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Theorem eldprd 16609
Description: A class  A is an internal direct product iff it is the (group) sum of an infinite, but finitely supported cartesian product of subgroups (which mutually commute and have trivial intersections). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdval.0  |-  .0.  =  ( 0g `  G )
dprdval.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
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 5826 . . . . 5  |-  ( A  e.  ( DProd  `  <. G ,  S >. )  -> 
<. G ,  S >.  e. 
dom DProd  )
2 df-ov 6204 . . . . 5  |-  ( G DProd 
S )  =  ( DProd  `  <. G ,  S >. )
31, 2eleq2s 2562 . . . 4  |-  ( A  e.  ( G DProd  S
)  ->  <. G ,  S >.  e.  dom DProd  )
4 df-br 4402 . . . 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 633 . 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 finSupp  .0.  }
97, 8dprdval 16608 . . . . . 6  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
109eleq2d 2524 . . . . 5  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <-> 
A  e.  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
11 eqid 2454 . . . . . 6  |-  ( f  e.  W  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) )
12 ovex 6226 . . . . . 6  |-  ( G 
gsumg  f )  e.  _V
1311, 12elrnmpti 5199 . . . . 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 453 . . 3  |-  ( ( dom  S  =  I  /\  G dom DProd  S )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1615pm5.32da 641 . 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 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803   <.cop 3992   class class class wbr 4401    |-> cmpt 4459   dom cdm 4949   ran crn 4950   ` cfv 5527  (class class class)co 6201   X_cixp 7374   finSupp cfsupp 7732   0gc0g 14498    gsumg cgsu 14499   DProd cdprd 16598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-ixp 7375  df-dprd 16600
This theorem is referenced by:  dprdssv  16629  eldprdi  16631  dprdsubg  16644  dprdss  16649  dmdprdsplitlem  16657  dprddisj2  16660  dpjidcl  16680
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