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Theorem tsmspropd 23991
Description: The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 18692 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tsmspropd.f (πœ‘ β†’ 𝐹 ∈ 𝑉)
tsmspropd.g (πœ‘ β†’ 𝐺 ∈ π‘Š)
tsmspropd.h (πœ‘ β†’ 𝐻 ∈ 𝑋)
tsmspropd.b (πœ‘ β†’ (Baseβ€˜πΊ) = (Baseβ€˜π»))
tsmspropd.p (πœ‘ β†’ (+gβ€˜πΊ) = (+gβ€˜π»))
tsmspropd.j (πœ‘ β†’ (TopOpenβ€˜πΊ) = (TopOpenβ€˜π»))
Assertion
Ref Expression
tsmspropd (πœ‘ β†’ (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))

Proof of Theorem tsmspropd
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmspropd.j . . . 4 (πœ‘ β†’ (TopOpenβ€˜πΊ) = (TopOpenβ€˜π»))
21oveq1d 7420 . . 3 (πœ‘ β†’ ((TopOpenβ€˜πΊ) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦}))) = ((TopOpenβ€˜π») fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦}))))
3 tsmspropd.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ 𝑉)
43resexd 6022 . . . . 5 (πœ‘ β†’ (𝐹 β†Ύ 𝑦) ∈ V)
5 tsmspropd.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ π‘Š)
6 tsmspropd.h . . . . 5 (πœ‘ β†’ 𝐻 ∈ 𝑋)
7 tsmspropd.b . . . . 5 (πœ‘ β†’ (Baseβ€˜πΊ) = (Baseβ€˜π»))
8 tsmspropd.p . . . . 5 (πœ‘ β†’ (+gβ€˜πΊ) = (+gβ€˜π»))
94, 5, 6, 7, 8gsumpropd 18611 . . . 4 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)) = (𝐻 Ξ£g (𝐹 β†Ύ 𝑦)))
109mpteq2dv 5243 . . 3 (πœ‘ β†’ (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Ξ£g (𝐹 β†Ύ 𝑦))))
112, 10fveq12d 6892 . 2 (πœ‘ β†’ (((TopOpenβ€˜πΊ) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))) = (((TopOpenβ€˜π») fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Ξ£g (𝐹 β†Ύ 𝑦)))))
12 eqid 2726 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
13 eqid 2726 . . 3 (TopOpenβ€˜πΊ) = (TopOpenβ€˜πΊ)
14 eqid 2726 . . 3 (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15 eqid 2726 . . 3 ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦}) = ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦})
16 eqidd 2727 . . 3 (πœ‘ β†’ dom 𝐹 = dom 𝐹)
1712, 13, 14, 15, 5, 3, 16tsmsval2 23989 . 2 (πœ‘ β†’ (𝐺 tsums 𝐹) = (((TopOpenβ€˜πΊ) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
18 eqid 2726 . . 3 (Baseβ€˜π») = (Baseβ€˜π»)
19 eqid 2726 . . 3 (TopOpenβ€˜π») = (TopOpenβ€˜π»)
2018, 19, 14, 15, 6, 3, 16tsmsval2 23989 . 2 (πœ‘ β†’ (𝐻 tsums 𝐹) = (((TopOpenβ€˜π») fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧 βŠ† 𝑦})))β€˜(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Ξ£g (𝐹 β†Ύ 𝑦)))))
2111, 17, 203eqtr4d 2776 1 (πœ‘ β†’ (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597   ↦ cmpt 5224  dom cdm 5669  ran crn 5670   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  Fincfn 8941  Basecbs 17153  +gcplusg 17206  TopOpenctopn 17376   Ξ£g cgsu 17395  filGencfg 21229   fLimf cflf 23794   tsums ctsu 23985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seq 13973  df-0g 17396  df-gsum 17397  df-tsms 23986
This theorem is referenced by: (None)
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