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Theorem tsmsval 23634
Description: Definition of the topological group sum(s) of a collection 𝐹(π‘₯) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsval.b 𝐡 = (Baseβ€˜πΊ)
tsmsval.j 𝐽 = (TopOpenβ€˜πΊ)
tsmsval.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
tsmsval.g (πœ‘ β†’ 𝐺 ∈ 𝑉)
tsmsval.a (πœ‘ β†’ 𝐴 ∈ π‘Š)
tsmsval.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Assertion
Ref Expression
tsmsval (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   πœ‘,𝑦,𝑧   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐡(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   π‘Š(𝑦,𝑧)

Proof of Theorem tsmsval
StepHypRef Expression
1 tsmsval.b . 2 𝐡 = (Baseβ€˜πΊ)
2 tsmsval.j . 2 𝐽 = (TopOpenβ€˜πΊ)
3 tsmsval.s . 2 𝑆 = (𝒫 𝐴 ∩ Fin)
4 tsmsval.l . 2 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
5 tsmsval.g . 2 (πœ‘ β†’ 𝐺 ∈ 𝑉)
6 tsmsval.f . . 3 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
7 tsmsval.a . . 3 (πœ‘ β†’ 𝐴 ∈ π‘Š)
81fvexi 6905 . . . 4 𝐡 ∈ V
98a1i 11 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
10 fex2 7923 . . 3 ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ π‘Š ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1371 . 2 (πœ‘ β†’ 𝐹 ∈ V)
126fdmd 6728 . 2 (πœ‘ β†’ dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 23633 1 (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Fincfn 8938  Basecbs 17143  TopOpenctopn 17366   Ξ£g cgsu 17385  filGencfg 20932   fLimf cflf 23438   tsums ctsu 23629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-tsms 23630
This theorem is referenced by:  eltsms  23636  haustsms  23639  tsmscls  23641  tsmsmhm  23649  tsmsadd  23650
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