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Theorem eldprdOLD 17150
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 dprdval 17147 as of 11-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dprdvalOLD.0  |-  .0.  =  ( 0g `  G )
dprdvalOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
eldprdOLD  |-  ( dom 
S  =  I  -> 
( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
Distinct variable groups:    f, h, i    A, f    f, I, h, i    S, f, h, i    f, G, h, i
Allowed substitution hints:    A( h, i)    W( f, h, i)    .0. ( f, h, i)

Proof of Theorem eldprdOLD
StepHypRef Expression
1 elfvdm 5800 . . . . 5  |-  ( A  e.  ( DProd  `  <. G ,  S >. )  -> 
<. G ,  S >.  e. 
dom DProd  )
2 df-ov 6199 . . . . 5  |-  ( G DProd 
S )  =  ( DProd  `  <. G ,  S >. )
31, 2eleq2s 2490 . . . 4  |-  ( A  e.  ( G DProd  S
)  ->  <. G ,  S >.  e.  dom DProd  )
4 df-br 4368 . . . 4  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
53, 4sylibr 212 . . 3  |-  ( A  e.  ( G DProd  S
)  ->  G dom DProd  S )
65pm4.71ri 631 . 2  |-  ( A  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  A  e.  ( G DProd  S ) ) )
7 dprdvalOLD.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
8 dprdvalOLD.w . . . . . . 7  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
97, 8dprdvalOLD 17149 . . . . . 6  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
109eleq2d 2452 . . . . 5  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <-> 
A  e.  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
11 eqid 2382 . . . . . 6  |-  ( f  e.  W  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) )
12 ovex 6224 . . . . . 6  |-  ( G 
gsumg  f )  e.  _V
1311, 12elrnmpti 5166 . . . . 5  |-  ( A  e.  ran  ( f  e.  W  |->  ( G 
gsumg  f ) )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) )
1410, 13syl6bb 261 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1514ancoms 451 . . 3  |-  ( ( dom  S  =  I  /\  G dom DProd  S )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1615pm5.32da 639 . 2  |-  ( dom 
S  =  I  -> 
( ( G dom DProd  S  /\  A  e.  ( G DProd  S ) )  <-> 
( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
176, 16syl5bb 257 1  |-  ( dom 
S  =  I  -> 
( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736   _Vcvv 3034    \ cdif 3386   {csn 3944   <.cop 3950   class class class wbr 4367    |-> cmpt 4425   `'ccnv 4912   dom cdm 4913   ran crn 4914   "cima 4916   ` cfv 5496  (class class class)co 6196   X_cixp 7388   Fincfn 7435   0gc0g 14847    gsumg cgsu 14848   DProd cdprd 17137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-supp 6818  df-ixp 7389  df-fsupp 7745  df-dprd 17139
This theorem is referenced by:  eldprdiOLD  17178  dmdprdsplitlemOLD  17198  dprddisj2OLD  17201  dpjidclOLD  17227
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