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Theorem eldprd 15255
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 5570 . . . . 5  |-  ( A  e.  ( DProd  `  <. G ,  S >. )  -> 
<. G ,  S >.  e. 
dom DProd  )
2 df-ov 5877 . . . . 5  |-  ( G DProd 
S )  =  ( DProd  `  <. G ,  S >. )
31, 2eleq2s 2388 . . . 4  |-  ( A  e.  ( G DProd  S
)  ->  <. G ,  S >.  e.  dom DProd  )
4 df-br 4040 . . . 4  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
53, 4sylibr 203 . . 3  |-  ( A  e.  ( G DProd  S
)  ->  G dom DProd  S )
65pm4.71ri 614 . 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 15254 . . . . . 6  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
109eleq2d 2363 . . . . 5  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <-> 
A  e.  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
11 eqid 2296 . . . . . 6  |-  ( f  e.  W  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) )
12 ovex 5899 . . . . . 6  |-  ( G 
gsumg  f )  e.  _V
1311, 12elrnmpti 4946 . . . . 5  |-  ( A  e.  ran  ( f  e.  W  |->  ( G 
gsumg  f ) )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) )
1410, 13syl6bb 252 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1514ancoms 439 . . 3  |-  ( ( dom  S  =  I  /\  G dom DProd  S )  ->  ( A  e.  ( G DProd  S )  <->  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
1615pm5.32da 622 . 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 248 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Fincfn 6879   0gc0g 13416    gsumg cgsu 13417   DProd cdprd 15247
This theorem is referenced by:  dprdssv  15267  eldprdi  15269  dprdsubg  15275  dprdss  15280  dmdprdsplitlem  15288  dprddisj2  15290  dpjidcl  15309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-dprd 15249
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