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Theorem tsmsval 17813
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  i^i  Fin )
tsmsval.l  |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )
tsmsval.g  |-  ( ph  ->  G  e.  V )
tsmsval.a  |-  ( ph  ->  A  e.  W )
tsmsval.f  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
tsmsval  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
Distinct variable groups:    y, z, F    y, G, z    ph, 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  i^i  Fin )
4 tsmsval.l . 2  |-  L  =  ran  ( z  e.  S  |->  { y  e.  S  |  z  C_  y } )
5 tsmsval.g . 2  |-  ( ph  ->  G  e.  V )
6 tsmsval.f . . 3  |-  ( ph  ->  F : A --> B )
7 tsmsval.a . . 3  |-  ( ph  ->  A  e.  W )
8 fvex 5539 . . . . 5  |-  ( Base `  G )  e.  _V
91, 8eqeltri 2353 . . . 4  |-  B  e. 
_V
109a1i 10 . . 3  |-  ( ph  ->  B  e.  _V )
11 fex2 5401 . . 3  |-  ( ( F : A --> B  /\  A  e.  W  /\  B  e.  _V )  ->  F  e.  _V )
126, 7, 10, 11syl3anc 1182 . 2  |-  ( ph  ->  F  e.  _V )
13 fdm 5393 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
146, 13syl 15 . 2  |-  ( ph  ->  dom  F  =  A )
151, 2, 3, 4, 5, 12, 14tsmsval2 17812 1  |-  ( ph  ->  ( G tsums  F )  =  ( ( J 
fLimf  ( S filGen L ) ) `  ( y  e.  S  |->  ( G 
gsumg  ( F  |`  y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   TopOpenctopn 13326    gsumg cgsu 13401   filGencfg 17519    fLimf cflf 17630   tsums ctsu 17808
This theorem is referenced by:  eltsms  17815  haustsms  17818  tsmscls  17820  tsmsmhm  17828  tsmsadd  17829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tsms 17809
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