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Theorem drngoi 23894
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 4059 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  <. g ,  h >.  =  <. ( 1st `  R ) ,  h >. )
21eleq1d 2509 . . . . 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 2493 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  g  =  G )
65rneqd 5067 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  ran  G )
7 drngi.3 . . . . . . . . . 10  |-  X  =  ran  G
86, 7syl6eqr 2493 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  X )
95fveq2d 5695 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  (GId `  G ) )
10 drngi.4 . . . . . . . . . . 11  |-  Z  =  (GId `  G )
119, 10syl6eqr 2493 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  Z )
1211sneqd 3889 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  { (GId `  g ) }  =  { Z } )
138, 12difeq12d 3475 . . . . . . . 8  |-  ( g  =  ( 1st `  R
)  ->  ( ran  g  \  { (GId `  g ) } )  =  ( X  \  { Z } ) )
14 xpeq2 4855 . . . . . . . . 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 4854 . . . . . . . . 9  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( X  \  { Z }
) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1614, 15eqtrd 2475 . . . . . . . 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 5110 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
1918eleq1d 2509 . . . . 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 4060 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  <. ( 1st `  R ) ,  h >.  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
2221eleq1d 2509 . . . . . 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 2494 . . . . . . . 8  |-  ( h  =  ( 2nd `  R
)  ->  H  =  h )
2726reseq1d 5109 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
2827eleq1d 2509 . . . . . 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 6635 . . 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 23893 . . 3  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
3331, 32eleq2s 2535 . 2  |-  ( R  e.  DivRingOps  ->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) )
3432relopabi 4965 . . . . 5  |-  Rel  DivRingOps
35 1st2nd 6620 . . . . 5  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
3634, 35mpan 670 . . . 4  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
3736eleq1d 2509 . . 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 1369    e. wcel 1756    \ cdif 3325   {csn 3877   <.cop 3883   {copab 4349    X. cxp 4838   ran crn 4841    |` cres 4842   Rel wrel 4845   ` cfv 5418   1stc1st 6575   2ndc2nd 6576   GrpOpcgr 23673  GIdcgi 23674   RingOpscrngo 23862   DivRingOpscdrng 23892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fv 5426  df-1st 6577  df-2nd 6578  df-drngo 23893
This theorem is referenced by:  dvrunz  23920  fldcrng  28804
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