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Theorem isdrng 15516
Description: The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
isdrng.b  |-  B  =  ( Base `  R
)
isdrng.u  |-  U  =  (Unit `  R )
isdrng.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
isdrng  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )

Proof of Theorem isdrng
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
2 isdrng.u . . . 4  |-  U  =  (Unit `  R )
31, 2syl6eqr 2333 . . 3  |-  ( r  =  R  ->  (Unit `  r )  =  U )
4 fveq2 5525 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
5 isdrng.b . . . . 5  |-  B  =  ( Base `  R
)
64, 5syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  B )
7 fveq2 5525 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
8 isdrng.z . . . . . 6  |-  .0.  =  ( 0g `  R )
97, 8syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
109sneqd 3653 . . . 4  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
116, 10difeq12d 3295 . . 3  |-  ( r  =  R  ->  (
( Base `  r )  \  { ( 0g `  r ) } )  =  ( B  \  {  .0.  } ) )
123, 11eqeq12d 2297 . 2  |-  ( r  =  R  ->  (
(Unit `  r )  =  ( ( Base `  r )  \  {
( 0g `  r
) } )  <->  U  =  ( B  \  {  .0.  } ) ) )
13 df-drng 15514 . 2  |-  DivRing  =  {
r  e.  Ring  |  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } ) }
1412, 13elrab2 2925 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   Ringcrg 15337  Unitcui 15421   DivRingcdr 15512
This theorem is referenced by:  drngunit  15517  drngui  15518  drngrng  15519  isdrng2  15522  drngprop  15523  drngid  15526  opprdrng  15536  drngpropd  15539  issubdrg  15570  drngdomn  16044  fidomndrng  16048  istdrg2  17860  cphreccllem  18614  cntzsdrg  27510
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-iota 5219  df-fv 5263  df-drng 15514
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