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Theorem isdrng 15841
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 5730 . . . 4  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
2 isdrng.u . . . 4  |-  U  =  (Unit `  R )
31, 2syl6eqr 2488 . . 3  |-  ( r  =  R  ->  (Unit `  r )  =  U )
4 fveq2 5730 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
5 isdrng.b . . . . 5  |-  B  =  ( Base `  R
)
64, 5syl6eqr 2488 . . . 4  |-  ( r  =  R  ->  ( Base `  r )  =  B )
7 fveq2 5730 . . . . . 6  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
8 isdrng.z . . . . . 6  |-  .0.  =  ( 0g `  R )
97, 8syl6eqr 2488 . . . . 5  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
109sneqd 3829 . . . 4  |-  ( r  =  R  ->  { ( 0g `  r ) }  =  {  .0.  } )
116, 10difeq12d 3468 . . 3  |-  ( r  =  R  ->  (
( Base `  r )  \  { ( 0g `  r ) } )  =  ( B  \  {  .0.  } ) )
123, 11eqeq12d 2452 . 2  |-  ( r  =  R  ->  (
(Unit `  r )  =  ( ( Base `  r )  \  {
( 0g `  r
) } )  <->  U  =  ( B  \  {  .0.  } ) ) )
13 df-drng 15839 . 2  |-  DivRing  =  {
r  e.  Ring  |  (Unit `  r )  =  ( ( Base `  r
)  \  { ( 0g `  r ) } ) }
1412, 13elrab2 3096 1  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319   {csn 3816   ` cfv 5456   Basecbs 13471   0gc0g 13725   Ringcrg 15662  Unitcui 15746   DivRingcdr 15837
This theorem is referenced by:  drngunit  15842  drngui  15843  drngrng  15844  isdrng2  15847  drngprop  15848  drngid  15851  opprdrng  15861  drngpropd  15864  issubdrg  15895  drngdomn  16365  fidomndrng  16369  istdrg2  18209  cphreccllem  19143  zrhunitpreima  24364  cntzsdrg  27489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-drng 15839
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