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Theorem eldprd 15521
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
dprdval.0  |-  .0.  =  ( 0g `  G )
dprdval.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
eldprd  |-  ( 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 eldprd
StepHypRef Expression
1 elfvdm 5720 . . . . 5  |-  ( A  e.  ( DProd  `  <. G ,  S >. )  -> 
<. G ,  S >.  e. 
dom DProd  )
2 df-ov 6047 . . . . 5  |-  ( G DProd 
S )  =  ( DProd  `  <. G ,  S >. )
31, 2eleq2s 2500 . . . 4  |-  ( A  e.  ( G DProd  S
)  ->  <. G ,  S >.  e.  dom DProd  )
4 df-br 4177 . . . 4  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
53, 4sylibr 204 . . 3  |-  ( A  e.  ( G DProd  S
)  ->  G dom DProd  S )
65pm4.71ri 615 . 2  |-  ( A  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  A  e.  ( G DProd  S ) ) )
7 dprdval.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
8 dprdval.w . . . . . . 7  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
97, 8dprdval 15520 . . . . . 6  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
109eleq2d 2475 . . . . 5  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <-> 
A  e.  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
11 eqid 2408 . . . . . 6  |-  ( f  e.  W  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) )
12 ovex 6069 . . . . . 6  |-  ( G 
gsumg  f )  e.  _V
1311, 12elrnmpti 5084 . . . . 5  |-  ( A  e.  ran  ( f  e.  W  |->  ( G 
gsumg  f ) )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) )
1410, 13syl6bb 253 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1514ancoms 440 . . 3  |-  ( ( dom  S  =  I  /\  G dom DProd  S )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1615pm5.32da 623 . 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 249 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2671   {crab 2674   _Vcvv 2920    \ cdif 3281   {csn 3778   <.cop 3781   class class class wbr 4176    e. cmpt 4230   `'ccnv 4840   dom cdm 4841   ran crn 4842   "cima 4844   ` cfv 5417  (class class class)co 6044   X_cixp 7026   Fincfn 7072   0gc0g 13682    gsumg cgsu 13683   DProd cdprd 15513
This theorem is referenced by:  dprdssv  15533  eldprdi  15535  dprdsubg  15541  dprdss  15546  dmdprdsplitlem  15554  dprddisj2  15556  dpjidcl  15575
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-ixp 7027  df-dprd 15515
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