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Theorem isdrngo1 26690
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 21089 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
21relopabi 4827 . . 3  |-  Rel  DivRingOps
3 1st2nd 6182 . . 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 21060 . . . 4  |-  Rel  RingOps
6 1st2nd 6182 . . . 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 3817 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
1211eqeq2i 2306 . . 3  |-  ( R  =  <. G ,  H >.  <-> 
R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
13 fvex 5555 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
1410, 13eqeltri 2366 . . . . . 6  |-  H  e. 
_V
15 isdivrngo 21114 . . . . . 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 3665 . . . . . . . . . 10  |-  { Z }  =  { (GId `  G ) }
2017, 19difeq12i 3305 . . . . . . . . 9  |-  ( X 
\  { Z }
)  =  ( ran 
G  \  { (GId `  G ) } )
2120, 20xpeq12i 4727 . . . . . . . 8  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) )
2221reseq2i 4968 . . . . . . 7  |-  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )
2322eleq1i 2359 . . . . . 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 2356 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  <. G ,  H >.  e.  DivRingOps
) )
27 eleq1 2356 . . . . . 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   {csn 3653   <.cop 3656    X. cxp 4703   ran crn 4706    |` cres 4707   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   RingOpscrngo 21058   DivRingOpscdrng 21088
This theorem is referenced by:  divrngcl  26691  isdrngo2  26692  divrngpr  26781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-rngo 21059  df-drngo 21089
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