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Theorem eldprd 17161
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 5898 . . . . 5  |-  ( A  e.  ( DProd  `  <. G ,  S >. )  -> 
<. G ,  S >.  e. 
dom DProd  )
2 df-ov 6299 . . . . 5  |-  ( G DProd 
S )  =  ( DProd  `  <. G ,  S >. )
31, 2eleq2s 2565 . . . 4  |-  ( A  e.  ( G DProd  S
)  ->  <. G ,  S >.  e.  dom DProd  )
4 df-br 4457 . . . 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 17160 . . . . . 6  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
109eleq2d 2527 . . . . 5  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <-> 
A  e.  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
11 eqid 2457 . . . . . 6  |-  ( f  e.  W  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) )
12 ovex 6324 . . . . . 6  |-  ( G 
gsumg  f )  e.  _V
1311, 12elrnmpti 5263 . . . . 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 1395    e. wcel 1819   E.wrex 2808   {crab 2811   <.cop 4038   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   ran crn 5009   ` cfv 5594  (class class class)co 6296   X_cixp 7488   finSupp cfsupp 7847   0gc0g 14856    gsumg cgsu 14857   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-ixp 7489  df-dprd 17152
This theorem is referenced by:  dprdssv  17182  eldprdi  17184  dprdsubg  17197  dprdss  17202  dmdprdsplitlem  17210  dprddisj2  17213  dpjidcl  17233
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