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Theorem isdrngo1 32153
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 26126 . . . 4  |-  DivRingOps  =  { <. g ,  h >.  |  ( <. g ,  h >.  e.  RingOps  /\  ( h  |`  ( ( ran  g  \  { (GId `  g
) } )  X.  ( ran  g  \  { (GId `  g ) } ) ) )  e.  GrpOp ) }
21relopabi 4976 . . 3  |-  Rel  DivRingOps
3 1st2nd 6851 . . 3  |-  ( ( Rel  DivRingOps  /\  R  e.  DivRingOps )  ->  R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >. )
42, 3mpan 675 . 2  |-  ( R  e.  DivRingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
5 relrngo 26097 . . . 4  |-  Rel  RingOps
6 1st2nd 6851 . . . 4  |-  ( ( Rel  RingOps  /\  R  e.  RingOps )  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
75, 6mpan 675 . . 3  |-  ( R  e.  RingOps  ->  R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
87adantr 467 . 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 4190 . . . 4  |-  <. G ,  H >.  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >.
1211eqeq2i 2441 . . 3  |-  ( R  =  <. G ,  H >.  <-> 
R  =  <. ( 1st `  R ) ,  ( 2nd `  R
) >. )
13 fvex 5889 . . . . . . 7  |-  ( 2nd `  R )  e.  _V
1410, 13eqeltri 2507 . . . . . 6  |-  H  e. 
_V
15 isdivrngo 26151 . . . . . 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 4008 . . . . . . . . . 10  |-  { Z }  =  { (GId `  G ) }
2017, 19difeq12i 3582 . . . . . . . . 9  |-  ( X 
\  { Z }
)  =  ( ran 
G  \  { (GId `  G ) } )
2120, 20xpeq12i 4873 . . . . . . . 8  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( ( ran  G  \  { (GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) )
2221reseq2i 5119 . . . . . . 7  |-  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( H  |`  ( ( ran  G  \  { (GId `  G
) } )  X.  ( ran  G  \  { (GId `  G ) } ) ) )
2322eleq1i 2500 . . . . . 6  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp 
<->  ( H  |`  (
( ran  G  \  {
(GId `  G ) } )  X.  ( ran  G  \  { (GId
`  G ) } ) ) )  e. 
GrpOp )
2423anbi2i 699 . . . . 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 256 . . . 4  |-  ( <. G ,  H >.  e.  DivRingOps  <->  (
<. G ,  H >.  e.  RingOps 
/\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  e.  GrpOp ) )
26 eleq1 2495 . . . . 5  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  <. G ,  H >.  e.  DivRingOps
) )
27 eleq1 2495 . . . . . 6  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  RingOps  <->  <. G ,  H >.  e.  RingOps ) )
2827anbi1d 710 . . . . 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 323 . . . 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 237 . . 3  |-  ( R  =  <. G ,  H >.  ->  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp ) ) )
3112, 30sylbir 217 . 2  |-  ( R  =  <. ( 1st `  R
) ,  ( 2nd `  R ) >.  ->  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) ) )
324, 8, 31pm5.21nii 355 1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   _Vcvv 3082    \ cdif 3434   {csn 3997   <.cop 4003    X. cxp 4849   ran crn 4852    |` cres 4853   Rel wrel 4856   ` cfv 5599   1stc1st 6803   2ndc2nd 6804   GrpOpcgr 25906  GIdcgi 25907   RingOpscrngo 26095   DivRingOpscdrng 26125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607  df-ov 6306  df-1st 6805  df-2nd 6806  df-rngo 26096  df-drngo 26126
This theorem is referenced by:  divrngcl  32154  isdrngo2  32155  divrngpr  32244
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