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Theorem tsmsval 18160
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 5742 . . . . 5  |-  ( Base `  G )  e.  _V
91, 8eqeltri 2506 . . . 4  |-  B  e. 
_V
109a1i 11 . . 3  |-  ( ph  ->  B  e.  _V )
11 fex2 5603 . . 3  |-  ( ( F : A --> B  /\  A  e.  W  /\  B  e.  _V )  ->  F  e.  _V )
126, 7, 10, 11syl3anc 1184 . 2  |-  ( ph  ->  F  e.  _V )
13 fdm 5595 . . 3  |-  ( F : A --> B  ->  dom  F  =  A )
146, 13syl 16 . 2  |-  ( ph  ->  dom  F  =  A )
151, 2, 3, 4, 5, 12, 14tsmsval2 18159 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 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799    e. cmpt 4266   dom cdm 4878   ran crn 4879    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469   TopOpenctopn 13649    gsumg cgsu 13724   filGencfg 16690    fLimf cflf 17967   tsums ctsu 18155
This theorem is referenced by:  eltsms  18162  haustsms  18165  tsmscls  18167  tsmsmhm  18175  tsmsadd  18176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tsms 18156
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