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Theorem tsmsval2 23625
Description: Definition of the topological group sum(s) of a collection 𝐹(π‘₯) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsval.b 𝐡 = (Baseβ€˜πΊ)
tsmsval.j 𝐽 = (TopOpenβ€˜πΊ)
tsmsval.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
tsmsval.g (πœ‘ β†’ 𝐺 ∈ 𝑉)
tsmsval2.f (πœ‘ β†’ 𝐹 ∈ π‘Š)
tsmsval2.a (πœ‘ β†’ dom 𝐹 = 𝐴)
Assertion
Ref Expression
tsmsval2 (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   πœ‘,𝑦,𝑧   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐡(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   π‘Š(𝑦,𝑧)
Proof of Theorem tsmsval2
Dummy variables 𝑓 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-tsms 23622 . . 3 tsums = (𝑀 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / π‘ β¦Œ(((TopOpenβ€˜π‘€) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ 𝑠 ↦ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦)))))
21a1i 11 . 2 (πœ‘ β†’ tsums = (𝑀 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫 dom 𝑓 ∩ Fin) / π‘ β¦Œ(((TopOpenβ€˜π‘€) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ 𝑠 ↦ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦))))))
3 vex 3478 . . . . . . 7 𝑓 ∈ V
43dmex 7898 . . . . . 6 dom 𝑓 ∈ V
54pwex 5377 . . . . 5 𝒫 dom 𝑓 ∈ V
65inex1 5316 . . . 4 (𝒫 dom 𝑓 ∩ Fin) ∈ V
76a1i 11 . . 3 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝒫 dom 𝑓 ∩ Fin) ∈ V)
8 simplrl 775 . . . . . . 7 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ 𝑀 = 𝐺)
98fveq2d 6892 . . . . . 6 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (TopOpenβ€˜π‘€) = (TopOpenβ€˜πΊ))
10 tsmsval.j . . . . . 6 𝐽 = (TopOpenβ€˜πΊ)
119, 10eqtr4di 2790 . . . . 5 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (TopOpenβ€˜π‘€) = 𝐽)
12 id 22 . . . . . . 7 (𝑠 = (𝒫 dom 𝑓 ∩ Fin) β†’ 𝑠 = (𝒫 dom 𝑓 ∩ Fin))
13 simprr 771 . . . . . . . . . . . 12 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑓 = 𝐹)
1413dmeqd 5903 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ dom 𝑓 = dom 𝐹)
15 tsmsval2.a . . . . . . . . . . . 12 (πœ‘ β†’ dom 𝐹 = 𝐴)
1615adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ dom 𝐹 = 𝐴)
1714, 16eqtrd 2772 . . . . . . . . . 10 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ dom 𝑓 = 𝐴)
1817pweqd 4618 . . . . . . . . 9 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝒫 dom 𝑓 = 𝒫 𝐴)
1918ineq1d 4210 . . . . . . . 8 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝒫 dom 𝑓 ∩ Fin) = (𝒫 𝐴 ∩ Fin))
20 tsmsval.s . . . . . . . 8 𝑆 = (𝒫 𝐴 ∩ Fin)
2119, 20eqtr4di 2790 . . . . . . 7 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝒫 dom 𝑓 ∩ Fin) = 𝑆)
2212, 21sylan9eqr 2794 . . . . . 6 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ 𝑠 = 𝑆)
2322rabeqdv 3447 . . . . . . . . 9 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦} = {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
2422, 23mpteq12dv 5238 . . . . . . . 8 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦}) = (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}))
2524rneqd 5935 . . . . . . 7 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦}) = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦}))
26 tsmsval.l . . . . . . 7 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
2725, 26eqtr4di 2790 . . . . . 6 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ ran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦}) = 𝐿)
2822, 27oveq12d 7423 . . . . 5 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦})) = (𝑆filGen𝐿))
2911, 28oveq12d 7423 . . . 4 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ ((TopOpenβ€˜π‘€) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦}))) = (𝐽 fLimf (𝑆filGen𝐿)))
30 simplrr 776 . . . . . . 7 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ 𝑓 = 𝐹)
3130reseq1d 5978 . . . . . 6 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (𝑓 β†Ύ 𝑦) = (𝐹 β†Ύ 𝑦))
328, 31oveq12d 7423 . . . . 5 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦)) = (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))
3322, 32mpteq12dv 5238 . . . 4 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (𝑦 ∈ 𝑠 ↦ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦))) = (𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦))))
3429, 33fveq12d 6895 . . 3 (((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) ∧ 𝑠 = (𝒫 dom 𝑓 ∩ Fin)) β†’ (((TopOpenβ€˜π‘€) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ 𝑠 ↦ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
357, 34csbied 3930 . 2 ((πœ‘ ∧ (𝑀 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ ⦋(𝒫 dom 𝑓 ∩ Fin) / π‘ β¦Œ(((TopOpenβ€˜π‘€) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ 𝑠 ↦ (𝑀 Ξ£g (𝑓 β†Ύ 𝑦)))) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
36 tsmsval.g . . 3 (πœ‘ β†’ 𝐺 ∈ 𝑉)
3736elexd 3494 . 2 (πœ‘ β†’ 𝐺 ∈ V)
38 tsmsval2.f . . 3 (πœ‘ β†’ 𝐹 ∈ π‘Š)
3938elexd 3494 . 2 (πœ‘ β†’ 𝐹 ∈ V)
40 fvexd 6903 . 2 (πœ‘ β†’ ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))) ∈ V)
412, 35, 37, 39, 40ovmpod 7556 1 (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474  β¦‹csb 3892   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601   ↦ cmpt 5230  dom cdm 5675  ran crn 5676   β†Ύ cres 5677  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8935  Basecbs 17140  TopOpenctopn 17363   Ξ£g cgsu 17382  filGencfg 20925   fLimf cflf 23430   tsums ctsu 23621
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-tsms 23622
This theorem is referenced by:  tsmsval  23626  tsmspropd  23627
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