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Theorem tsmsval 21928
Description: 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.)
Hypotheses
Ref Expression
tsmsval.b 𝐵 = (Base‘𝐺)
tsmsval.j 𝐽 = (TopOpen‘𝐺)
tsmsval.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
tsmsval.g (𝜑𝐺𝑉)
tsmsval.a (𝜑𝐴𝑊)
tsmsval.f (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
tsmsval (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   𝜑,𝑦,𝑧   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐵(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem tsmsval
StepHypRef Expression
1 tsmsval.b . 2 𝐵 = (Base‘𝐺)
2 tsmsval.j . 2 𝐽 = (TopOpen‘𝐺)
3 tsmsval.s . 2 𝑆 = (𝒫 𝐴 ∩ Fin)
4 tsmsval.l . 2 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
5 tsmsval.g . 2 (𝜑𝐺𝑉)
6 tsmsval.f . . 3 (𝜑𝐹:𝐴𝐵)
7 tsmsval.a . . 3 (𝜑𝐴𝑊)
8 fvex 6199 . . . . 5 (Base‘𝐺) ∈ V
91, 8eqeltri 2696 . . . 4 𝐵 ∈ V
109a1i 11 . . 3 (𝜑𝐵 ∈ V)
11 fex2 7118 . . 3 ((𝐹:𝐴𝐵𝐴𝑊𝐵 ∈ V) → 𝐹 ∈ V)
126, 7, 10, 11syl3anc 1325 . 2 (𝜑𝐹 ∈ V)
13 fdm 6049 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
146, 13syl 17 . 2 (𝜑 → dom 𝐹 = 𝐴)
151, 2, 3, 4, 5, 12, 14tsmsval2 21927 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  {crab 2915  Vcvv 3198  cin 3571  wss 3572  𝒫 cpw 4156  cmpt 4727  dom cdm 5112  ran crn 5113  cres 5114  wf 5882  cfv 5886  (class class class)co 6647  Fincfn 7952  Basecbs 15851  TopOpenctopn 16076   Σg cgsu 16095  filGencfg 19729   fLimf cflf 21733   tsums ctsu 21923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-tsms 21924
This theorem is referenced by:  eltsms  21930  haustsms  21933  tsmscls  21935  tsmsmhm  21943  tsmsadd  21944
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