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Theorem eldprdi 15497
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.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
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 2381 . . 3  |-  ( G 
gsumg  F )  =  ( G  gsumg  F )
4 oveq2 6022 . . . . 5  |-  ( f  =  F  ->  ( G  gsumg  f )  =  ( G  gsumg  F ) )
54eqeq2d 2392 . . . 4  |-  ( f  =  F  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  F ) ) )
65rspcev 2989 . . 3  |-  ( ( F  e.  W  /\  ( G  gsumg  F )  =  ( G  gsumg  F ) )  ->  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) )
72, 3, 6sylancl 644 . 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 " ( _V  \  {  .0.  } ) )  e. 
Fin }
119, 10eldprd 15483 . . 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 889 1  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2644   {crab 2647   _Vcvv 2893    \ cdif 3254   {csn 3751   class class class wbr 4147   `'ccnv 4811   dom cdm 4812   "cima 4815   ` cfv 5388  (class class class)co 6014   X_cixp 6993   Fincfn 7039   0gc0g 13644    gsumg cgsu 13645   DProd cdprd 15475
This theorem is referenced by:  dprdfsub  15500  dprdf11  15502  dprdsubg  15503  dprdub  15504  dpjidcl  15537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-ixp 6994  df-dprd 15477
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