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Theorem isdrngo1 26572
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 21994 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
21relopabi 5000 . . 3  |-  Rel  DivRingOps
3 1st2nd 6393 . . 3  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
42, 3mpan 652 . 2  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 relrngo 21965 . . . 4  |-  Rel  RingOps
6 1st2nd 6393 . . . 4  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
75, 6mpan 652 . . 3  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
87adantr 452 . 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 3989 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
1211eqeq2i 2446 . . 3  |-  ( R  =  <. G ,  H >.  <-> 
R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
13 fvex 5742 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
1410, 13eqeltri 2506 . . . . . 6  |-  H  e. 
_V
15 isdivrngo 22019 . . . . . 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 3826 . . . . . . . . . 10  |-  { Z }  =  { (GId `  G ) }
2017, 19difeq12i 3463 . . . . . . . . 9  |-  ( X 
\  { Z }
)  =  ( ran 
G  \  { (GId `  G ) } )
2120, 20xpeq12i 4900 . . . . . . . 8  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) )
2221reseq2i 5143 . . . . . . 7  |-  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )
2322eleq1i 2499 . . . . . 6  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( H  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp )
2423anbi2i 676 . . . . 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 244 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
26 eleq1 2496 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  <. G ,  H >.  e.  DivRingOps
) )
27 eleq1 2496 . . . . . 6  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
2827anbi1d 686 . . . . 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 313 . . . 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 225 . . 3  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3112, 30sylbir 205 . 2  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
324, 8, 31pm5.21nii 343 1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317   {csn 3814   <.cop 3817    X. cxp 4876   ran crn 4879    |` cres 4880   Rel wrel 4883   ` cfv 5454   1stc1st 6347   2ndc2nd 6348   GrpOpcgr 21774  GIdcgi 21775   RingOpscrngo 21963   DivRingOpscdrng 21993
This theorem is referenced by:  divrngcl  26573  isdrngo2  26574  divrngpr  26663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-rngo 21964  df-drngo 21994
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