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Theorem drngoi 25232
Description: The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
drngi.1  |-  G  =  ( 1st `  R
)
drngi.2  |-  H  =  ( 2nd `  R
)
drngi.3  |-  X  =  ran  G
drngi.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
drngoi  |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )

Proof of Theorem drngoi
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4219 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  <. g ,  h >.  =  <. ( 1st `  R ) ,  h >. )
21eleq1d 2536 . . . . 5  |-  ( g  =  ( 1st `  R
)  ->  ( <. g ,  h >.  e.  RingOps  <->  <. ( 1st `  R ) ,  h >.  e.  RingOps ) )
3 id 22 . . . . . . . . . . . 12  |-  ( g  =  ( 1st `  R
)  ->  g  =  ( 1st `  R ) )
4 drngi.1 . . . . . . . . . . . 12  |-  G  =  ( 1st `  R
)
53, 4syl6eqr 2526 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  g  =  G )
65rneqd 5236 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  ran  G )
7 drngi.3 . . . . . . . . . 10  |-  X  =  ran  G
86, 7syl6eqr 2526 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  X )
95fveq2d 5876 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  (GId `  G ) )
10 drngi.4 . . . . . . . . . . 11  |-  Z  =  (GId `  G )
119, 10syl6eqr 2526 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  Z )
1211sneqd 4045 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  { (GId `  g ) }  =  { Z } )
138, 12difeq12d 3628 . . . . . . . 8  |-  ( g  =  ( 1st `  R
)  ->  ( ran  g  \  { (GId `  g ) } )  =  ( X  \  { Z } ) )
14 xpeq2 5020 . . . . . . . . 9  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( ran  g  \  { (GId `  g ) } )  X.  ( X  \  { Z }
) ) )
15 xpeq1 5019 . . . . . . . . 9  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( X  \  { Z }
) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1614, 15eqtrd 2508 . . . . . . . 8  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1713, 16syl 16 . . . . . . 7  |-  ( g  =  ( 1st `  R
)  ->  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1817reseq2d 5279 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
1918eleq1d 2536 . . . . 5  |-  ( g  =  ( 1st `  R
)  ->  ( (
h  |`  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) ) )  e. 
GrpOp 
<->  ( h  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
202, 19anbi12d 710 . . . 4  |-  ( g  =  ( 1st `  R
)  ->  ( ( <. g ,  h >.  e.  RingOps 
/\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp )  <->  ( <. ( 1st `  R ) ,  h >.  e.  RingOps  /\  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
21 opeq2 4220 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  <. ( 1st `  R ) ,  h >.  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
2221eleq1d 2536 . . . . . 6  |-  ( h  =  ( 2nd `  R
)  ->  ( <. ( 1st `  R ) ,  h >.  e.  RingOps  <->  <. ( 1st `  R ) ,  ( 2nd `  R )
>.  e.  RingOps ) )
2322anbi1d 704 . . . . 5  |-  ( h  =  ( 2nd `  R
)  ->  ( ( <. ( 1st `  R
) ,  h >.  e.  RingOps 
/\  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp )  <->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  (
h  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
24 id 22 . . . . . . . . 9  |-  ( h  =  ( 2nd `  R
)  ->  h  =  ( 2nd `  R ) )
25 drngi.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
2624, 25syl6reqr 2527 . . . . . . . 8  |-  ( h  =  ( 2nd `  R
)  ->  H  =  h )
2726reseq1d 5278 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
2827eleq1d 2536 . . . . . 6  |-  ( h  =  ( 2nd `  R
)  ->  ( ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( h  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
2928anbi2d 703 . . . . 5  |-  ( h  =  ( 2nd `  R
)  ->  ( ( <. ( 1st `  R
) ,  ( 2nd `  R ) >.  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp )  <->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  (
h  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3023, 29bitr4d 256 . . . 4  |-  ( h  =  ( 2nd `  R
)  ->  ( ( <. ( 1st `  R
) ,  h >.  e.  RingOps 
/\  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp )  <->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3120, 30elopabi 6856 . . 3  |-  ( R  e.  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps 
/\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }  ->  (
<. ( 1st `  R
) ,  ( 2nd `  R ) >.  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
32 df-drngo 25231 . . 3  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
3331, 32eleq2s 2575 . 2  |-  ( R  e.  DivRingOps  ->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) )
3432relopabi 5134 . . . . 5  |-  Rel  DivRingOps
35 1st2nd 6841 . . . . 5  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
3634, 35mpan 670 . . . 4  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
3736eleq1d 2536 . . 3  |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps  <->  <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  RingOps ) )
3837anbi1d 704 . 2  |-  ( R  e.  DivRingOps  ->  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp )  <->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3933, 38mpbird 232 1  |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3478   {csn 4033   <.cop 4039   {copab 4510    X. cxp 5003   ran crn 5006    |` cres 5007   Rel wrel 5010   ` cfv 5594   1stc1st 6793   2ndc2nd 6794   GrpOpcgr 25011  GIdcgi 25012   RingOpscrngo 25200   DivRingOpscdrng 25230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6795  df-2nd 6796  df-drngo 25231
This theorem is referenced by:  dvrunz  25258  fldcrng  30328
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