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Theorem tsmspropd 22306
 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 17670 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 6921 . . 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 5680 . . . . . 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 17626 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝑦)) = (𝐻 Σg (𝐹𝑦)))
1110mpteq2dv 4969 . . 3 (𝜑 → (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑦))) = (𝑦 ∈ (𝒫 dom 𝐹 ∩ Fin) ↦ (𝐻 Σg (𝐹𝑦))))
122, 11fveq12d 6441 . 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 2826 . . 3 (Base‘𝐺) = (Base‘𝐺)
14 eqid 2826 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
15 eqid 2826 . . 3 (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
16 eqid 2826 . . 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 2827 . . 3 (𝜑 → dom 𝐹 = dom 𝐹)
1813, 14, 15, 16, 6, 3, 17tsmsval2 22304 . 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 2826 . . 3 (Base‘𝐻) = (Base‘𝐻)
20 eqid 2826 . . 3 (TopOpen‘𝐻) = (TopOpen‘𝐻)
2119, 20, 15, 16, 7, 3, 17tsmsval2 22304 . 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 2872 1 (𝜑 → (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  {crab 3122  Vcvv 3415   ∩ cin 3798   ⊆ wss 3799  𝒫 cpw 4379   ↦ cmpt 4953  dom cdm 5343  ran crn 5344   ↾ cres 5345  ‘cfv 6124  (class class class)co 6906  Fincfn 8223  Basecbs 16223  +gcplusg 16306  TopOpenctopn 16436   Σg cgsu 16455  filGencfg 20096   fLimf cflf 22110   tsums ctsu 22300 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-seq 13097  df-0g 16456  df-gsum 16457  df-tsms 22301 This theorem is referenced by: (None)
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