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Theorem tsmsval2 24028
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 24025 . . 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 3474 . . . . . . 7 𝑓 ∈ V
43dmex 7912 . . . . . 6 dom 𝑓 ∈ V
54pwex 5375 . . . . 5 𝒫 dom 𝑓 ∈ V
65inex1 5312 . . . 4 (𝒫 dom 𝑓 ∩ Fin) ∈ V
76a1i 11 . . 3 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8 simplrl 776 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
98fveq2d 6896 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = (TopOpen‘𝐺))
10 tsmsval.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
119, 10eqtr4di 2786 . . . . 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 772 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
1413dmeqd 5903 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15 tsmsval2.a . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
1615adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
1714, 16eqtrd 2768 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
1817pweqd 4616 . . . . . . . . 9 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
1918ineq1d 4208 . . . . . . . 8 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20 tsmsval.s . . . . . . . 8 𝑆 = (𝒫 𝐴 ∩ Fin)
2119, 20eqtr4di 2786 . . . . . . 7 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
2212, 21sylan9eqr 2790 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
2322rabeqdv 3443 . . . . . . . . 9 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2422, 23mpteq12dv 5234 . . . . . . . 8 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
2524rneqd 5935 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
26 tsmsval.l . . . . . . 7 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
2725, 26eqtr4di 2786 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = 𝐿)
2822, 27oveq12d 7433 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})) = (𝑆filGen𝐿))
2911, 28oveq12d 7433 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
30 simplrr 777 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
3130reseq1d 5979 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓𝑦) = (𝐹𝑦))
328, 31oveq12d 7433 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓𝑦)) = (𝐺 Σg (𝐹𝑦)))
3322, 32mpteq12dv 5234 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦))) = (𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦))))
3429, 33fveq12d 6899 . . 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)))))
357, 34csbied 3928 . 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)))))
36 tsmsval.g . . 3 (𝜑𝐺𝑉)
3736elexd 3491 . 2 (𝜑𝐺 ∈ V)
38 tsmsval2.f . . 3 (𝜑𝐹𝑊)
3938elexd 3491 . 2 (𝜑𝐹 ∈ V)
40 fvexd 6907 . 2 (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))) ∈ V)
412, 35, 37, 39, 40ovmpod 7568 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {crab 3428  Vcvv 3470  csb 3890  cin 3944  wss 3945  𝒫 cpw 4599  cmpt 5226  dom cdm 5673  ran crn 5674  cres 5675  cfv 6543  (class class class)co 7415  cmpo 7417  Fincfn 8958  Basecbs 17174  TopOpenctopn 17397   Σg cgsu 17416  filGencfg 21262   fLimf cflf 23833   tsums ctsu 24024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-tsms 24025
This theorem is referenced by:  tsmsval  24029  tsmspropd  24030
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