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Theorem dprdgrp 15491
Description: Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdgrp  |-  ( G dom DProd  S  ->  G  e. 
Grp )

Proof of Theorem dprdgrp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldmdprd 15486 . . . . . 6  |-  Rel  dom DProd
21brrelex2i 4860 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
3 dmexg 5071 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
42, 3syl 16 . . . 4  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
5 eqid 2388 . . . 4  |-  dom  S  =  dom  S
6 eqid 2388 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
7 eqid 2388 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
8 eqid 2388 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
96, 7, 8dmdprd 15487 . . . 4  |-  ( ( dom  S  e.  _V  /\ 
dom  S  =  dom  S )  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
104, 5, 9sylancl 644 . . 3  |-  ( G dom DProd  S  ->  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  { x } ) ( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) ) )
1110ibi 233 . 2  |-  ( G dom DProd  S  ->  ( G  e.  Grp  /\  S : dom  S --> (SubGrp `  G )  /\  A. x  e.  dom  S ( A. y  e.  ( dom  S  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " ( dom  S  \  { x } ) ) ) )  =  { ( 0g `  G ) } ) ) )
1211simp1d 969 1  |-  ( G dom DProd  S  ->  G  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900    \ cdif 3261    i^i cin 3263    C_ wss 3264   {csn 3758   U.cuni 3958   class class class wbr 4154   dom cdm 4819   "cima 4822   -->wf 5391   ` cfv 5395   0gc0g 13651  mrClscmrc 13736   Grpcgrp 14613  SubGrpcsubg 14866  Cntzccntz 15042   DProd cdprd 15482
This theorem is referenced by:  dprdssv  15502  dprdfid  15503  dprdfinv  15505  dprdfadd  15506  dprdfsub  15507  dprdfeq0  15508  dprdf11  15509  dprdsubg  15510  dprdlub  15512  dprdspan  15513  dprdres  15514  dprdss  15515  dprdf1o  15518  dmdprdsplitlem  15523  dprdcntz2  15524  dprddisj2  15525  dprd2dlem1  15527  dprd2da  15528  dmdprdsplit2lem  15531  dmdprdsplit2  15532  dpjfval  15541  dpjidcl  15544
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-ixp 7001  df-dprd 15484
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