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Theorem tsmsval 21928
 Description: Definition of the topological group sum(s) of a collection 𝐹(𝑥) of values in the group with index set 𝐴. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tsmsval.b 𝐵 = (Base‘𝐺)
tsmsval.j 𝐽 = (TopOpen‘𝐺)
tsmsval.s 𝑆 = (𝒫 𝐴 ∩ Fin)
tsmsval.l 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
tsmsval.g (𝜑𝐺𝑉)
tsmsval.a (𝜑𝐴𝑊)
tsmsval.f (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
tsmsval (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
Distinct variable groups:   𝑦,𝑧,𝐹   𝑦,𝐺,𝑧   𝜑,𝑦,𝑧   𝑦,𝑆
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝐵(𝑦,𝑧)   𝑆(𝑧)   𝐽(𝑦,𝑧)   𝐿(𝑦,𝑧)   𝑉(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem tsmsval
StepHypRef Expression
1 tsmsval.b . 2 𝐵 = (Base‘𝐺)
2 tsmsval.j . 2 𝐽 = (TopOpen‘𝐺)
3 tsmsval.s . 2 𝑆 = (𝒫 𝐴 ∩ Fin)
4 tsmsval.l . 2 𝐿 = ran (𝑧𝑆 ↦ {𝑦𝑆𝑧𝑦})
5 tsmsval.g . 2 (𝜑𝐺𝑉)
6 tsmsval.f . . 3 (𝜑𝐹:𝐴𝐵)
7 tsmsval.a . . 3 (𝜑𝐴𝑊)
8 fvex 6199 . . . . 5 (Base‘𝐺) ∈ V
91, 8eqeltri 2696 . . . 4 𝐵 ∈ V
109a1i 11 . . 3 (𝜑𝐵 ∈ V)
11 fex2 7118 . . 3 ((𝐹:𝐴𝐵𝐴𝑊𝐵 ∈ V) → 𝐹 ∈ V)
126, 7, 10, 11syl3anc 1325 . 2 (𝜑𝐹 ∈ V)
13 fdm 6049 . . 3 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
146, 13syl 17 . 2 (𝜑 → dom 𝐹 = 𝐴)
151, 2, 3, 4, 5, 12, 14tsmsval2 21927 1 (𝜑 → (𝐺 tsums 𝐹) = ((𝐽 fLimf (𝑆filGen𝐿))‘(𝑦𝑆 ↦ (𝐺 Σg (𝐹𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1482   ∈ wcel 1989  {crab 2915  Vcvv 3198   ∩ cin 3571   ⊆ wss 3572  𝒫 cpw 4156   ↦ cmpt 4727  dom cdm 5112  ran crn 5113   ↾ cres 5114  ⟶wf 5882  ‘cfv 5886  (class class class)co 6647  Fincfn 7952  Basecbs 15851  TopOpenctopn 16076   Σg cgsu 16095  filGencfg 19729   fLimf cflf 21733   tsums ctsu 21923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-tsms 21924 This theorem is referenced by:  eltsms  21930  haustsms  21933  tsmscls  21935  tsmsmhm  21943  tsmsadd  21944
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