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Theorem tsmsval 23855
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 6904 . . . 4 𝐡 ∈ V
98a1i 11 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
10 fex2 7926 . . 3 ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ π‘Š ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1369 . 2 (πœ‘ β†’ 𝐹 ∈ V)
126fdmd 6727 . 2 (πœ‘ β†’ dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 23854 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 1539   ∈ wcel 2104  {crab 3430  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Fincfn 8941  Basecbs 17148  TopOpenctopn 17371   Ξ£g cgsu 17390  filGencfg 21133   fLimf cflf 23659   tsums ctsu 23850
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-tsms 23851
This theorem is referenced by:  eltsms  23857  haustsms  23860  tsmscls  23862  tsmsmhm  23870  tsmsadd  23871
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