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Theorem dprdgrp 15555
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 15550 . . . . . 6  |-  Rel  dom DProd
21brrelex2i 4911 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
3 dmexg 5122 . . . . 5  |-  ( S  e.  _V  ->  dom  S  e.  _V )
42, 3syl 16 . . . 4  |-  ( G dom DProd  S  ->  dom  S  e.  _V )
5 eqid 2435 . . . 4  |-  dom  S  =  dom  S
6 eqid 2435 . . . . 5  |-  (Cntz `  G )  =  (Cntz `  G )
7 eqid 2435 . . . . 5  |-  ( 0g
`  G )  =  ( 0g `  G
)
8 eqid 2435 . . . . 5  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
96, 7, 8dmdprd 15551 . . . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   U.cuni 4007   class class class wbr 4204   dom cdm 4870   "cima 4873   -->wf 5442   ` cfv 5446   0gc0g 13715  mrClscmrc 13800   Grpcgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546
This theorem is referenced by:  dprdssv  15566  dprdfid  15567  dprdfinv  15569  dprdfadd  15570  dprdfsub  15571  dprdfeq0  15572  dprdf11  15573  dprdsubg  15574  dprdlub  15576  dprdspan  15577  dprdres  15578  dprdss  15579  dprdf1o  15582  dmdprdsplitlem  15587  dprdcntz2  15588  dprddisj2  15589  dprd2dlem1  15591  dprd2da  15592  dmdprdsplit2lem  15595  dmdprdsplit2  15596  dpjfval  15605  dpjidcl  15608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ixp 7056  df-dprd 15548
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