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Theorem isdrngo1 26587
Description: The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1  |-  G  =  ( 1st `  R
)
isdivrng1.2  |-  H  =  ( 2nd `  R
)
isdivrng1.3  |-  Z  =  (GId `  G )
isdivrng1.4  |-  X  =  ran  G
Assertion
Ref Expression
isdrngo1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )

Proof of Theorem isdrngo1
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-drngo 21073 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
21relopabi 4811 . . 3  |-  Rel  DivRingOps
3 1st2nd 6166 . . 3  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
42, 3mpan 651 . 2  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 relrngo 21044 . . . 4  |-  Rel  RingOps
6 1st2nd 6166 . . . 4  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
75, 6mpan 651 . . 3  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
87adantr 451 . 2  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  R  = 
<. ( 1st `  R
) ,  ( 2nd `  R ) >. )
9 isdivrng1.1 . . . . 5  |-  G  =  ( 1st `  R
)
10 isdivrng1.2 . . . . 5  |-  H  =  ( 2nd `  R
)
119, 10opeq12i 3801 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
1211eqeq2i 2293 . . 3  |-  ( R  =  <. G ,  H >.  <-> 
R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
13 fvex 5539 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
1410, 13eqeltri 2353 . . . . . 6  |-  H  e. 
_V
15 isdivrngo 21098 . . . . . 6  |-  ( H  e.  _V  ->  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) ) )
1614, 15ax-mp 8 . . . . 5  |-  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )  e.  GrpOp ) )
17 isdivrng1.4 . . . . . . . . . 10  |-  X  =  ran  G
18 isdivrng1.3 . . . . . . . . . . 11  |-  Z  =  (GId `  G )
1918sneqi 3652 . . . . . . . . . 10  |-  { Z }  =  { (GId `  G ) }
2017, 19difeq12i 3292 . . . . . . . . 9  |-  ( X 
\  { Z }
)  =  ( ran 
G  \  { (GId `  G ) } )
2120, 20xpeq12i 4711 . . . . . . . 8  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) )
2221reseq2i 4952 . . . . . . 7  |-  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )
2322eleq1i 2346 . . . . . 6  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( H  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp )
2423anbi2i 675 . . . . 5  |-  ( (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp )  <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp ) )
2516, 24bitr4i 243 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
26 eleq1 2343 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  <. G ,  H >.  e.  DivRingOps
) )
27 eleq1 2343 . . . . . 6  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
2827anbi1d 685 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp )  <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
2926, 28bibi12d 312 . . . 4  |-  ( R  =  <. G ,  H >.  ->  ( ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )  <->  ( <. G ,  H >.  e.  DivRingOps  <->  ( <. G ,  H >.  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) ) )
3025, 29mpbiri 224 . . 3  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3112, 30sylbir 204 . 2  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
324, 8, 31pm5.21nii 342 1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   {csn 3640   <.cop 3643    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:  divrngcl  26588  isdrngo2  26589  divrngpr  26678
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-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-rngo 21043  df-drngo 21073
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