MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tsmsval2 Structured version   Visualization version   GIF version

Theorem tsmsval2 22153
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 22150 . . 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 3354 . . . . . . 7 𝑓 ∈ V
43dmex 7250 . . . . . 6 dom 𝑓 ∈ V
54pwex 4982 . . . . 5 𝒫 dom 𝑓 ∈ V
65inex1 4934 . . . 4 (𝒫 dom 𝑓 ∩ Fin) ∈ V
76a1i 11 . . 3 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8 simplrl 762 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑤 = 𝐺)
98fveq2d 6337 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (TopOpen‘𝑤) = (TopOpen‘𝐺))
10 tsmsval.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
119, 10syl6eqr 2823 . . . . 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 756 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
1413dmeqd 5463 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = dom 𝐹)
15 tsmsval2.a . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝐴)
1615adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝐹 = 𝐴)
1714, 16eqtrd 2805 . . . . . . . . . 10 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → dom 𝑓 = 𝐴)
1817pweqd 4303 . . . . . . . . 9 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → 𝒫 dom 𝑓 = 𝒫 𝐴)
1918ineq1d 3964 . . . . . . . 8 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20 tsmsval.s . . . . . . . 8 𝑆 = (𝒫 𝐴 ∩ Fin)
2119, 20syl6eqr 2823 . . . . . . 7 ((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) → (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
2212, 21sylan9eqr 2827 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑠 = 𝑆)
23 rabeq 3342 . . . . . . . . . 10 (𝑠 = 𝑆 → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2422, 23syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → {𝑦𝑠𝑧𝑦} = {𝑦𝑆𝑧𝑦})
2522, 24mpteq12dv 4868 . . . . . . . 8 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦}))
2625rneqd 5490 . . . . . . 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 2823 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}) = 𝐿)
2922, 28oveq12d 6814 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦})) = (𝑆filGen𝐿))
3011, 29oveq12d 6814 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → ((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧𝑠 ↦ {𝑦𝑠𝑧𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
31 simplrr 763 . . . . . . 7 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → 𝑓 = 𝐹)
3231reseq1d 5532 . . . . . 6 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑓𝑦) = (𝐹𝑦))
338, 32oveq12d 6814 . . . . 5 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑤 Σg (𝑓𝑦)) = (𝐺 Σg (𝐹𝑦)))
3422, 33mpteq12dv 4868 . . . 4 (((𝜑 ∧ (𝑤 = 𝐺𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) → (𝑦𝑠 ↦ (𝑤 Σg (𝑓𝑦))) = (𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦))))
3530, 34fveq12d 6340 . . 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 3709 . 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 3364 . . 3 (𝐺𝑉𝐺 ∈ V)
3937, 38syl 17 . 2 (𝜑𝐺 ∈ V)
40 tsmsval2.f . . 3 (𝜑𝐹𝑊)
41 elex 3364 . . 3 (𝐹𝑊𝐹 ∈ V)
4240, 41syl 17 . 2 (𝜑𝐹 ∈ V)
43 fvexd 6346 . 2 (𝜑 → ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))) ∈ V)
442, 36, 39, 42, 43ovmpt2d 6939 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  csb 3682  cin 3722  wss 3723  𝒫 cpw 4298  cmpt 4864  dom cdm 5250  ran crn 5251  cres 5252  cfv 6030  (class class class)co 6796  cmpt2 6798  Fincfn 8113  Basecbs 16064  TopOpenctopn 16290   Σg cgsu 16309  filGencfg 19950   fLimf cflf 21959   tsums ctsu 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-tsms 22150
This theorem is referenced by:  tsmsval  22154  tsmspropd  22155
  Copyright terms: Public domain W3C validator