 Home Metamath Proof ExplorerTheorem List (p. 220 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

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}))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >