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Theorem tsmspropd 24054
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 18718 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 7431 . . 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 6027 . . . . 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 18637 . . . 4 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)) = (𝐻 Ξ£g (𝐹 β†Ύ 𝑦)))
109mpteq2dv 5245 . . 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 6899 . 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 2725 . . 3 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
13 eqid 2725 . . 3 (TopOpenβ€˜πΊ) = (TopOpenβ€˜πΊ)
14 eqid 2725 . . 3 (𝒫 dom 𝐹 ∩ Fin) = (𝒫 dom 𝐹 ∩ Fin)
15 eqid 2725 . . 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 2726 . . 3 (πœ‘ β†’ dom 𝐹 = dom 𝐹)
1712, 13, 14, 15, 5, 3, 16tsmsval2 24052 . 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 2725 . . 3 (Baseβ€˜π») = (Baseβ€˜π»)
19 eqid 2725 . . 3 (TopOpenβ€˜π») = (TopOpenβ€˜π»)
2018, 19, 14, 15, 6, 3, 16tsmsval2 24052 . 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 2775 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 3419  Vcvv 3463   ∩ cin 3938   βŠ† wss 3939  π’« cpw 4598   ↦ cmpt 5226  dom cdm 5672  ran crn 5673   β†Ύ cres 5674  β€˜cfv 6543  (class class class)co 7416  Fincfn 8962  Basecbs 17179  +gcplusg 17232  TopOpenctopn 17402   Ξ£g cgsu 17421  filGencfg 21272   fLimf cflf 23857   tsums ctsu 24048
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seq 13999  df-0g 17422  df-gsum 17423  df-tsms 24049
This theorem is referenced by: (None)
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