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Theorem tsmsval2 22735
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 22732 . . 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 3444 . . . . . . 7 𝑓 ∈ V
43dmex 7598 . . . . . 6 dom 𝑓 ∈ V
54pwex 5246 . . . . 5 𝒫 dom 𝑓 ∈ V
65inex1 5185 . . . 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 6649 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = (TopOpen‘𝐺))
10 tsmsval.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
119, 10eqtr4di 2851 . . . . 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 5738 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15 tsmsval2.a . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
1615adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
1714, 16eqtrd 2833 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
1817pweqd 4516 . . . . . . . . 9 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
1918ineq1d 4138 . . . . . . . 8 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20 tsmsval.s . . . . . . . 8 𝑆 = (𝒫 𝐴 ∩ Fin)
2119, 20eqtr4di 2851 . . . . . . 7 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
2212, 21sylan9eqr 2855 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
2322rabeqdv 3432 . . . . . . . . 9 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2422, 23mpteq12dv 5115 . . . . . . . 8 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
2524rneqd 5772 . . . . . . 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 2851 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = 𝐿)
2822, 27oveq12d 7153 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})) = (𝑆filGen𝐿))
2911, 28oveq12d 7153 . . . 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 5817 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓𝑦) = (𝐹𝑦))
328, 31oveq12d 7153 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓𝑦)) = (𝐺 Σg (𝐹𝑦)))
3322, 32mpteq12dv 5115 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦))) = (𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦))))
3429, 33fveq12d 6652 . . 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 3864 . 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 3461 . 2 (𝜑𝐺 ∈ V)
38 tsmsval2.f . . 3 (𝜑𝐹𝑊)
3938elexd 3461 . 2 (𝜑𝐹 ∈ V)
40 fvexd 6660 . 2 (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))) ∈ V)
412, 35, 37, 39, 40ovmpod 7281 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {crab 3110  Vcvv 3441  csb 3828  cin 3880  wss 3881  𝒫 cpw 4497  cmpt 5110  dom cdm 5519  ran crn 5520  cres 5521  cfv 6324  (class class class)co 7135  cmpo 7137  Fincfn 8492  Basecbs 16475  TopOpenctopn 16687   Σg cgsu 16706  filGencfg 20080   fLimf cflf 22540   tsums ctsu 22731
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-tsms 22732
This theorem is referenced by:  tsmsval  22736  tsmspropd  22737
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