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Theorem eldprdiOLD 16513
Description: The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of eldprdi 16506 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
eldprdiOLD  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Distinct variable groups:    h, F    h, i, G    h, I,
i    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    W( h, i)    .0. ( i)

Proof of Theorem eldprdiOLD
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.1 . 2  |-  ( ph  ->  G dom DProd  S )
2 eldprdiOLD.3 . . 3  |-  ( ph  ->  F  e.  W )
3 eqid 2441 . . 3  |-  ( G 
gsumg  F )  =  ( G  gsumg  F )
4 oveq2 6097 . . . . 5  |-  ( f  =  F  ->  ( G  gsumg  f )  =  ( G  gsumg  F ) )
54eqeq2d 2452 . . . 4  |-  ( f  =  F  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  F ) ) )
65rspcev 3071 . . 3  |-  ( ( F  e.  W  /\  ( G  gsumg  F )  =  ( G  gsumg  F ) )  ->  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) )
72, 3, 6sylancl 662 . 2  |-  ( ph  ->  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) )
8 eldprdiOLD.2 . . 3  |-  ( ph  ->  dom  S  =  I )
9 eldprdiOLD.0 . . . 4  |-  .0.  =  ( 0g `  G )
10 eldprdiOLD.w . . . 4  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
119, 10eldprdOLD 16486 . . 3  |-  ( dom 
S  =  I  -> 
( ( G  gsumg  F )  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) ) ) )
128, 11syl 16 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) ) ) )
131, 7, 12mpbir2and 913 1  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714   {crab 2717   _Vcvv 2970    \ cdif 3323   {csn 3875   class class class wbr 4290   `'ccnv 4837   dom cdm 4838   "cima 4841   ` cfv 5416  (class class class)co 6089   X_cixp 7261   Fincfn 7308   0gc0g 14376    gsumg cgsu 14377   DProd cdprd 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-supp 6689  df-ixp 7262  df-fsupp 7619  df-dprd 16475
This theorem is referenced by:  dprdfsubOLD  16516  dprdf11OLD  16518  dpjidclOLD  16562
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