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Theorem tsmsval2 22130
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)
tsmsval2.f (𝜑𝐹𝑊)
tsmsval2.a (𝜑 → dom 𝐹 = 𝐴)
Assertion
Ref Expression
tsmsval2 (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 tsmsval2
Dummy variables 𝑓 𝑠 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tsms 22127 . . 3 tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ (𝒫 dom 𝑓 ∩ Fin) / 𝑠(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦)))))
21a1i 11 . 2 (𝜑 → tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ (𝒫 dom 𝑓 ∩ Fin) / 𝑠(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦))))))
3 vex 3339 . . . . . . 7 𝑓 ∈ V
43dmex 7260 . . . . . 6 dom 𝑓 ∈ V
54pwex 4993 . . . . 5 𝒫 dom 𝑓 ∈ V
65inex1 4947 . . . 4 (𝒫 dom 𝑓 ∩ Fin) ∈ V
76a1i 11 . . 3 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8 simplrl 819 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
98fveq2d 6352 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = (TopOpen‘𝐺))
10 tsmsval.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
119, 10syl6eqr 2808 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = 𝐽)
12 id 22 . . . . . . 7 (𝑠 = (𝒫 dom 𝑓 ∩ Fin) → 𝑠 = (𝒫 dom 𝑓 ∩ Fin))
13 simprr 813 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
1413dmeqd 5477 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15 tsmsval2.a . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
1615adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
1714, 16eqtrd 2790 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
1817pweqd 4303 . . . . . . . . 9 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
1918ineq1d 3952 . . . . . . . 8 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20 tsmsval.s . . . . . . . 8 𝑆 = (𝒫 𝐴 ∩ Fin)
2119, 20syl6eqr 2808 . . . . . . 7 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
2212, 21sylan9eqr 2812 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23 rabeq 3328 . . . . . . . . . 10 (𝑠 = 𝑆 → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2422, 23syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2522, 24mpteq12dv 4881 . . . . . . . 8 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
2625rneqd 5504 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
27 tsmsval.l . . . . . . 7 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
2826, 27syl6eqr 2808 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = 𝐿)
2922, 28oveq12d 6827 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})) = (𝑆filGen𝐿))
3011, 29oveq12d 6827 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
31 simplrr 820 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
3231reseq1d 5546 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓𝑦) = (𝐹𝑦))
338, 32oveq12d 6827 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓𝑦)) = (𝐺 Σg (𝐹𝑦)))
3422, 33mpteq12dv 4881 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦))) = (𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦))))
3530, 34fveq12d 6354 . . 3 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
367, 35csbied 3697 . 2 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) / 𝑠(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})))‘(𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
37 tsmsval.g . . 3 (𝜑𝐺𝑉)
38 elex 3348 . . 3 (𝐺𝑉𝐺 ∈ V)
3937, 38syl 17 . 2 (𝜑𝐺 ∈ V)
40 tsmsval2.f . . 3 (𝜑𝐹𝑊)
41 elex 3348 . . 3 (𝐹𝑊𝐹 ∈ V)
4240, 41syl 17 . 2 (𝜑𝐹 ∈ V)
43 fvexd 6360 . 2 (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))) ∈ V)
442, 36, 39, 42, 43ovmpt2d 6949 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  {crab 3050  Vcvv 3336  csb 3670  cin 3710  wss 3711  𝒫 cpw 4298  cmpt 4877  dom cdm 5262  ran crn 5263  cres 5264  cfv 6045  (class class class)co 6809  cmpt2 6811  Fincfn 8117  Basecbs 16055  TopOpenctopn 16280   Σg cgsu 16299  filGencfg 19933   fLimf cflf 21936   tsums ctsu 22126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-iota 6008  df-fun 6047  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-tsms 22127
This theorem is referenced by:  tsmsval  22131  tsmspropd  22132
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