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Theorem isdrngo1 31641
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 25822 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
21relopabi 4948 . . 3  |-  Rel  DivRingOps
3 1st2nd 6830 . . 3  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
42, 3mpan 668 . 2  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 relrngo 25793 . . . 4  |-  Rel  RingOps
6 1st2nd 6830 . . . 4  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
75, 6mpan 668 . . 3  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
87adantr 463 . 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 4164 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
1211eqeq2i 2420 . . 3  |-  ( R  =  <. G ,  H >.  <-> 
R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
13 fvex 5859 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
1410, 13eqeltri 2486 . . . . . 6  |-  H  e. 
_V
15 isdivrngo 25847 . . . . . 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 5 . . . . 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 3983 . . . . . . . . . 10  |-  { Z }  =  { (GId `  G ) }
2017, 19difeq12i 3559 . . . . . . . . 9  |-  ( X 
\  { Z }
)  =  ( ran 
G  \  { (GId `  G ) } )
2120, 20xpeq12i 4845 . . . . . . . 8  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) )
2221reseq2i 5091 . . . . . . 7  |-  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )
2322eleq1i 2479 . . . . . 6  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( H  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp )
2423anbi2i 692 . . . . 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 252 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
26 eleq1 2474 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  <. G ,  H >.  e.  DivRingOps
) )
27 eleq1 2474 . . . . . 6  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
2827anbi1d 703 . . . . 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 319 . . . 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 233 . . 3  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3112, 30sylbir 213 . 2  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
324, 8, 31pm5.21nii 351 1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    \ cdif 3411   {csn 3972   <.cop 3978    X. cxp 4821   ran crn 4824    |` cres 4825   Rel wrel 4828   ` cfv 5569   1stc1st 6782   2ndc2nd 6783   GrpOpcgr 25602  GIdcgi 25603   RingOpscrngo 25791   DivRingOpscdrng 25821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577  df-ov 6281  df-1st 6784  df-2nd 6785  df-rngo 25792  df-drngo 25822
This theorem is referenced by:  divrngcl  31642  isdrngo2  31643  divrngpr  31732
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