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Theorem tsmspropd 22740
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 17936 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
tsmspropd.f (𝜑𝐹𝑉)
tsmspropd.g (𝜑𝐺𝑊)
tsmspropd.h (𝜑𝐻𝑋)
tsmspropd.b (𝜑 → (Base‘𝐺) = (Base‘𝐻))
tsmspropd.p (𝜑 → (+g𝐺) = (+g𝐻))
tsmspropd.j (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻))
Assertion
Ref Expression
tsmspropd (𝜑 → (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 (𝜑 → (TopOpen‘𝐺) = (TopOpen‘𝐻))
21oveq1d 7164 . . 3 (𝜑 → ((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦}))) = ((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦}))))
3 tsmspropd.f . . . . . 6 (𝜑𝐹𝑉)
4 resexg 5885 . . . . . 6 (𝐹𝑉 → (𝐹𝑦) ∈ V)
53, 4syl 17 . . . . 5 (𝜑 → (𝐹𝑦) ∈ V)
6 tsmspropd.g . . . . 5 (𝜑𝐺𝑊)
7 tsmspropd.h . . . . 5 (𝜑𝐻𝑋)
8 tsmspropd.b . . . . 5 (𝜑 → (Base‘𝐺) = (Base‘𝐻))
9 tsmspropd.p . . . . 5 (𝜑 → (+g𝐺) = (+g𝐻))
105, 6, 7, 8, 9gsumpropd 17888 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝑦)) = (𝐻 Σg (𝐹𝑦)))
1110mpteq2dv 5148 . . 3 (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹𝑦))))
122, 11fveq12d 6668 . 2 (𝜑 → (((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑦)))) = (((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹𝑦)))))
13 eqid 2824 . . 3 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2824 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
15 eqid 2824 . . 3 (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
16 eqid 2824 . . 3 ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦}) = ran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦})
17 eqidd 2825 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
1813, 14, 15, 16, 6, 3, 17tsmsval2 22738 . 2 (𝜑 → (𝐺 tsums 𝐹) = (((TopOpen‘𝐺) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑦)))))
19 eqid 2824 . . 3 (Base‘𝐻) = (Base‘𝐻)
20 eqid 2824 . . 3 (TopOpen‘𝐻) = (TopOpen‘𝐻)
2119, 20, 15, 16, 7, 3, 17tsmsval2 22738 . 2 (𝜑 → (𝐻 tsums 𝐹) = (((TopOpen‘𝐻) fLimf ((𝒫 dom 𝐹 ∩ Fin)filGenran (𝑧 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ {𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ∣ 𝑧𝑦})))‘(𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹𝑦)))))
2212, 18, 213eqtr4d 2869 1 (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  cin 3918  wss 3919  𝒫 cpw 4522  cmpt 5132  dom cdm 5542  ran crn 5543  cres 5544  cfv 6343  (class class class)co 7149  Fincfn 8505  Basecbs 16483  +gcplusg 16565  TopOpenctopn 16695   Σg cgsu 16714  filGencfg 20534   fLimf cflf 22543   tsums ctsu 22734
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seq 13374  df-0g 16715  df-gsum 16716  df-tsms 22735
This theorem is referenced by: (None)
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