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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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