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Theorem drngunit 15840
Description: Elementhood in the set of units when  R is a division ring. (Contributed 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
drngunit  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  ( X  e.  B  /\  X  =/= 
.0.  ) ) )

Proof of Theorem drngunit
StepHypRef Expression
1 isdrng.b . . . . 5  |-  B  =  ( Base `  R
)
2 isdrng.u . . . . 5  |-  U  =  (Unit `  R )
3 isdrng.z . . . . 5  |-  .0.  =  ( 0g `  R )
41, 2, 3isdrng 15839 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
54simprbi 451 . . 3  |-  ( R  e.  DivRing  ->  U  =  ( B  \  {  .0.  } ) )
65eleq2d 2503 . 2  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  X  e.  ( B  \  {  .0.  }
) ) )
7 eldifsn 3927 . 2  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
86, 7syl6bb 253 1  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  ( X  e.  B  /\  X  =/= 
.0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814   ` cfv 5454   Basecbs 13469   0gc0g 13723   Ringcrg 15660  Unitcui 15744   DivRingcdr 15835
This theorem is referenced by:  drngunz  15850  drnginvrcl  15852  drnginvrn0  15853  drnginvrl  15854  drnginvrr  15855  issubdrg  15893  abvdiv  15925  qsssubdrg  16758  drnguc1p  20093  lgseisenlem3  21135  redvr  24277  qqhval2lem  24365  qqhf  24370
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-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-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-iota 5418  df-fv 5462  df-drng 15837
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