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Theorem tsmsval 23616
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 6901 . . . 4 𝐵 ∈ V
98a1i 11 . . 3 (𝜑𝐵 ∈ V)
10 fex2 7918 . . 3 ((𝐹:𝐴𝐵𝐴𝑊𝐵 ∈ V) → 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1372 . 2 (𝜑𝐹 ∈ V)
126fdmd 6724 . 2 (𝜑 → dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 23615 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cin 3945  wss 3946  𝒫 cpw 4600  cmpt 5229  ran crn 5675  cres 5676  wf 6535  cfv 6539  (class class class)co 7403  Fincfn 8934  Basecbs 17139  TopOpenctopn 17362   Σg cgsu 17381  filGencfg 20917   fLimf cflf 23420   tsums ctsu 23611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-tsms 23612
This theorem is referenced by:  eltsms  23618  haustsms  23621  tsmscls  23623  tsmsmhm  23631  tsmsadd  23632
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