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Theorem tsmspropd 23499
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 18586 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 7373 . . 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 5985 . . . . 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 18538 . . . 4 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)) = (𝐻 Ξ£g (𝐹 β†Ύ 𝑦)))
109mpteq2dv 5208 . . 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 6850 . 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 2733 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
13 eqid 2733 . . 3 (TopOpenβ€˜πΊ) = (TopOpenβ€˜πΊ)
14 eqid 2733 . . 3 (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15 eqid 2733 . . 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 2734 . . 3 (πœ‘ β†’ dom 𝐹 = dom 𝐹)
1712, 13, 14, 15, 5, 3, 16tsmsval2 23497 . 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 2733 . . 3 (Baseβ€˜π») = (Baseβ€˜π»)
19 eqid 2733 . . 3 (TopOpenβ€˜π») = (TopOpenβ€˜π»)
2018, 19, 14, 15, 6, 3, 16tsmsval2 23497 . 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 2783 1 (πœ‘ β†’ (𝐺 tsums 𝐹) = (𝐻 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561   ↦ cmpt 5189  dom cdm 5634  ran crn 5635   β†Ύ cres 5636  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  Basecbs 17088  +gcplusg 17138  TopOpenctopn 17308   Ξ£g cgsu 17327  filGencfg 20801   fLimf cflf 23302   tsums ctsu 23493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seq 13913  df-0g 17328  df-gsum 17329  df-tsms 23494
This theorem is referenced by: (None)
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