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Theorem tsmsval 24139
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)
81fvexi 6920 . . . 4 𝐵 ∈ V
98a1i 11 . . 3 (𝜑𝐵 ∈ V)
10 fex2 7958 . . 3 ((𝐹:𝐴𝐵𝐴𝑊𝐵 ∈ V) → 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1373 . 2 (𝜑𝐹 ∈ V)
126fdmd 6746 . 2 (𝜑 → dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 24138 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600  cmpt 5225  ran crn 5686  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  Fincfn 8985  Basecbs 17247  TopOpenctopn 17466   Σg cgsu 17485  filGencfg 21353   fLimf cflf 23943   tsums ctsu 24134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tsms 24135
This theorem is referenced by:  eltsms  24141  haustsms  24144  tsmscls  24146  tsmsmhm  24154  tsmsadd  24155
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