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Theorem drngunit 16959
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 16958 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
54simprbi 464 . . 3  |-  ( R  e.  DivRing  ->  U  =  ( B  \  {  .0.  } ) )
65eleq2d 2524 . 2  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  X  e.  ( B  \  {  .0.  }
) ) )
7 eldifsn 4107 . 2  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
86, 7syl6bb 261 1  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  ( X  e.  B  /\  X  =/= 
.0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647    \ cdif 3432   {csn 3984   ` cfv 5525   Basecbs 14291   0gc0g 14496   Ringcrg 16767  Unitcui 16853   DivRingcdr 16954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-iota 5488  df-fv 5533  df-drng 16956
This theorem is referenced by:  drngunz  16969  drnginvrcl  16971  drnginvrn0  16972  drnginvrl  16973  drnginvrr  16974  issubdrg  17012  abvdiv  17044  qsssubdrg  17996  redvr  18171  drnguc1p  21774  lgseisenlem3  22822  ornglmullt  26419  orngrmullt  26420  isarchiofld  26429  qqhval2lem  26554  qqhf  26559  lincreslvec3  31134  isldepslvec2  31137
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