity element, is a closed (conncompcld 21285) 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 21964* 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 21965 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 21966 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 21967 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 21968 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 21969 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 21970* 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 21971* Lemma for qustgp 21972. (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 21972 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 21973 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 21974 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 21975 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 21976 Extend class notation to include infinite group sums in a topological group.
class tsums
 
Definitiondf-tsms 21977* 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 14461) and Σg (df-gsum 16150), 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 21978* 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 21979 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 21980* 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 21981* 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 21982 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 17363 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 21983* 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 21984* 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 21985 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 21986* 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 21987 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 21988 One half of tgptsmscls 22000, 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 21989 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 21990 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 21991 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 21992* 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 21993 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 21994 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 21995 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 21996 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 21997 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 21998 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 21999 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 22000 A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 21960, 0nsg 17686. (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}))
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