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Theorem tsmsval 23498
Description: Definition of the topological group sum(s) of a collection 𝐹(π‘₯) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsval.b 𝐡 = (Baseβ€˜πΊ)
tsmsval.j 𝐽 = (TopOpenβ€˜πΊ)
tsmsval.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
tsmsval.g (πœ‘ β†’ 𝐺 ∈ 𝑉)
tsmsval.a (πœ‘ β†’ 𝐴 ∈ π‘Š)
tsmsval.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
Assertion
Ref Expression
tsmsval (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   πœ‘,𝑦,𝑧   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐡(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   π‘Š(𝑦,𝑧)

Proof of Theorem tsmsval
StepHypRef Expression
1 tsmsval.b . 2 𝐡 = (Baseβ€˜πΊ)
2 tsmsval.j . 2 𝐽 = (TopOpenβ€˜πΊ)
3 tsmsval.s . 2 𝑆 = (𝒫 𝐴 ∩ Fin)
4 tsmsval.l . 2 𝐿 = ran (𝑧 ∈ 𝑆 ↦ {𝑦 ∈ 𝑆 ∣ 𝑧 βŠ† 𝑦})
5 tsmsval.g . 2 (πœ‘ β†’ 𝐺 ∈ 𝑉)
6 tsmsval.f . . 3 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
7 tsmsval.a . . 3 (πœ‘ β†’ 𝐴 ∈ π‘Š)
81fvexi 6857 . . . 4 𝐡 ∈ V
98a1i 11 . . 3 (πœ‘ β†’ 𝐡 ∈ V)
10 fex2 7871 . . 3 ((𝐹:𝐴⟢𝐡 ∧ 𝐴 ∈ π‘Š ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
116, 7, 9, 10syl3anc 1372 . 2 (πœ‘ β†’ 𝐹 ∈ V)
126fdmd 6680 . 2 (πœ‘ β†’ dom 𝐹 = 𝐴)
131, 2, 3, 4, 5, 11, 12tsmsval2 23497 1 (πœ‘ β†’ (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))β€˜(𝑦 ∈ 𝑆 ↦ (𝐺 Ξ£g (𝐹 β†Ύ 𝑦)))))
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  ran crn 5635   β†Ύ cres 5636  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  Basecbs 17088  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-tsms 23494
This theorem is referenced by:  eltsms  23500  haustsms  23503  tsmscls  23505  tsmsmhm  23513  tsmsadd  23514
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