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Theorem drngoi 21074
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 3796 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  <. g ,  h >.  =  <. ( 1st `  R ) ,  h >. )
21eleq1d 2349 . . . . 5  |-  ( g  =  ( 1st `  R
)  ->  ( <. g ,  h >.  e.  RingOps  <->  <. ( 1st `  R ) ,  h >.  e.  RingOps ) )
3 id 19 . . . . . . . . . . . 12  |-  ( g  =  ( 1st `  R
)  ->  g  =  ( 1st `  R ) )
4 drngi.1 . . . . . . . . . . . 12  |-  G  =  ( 1st `  R
)
53, 4syl6eqr 2333 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  g  =  G )
65rneqd 4906 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  ran  G )
7 drngi.3 . . . . . . . . . 10  |-  X  =  ran  G
86, 7syl6eqr 2333 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  ran  g  =  X )
95fveq2d 5529 . . . . . . . . . . 11  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  (GId `  G ) )
10 drngi.4 . . . . . . . . . . 11  |-  Z  =  (GId `  G )
119, 10syl6eqr 2333 . . . . . . . . . 10  |-  ( g  =  ( 1st `  R
)  ->  (GId `  g
)  =  Z )
1211sneqd 3653 . . . . . . . . 9  |-  ( g  =  ( 1st `  R
)  ->  { (GId `  g ) }  =  { Z } )
138, 12difeq12d 3295 . . . . . . . 8  |-  ( g  =  ( 1st `  R
)  ->  ( ran  g  \  { (GId `  g ) } )  =  ( X  \  { Z } ) )
14 xpeq2 4704 . . . . . . . . 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 4703 . . . . . . . . 9  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( X  \  { Z }
) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1614, 15eqtrd 2315 . . . . . . . 8  |-  ( ( ran  g  \  {
(GId `  g ) } )  =  ( X  \  { Z } )  ->  (
( ran  g  \  { (GId `  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1713, 16syl 15 . . . . . . 7  |-  ( g  =  ( 1st `  R
)  ->  ( ( ran  g  \  { (GId
`  g ) } )  X.  ( ran  g  \  { (GId
`  g ) } ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )
1817reseq2d 4955 . . . . . 6  |-  ( g  =  ( 1st `  R
)  ->  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
1918eleq1d 2349 . . . . 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 691 . . . 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 3797 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  <. ( 1st `  R ) ,  h >.  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
2221eleq1d 2349 . . . . . 6  |-  ( h  =  ( 2nd `  R
)  ->  ( <. ( 1st `  R ) ,  h >.  e.  RingOps  <->  <. ( 1st `  R ) ,  ( 2nd `  R )
>.  e.  RingOps ) )
2322anbi1d 685 . . . . 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 19 . . . . . . . . 9  |-  ( h  =  ( 2nd `  R
)  ->  h  =  ( 2nd `  R ) )
25 drngi.2 . . . . . . . . 9  |-  H  =  ( 2nd `  R
)
2624, 25syl6reqr 2334 . . . . . . . 8  |-  ( h  =  ( 2nd `  R
)  ->  H  =  h )
2726reseq1d 4954 . . . . . . 7  |-  ( h  =  ( 2nd `  R
)  ->  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( h  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
2827eleq1d 2349 . . . . . 6  |-  ( h  =  ( 2nd `  R
)  ->  ( ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( h  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
2928anbi2d 684 . . . . 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 247 . . . 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 6185 . . 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 21073 . . 3  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
3331, 32eleq2s 2375 . 2  |-  ( R  e.  DivRingOps  ->  ( <. ( 1st `  R ) ,  ( 2nd `  R
) >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) )
3432relopabi 4811 . . . . 5  |-  Rel  DivRingOps
35 1st2nd 6166 . . . . 5  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
3634, 35mpan 651 . . . 4  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
3736eleq1d 2349 . . 3  |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps  <->  <.
( 1st `  R
) ,  ( 2nd `  R ) >.  e.  RingOps ) )
3837anbi1d 685 . 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 223 1  |-  ( R  e.  DivRingOps  ->  ( R  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640   <.cop 3643   {copab 4076    X. cxp 4687   ran crn 4690    |` cres 4691   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072
This theorem is referenced by:  dvrunz  21100  fldi  25427  fldcrng  26629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123  df-drngo 21073
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