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Theorem eldprdi 16523
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.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
eldprdi  |-  ( 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 eldprdi
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2  |-  ( ph  ->  G dom DProd  S )
2 eldprdi.3 . . 3  |-  ( ph  ->  F  e.  W )
3 eqid 2443 . . 3  |-  ( G 
gsumg  F )  =  ( G  gsumg  F )
4 oveq2 6114 . . . . 5  |-  ( f  =  F  ->  ( G  gsumg  f )  =  ( G  gsumg  F ) )
54eqeq2d 2454 . . . 4  |-  ( f  =  F  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  F ) ) )
65rspcev 3088 . . 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 eldprdi.2 . . 3  |-  ( ph  ->  dom  S  =  I )
9 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
10 eldprdi.w . . . 4  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
119, 10eldprd 16501 . . 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 2731   {crab 2734   class class class wbr 4307   dom cdm 4855   ` cfv 5433  (class class class)co 6106   X_cixp 7278   finSupp cfsupp 7635   0gc0g 14393    gsumg cgsu 14394   DProd cdprd 16490
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 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-ixp 7279  df-dprd 16492
This theorem is referenced by:  dprdfsub  16526  dprdf11  16528  dprdsubg  16536  dprdub  16537  dpjidcl  16572
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