P. 218 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnmpt1plusg 21701*	Continuity of the group sum; analogue of cnmpt12f 21279 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Theorem	cnmpt2plusg 21702*	Continuity of the group sum; analogue of cnmpt22f 21288 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Theorem	tmdcn2 21703*	Write out the definition of continuity of +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ TopMnd ∧ 𝑈 ∈ 𝐽) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 + 𝑌) ∈ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ ∀𝑥 ∈ 𝑢 ∀𝑦 ∈ 𝑣 (𝑥 + 𝑦) ∈ 𝑈))
Theorem	tgpsubcn 21704	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)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → − ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
Theorem	istgp2 21705	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)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
Theorem	tmdmulg 21706*	In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))
Theorem	tgpmulg 21707*	In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ 𝐵 ↦ (𝑁 · 𝑥)) ∈ (𝐽 Cn 𝐽))
Theorem	tgpmulg2 21708	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 22427 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → · ∈ ((𝒫 ℤ ×t 𝐽) Cn 𝐽))
Theorem	tmdgsum 21709*	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 ∈ (𝐵 ↑𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
Theorem	tmdgsum2 21710*	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)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → ((#‘𝐴) · 𝑋) ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 (𝑋 ∈ 𝑢 ∧ ∀𝑓 ∈ (𝑢 ↑𝑚 𝐴)(𝐺 Σg 𝑓) ∈ 𝑈))
Theorem	oppgtmd 21711	The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopMnd → 𝑂 ∈ TopMnd)
Theorem	oppgtgp 21712	The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝑂 ∈ TopGrp)
Theorem	distgp 21713	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)
Theorem	indistgp 21714	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)
Theorem	symgtgp 21715	The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐺 ∈ TopGrp)
Theorem	tmdlactcn 21716*	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 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐽))
Theorem	tgplacthmeo 21717*	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 = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽Homeo𝐽))
Theorem	submtmd 21718	A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
Theorem	subgtgp 21719	A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Theorem	subgntr 21720	A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21722, 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)) → 𝑆 ∈ 𝐽)
Theorem	opnsubg 21721	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) ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ (Clsd‘𝐽))
Theorem	clssubg 21722	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))
Theorem	clsnsg 21723	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))
Theorem	cldsubg 21724	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) ↔ 𝑆 ∈ 𝐽))
Theorem	tgpconcompeqg 21725*	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 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)})
Theorem	tgpconcomp 21726*	The identity component, the connected component containing the identity element, is a closed (concompcld 21047) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}    ⇒   ⊢ (𝐺 ∈ TopGrp → 𝑆 ∈ (NrmSGrp‘𝐺))
Theorem	tgpconcompss 21727*	The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ ( 0 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ 𝐽) → 𝑆 ⊆ 𝑇)
Theorem	ghmcnp 21728	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) ↔ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
Theorem	snclseqg 21729	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 = (0g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝑆)    &   ⊢ 𝑆 = ((cls‘𝐽)‘{ 0 })    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((cls‘𝐽)‘{𝐴}))
Theorem	tgphaus 21730	A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ { 0 } ∈ (Clsd‘𝐽)))
Theorem	tgpt1 21731	Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
Theorem	tgpt0 21732	Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝐺 ∈ TopGrp → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Theorem	qustgpopn 21733*	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 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑆 ∈ 𝐽) → (𝐹 “ 𝑆) ∈ 𝐾)
Theorem	qustgplem 21734*	Lemma for qustgp 21735. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐾 = (TopOpen‘𝐻)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))    &   ⊢ − = (𝑧 ∈ 𝑋, 𝑤 ∈ 𝑋 ↦ [(𝑧(-g‘𝐺)𝑤)](𝐺 ~QG 𝑌))    ⇒   ⊢ ((𝐺 ∈ TopGrp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Theorem	qustgp 21735	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)
Theorem	qustgphaus 21736	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)
Theorem	prdstmdd 21737	The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)    ⇒   ⊢ (𝜑 → 𝑌 ∈ TopMnd)
Theorem	prdstgpd 21738	The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶TopGrp)    ⇒   ⊢ (𝜑 → 𝑌 ∈ TopGrp)
12.2.7  Infinite group sum on topological groups
Syntax	ctsu 21739	Extend class notation to include infinite group sums in a topological group.
class tsums
Definition	df-tsms 21740*	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 14265) and Σg (df-gsum 15926), 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 (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦)))))
Theorem	tsmsfbas 21741*	The collection of all sets of the form 𝐹(𝑧) = {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦}, 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 = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})    &   ⊢ 𝐿 = ran 𝐹    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (fBas‘𝑆))
Theorem	tsmslem1 21742	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)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺 Σg (𝐹 ↾ 𝑋)) ∈ 𝐵)
Theorem	tsmsval2 21743*	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 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → dom 𝐹 = 𝐴)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))
Theorem	tsmsval 21744*	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 (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 ⊆ 𝑦})    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦 ∈ 𝑆 ↦ (𝐺 Σg (𝐹 ↾ 𝑦)))))
Theorem	tsmspropd 21745	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 17139 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))    &   ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻))    &   ⊢ (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))
Theorem	eltsms 21746*	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)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ (𝐶 ∈ 𝐵 ∧ ∀𝑢 ∈ 𝐽 (𝐶 ∈ 𝑢 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑢)))))
Theorem	tsmsi 21747*	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)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐽)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑧 ⊆ 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝑈))
Theorem	tsmscl 21748	A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
Theorem	haustsms 21749*	In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    ⇒   ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ (𝐺 tsums 𝐹))
Theorem	haustsms2 21750	In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) = {𝑋}))
Theorem	tsmscls 21751	One half of tgptsmscls 21763, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
Theorem	tsmsgsum 21752	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ 𝐽 = (TopOpen‘𝐺)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{(𝐺 Σg 𝐹)}))
Theorem	tsmsid 21753	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 tsums 𝐹))
Theorem	haustsmsid 21754	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ Haus)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = {(𝐺 Σg 𝐹)})
Theorem	tsms0 21755*	The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 0 ∈ (𝐺 tsums (𝑥 ∈ 𝐴 ↦ 0 )))
Theorem	tsmssubm 21756	Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (𝜑 → (𝐻 tsums 𝐹) = ((𝐺 tsums 𝐹) ∩ 𝑆))
Theorem	tsmsres 21757	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑊)    ⇒   ⊢ (𝜑 → (𝐺 tsums (𝐹 ↾ 𝑊)) = (𝐺 tsums 𝐹))
Theorem	tsmsf1o 21758	Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐶–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = (𝐺 tsums (𝐹 ∘ 𝐻)))
Theorem	tsmsmhm 21759	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)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopSp)    &   ⊢ (𝜑 → 𝐻 ∈ CMnd)    &   ⊢ (𝜑 → 𝐻 ∈ TopSp)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 MndHom 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐶‘𝑋) ∈ (𝐻 tsums (𝐶 ∘ 𝐹)))
Theorem	tsmsadd 21760	The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums (𝐹 ∘𝑓 + 𝐻)))
Theorem	tsmsinv 21761	Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝐺 tsums (𝐼 ∘ 𝐹)))
Theorem	tsmssub 21762	The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums 𝐻))    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐺 tsums (𝐹 ∘𝑓 − 𝐻)))
Theorem	tgptsmscls 21763	A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 21723, 0nsg 17462. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Theorem	tgptsmscld 21764	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)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))
Theorem	tsmssplit 21765	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐺 tsums (𝐹 ↾ 𝐶)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 tsums (𝐹 ↾ 𝐷)))    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐺 tsums 𝐹))
Theorem	tsmsxplem1 21766*	Lemma for tsmsxp 21768. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 0 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → dom 𝐷 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷 ⊆ 𝑛 ∧ ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
Theorem	tsmsxplem2 21767*	Lemma for tsmsxp 21768. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 0 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → ∀𝑐 ∈ 𝑆 ∀𝑑 ∈ 𝑇 (𝑐 + 𝑑) ∈ 𝑈)    &   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝐶 ∩ Fin))    &   ⊢ (𝜑 → 𝐷 ⊆ (𝐾 × 𝑁))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ((𝐻‘𝑥) − (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑁)))) ∈ 𝐿)    &   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐾 × 𝑁))) ∈ 𝑆)    &   ⊢ (𝜑 → ∀𝑔 ∈ (𝐿 ↑𝑚 𝐾)(𝐺 Σg 𝑔) ∈ 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐻 ↾ 𝐾)) ∈ 𝑈)
Theorem	tsmsxp 21768*	Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 18198 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21766 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐺 ∈ TopGrp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐶)⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐻‘𝑗) ∈ (𝐺 tsums (𝑘 ∈ 𝐶 ↦ (𝑗𝐹𝑘))))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ (𝐺 tsums 𝐻))
12.2.8  Topological rings, fields, vector spaces
Syntax	ctrg 21769	The class of all topological division rings.
class TopRing
Syntax	ctdrg 21770	The class of all topological division rings.
class TopDRing
Syntax	ctlm 21771	The class of all topological modules.
class TopMod
Syntax	ctvc 21772	The class of all topological vector spaces.
class TopVec
Definition	df-trg 21773	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}
Definition	df-tdrg 21774	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}
Definition	df-tlm 21775	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 ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t (TopOpen‘𝑤)) Cn (TopOpen‘𝑤)))}
Definition	df-tvc 21776	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}
Theorem	istrg 21777	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))
Theorem	trgtmd 21778	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)
Theorem	istdrg 21779	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 ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
Theorem	tdrgunit 21780	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 → (𝑀 ↾s 𝑈) ∈ TopGrp)
Theorem	trgtgp 21781	A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
Theorem	trgtmd2 21782	A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
Theorem	trgtps 21783	A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)
Theorem	trgring 21784	A topological ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
Theorem	trggrp 21785	A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)
Theorem	tdrgtrg 21786	A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
Theorem	tdrgdrng 21787	A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)
Theorem	tdrgring 21788	A topological division ring is a ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring)
Theorem	tdrgtmd 21789	A topological division ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopMnd)
Theorem	tdrgtps 21790	A topological division ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp)
Theorem	istdrg2 21791	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 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s (𝐵 ∖ { 0 })) ∈ TopGrp))
Theorem	mulrcn 21792	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))
Theorem	invrcn2 21793	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 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)))
Theorem	invrcn 21794	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 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽))
Theorem	cnmpt1mulr 21795*	Continuity of ring multiplication; analogue of cnmpt12f 21279 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐾 Cn 𝐽))
Theorem	cnmpt2mulr 21796*	Continuity of ring multiplication; analogue of cnmpt22f 21288 which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Theorem	dvrcn 21797	The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽))
Theorem	istlm 21798	The predicate "𝑊 is a topological left module". (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod ↔ ((𝑊 ∈ TopMnd ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ TopRing) ∧ · ∈ ((𝐾 ×t 𝐽) Cn 𝐽)))
Theorem	vscacn 21799	The scalar multiplication is continuous in a topological module. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ TopMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Theorem	tlmtmd 21800	A topological module is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopMnd)