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Theorem tsmsval 24155
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 6921 . . . 4 𝐵 ∈ V
98a1i 11 . . 3 (𝜑𝐵 ∈ V)
10 fex2 7957 . . 3 ((𝐹:𝐴𝐵𝐴𝑊𝐵 ∈ V) → 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1370 . 2 (𝜑𝐹 ∈ V)
126fdmd 6747 . 2 (𝜑 → dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 24154 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605  cmpt 5231  ran crn 5690  cres 5691  wf 6559  cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245  TopOpenctopn 17468   Σg cgsu 17487  filGencfg 21371   fLimf cflf 23959   tsums ctsu 24150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tsms 24151
This theorem is referenced by:  eltsms  24157  haustsms  24160  tsmscls  24162  tsmsmhm  24170  tsmsadd  24171
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