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Theorem eldprdiOLD 17191
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 17184 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 2457 . . 3  |-  ( G 
gsumg  F )  =  ( G  gsumg  F )
4 oveq2 6304 . . . . 5  |-  ( f  =  F  ->  ( G  gsumg  f )  =  ( G  gsumg  F ) )
54eqeq2d 2471 . . . 4  |-  ( f  =  F  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  F ) ) )
65rspcev 3210 . . 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 17163 . . 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 922 1  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468   {csn 4032   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   "cima 5011   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   0gc0g 14856    gsumg cgsu 14857   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-supp 6918  df-ixp 7489  df-fsupp 7848  df-dprd 17152
This theorem is referenced by:  dprdfsubOLD  17194  dprdf11OLD  17196  dpjidclOLD  17240
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