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Theorem drngunit 15533
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 15532 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
54simprbi 450 . . 3  |-  ( R  e.  DivRing  ->  U  =  ( B  \  {  .0.  } ) )
65eleq2d 2363 . 2  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  X  e.  ( B  \  {  .0.  }
) ) )
7 eldifsn 3762 . 2  |-  ( X  e.  ( B  \  {  .0.  } )  <->  ( X  e.  B  /\  X  =/= 
.0.  ) )
86, 7syl6bb 252 1  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  ( X  e.  B  /\  X  =/= 
.0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   Ringcrg 15353  Unitcui 15437   DivRingcdr 15528
This theorem is referenced by:  drngunz  15543  drnginvrcl  15545  drnginvrn0  15546  drnginvrl  15547  drnginvrr  15548  issubdrg  15586  abvdiv  15618  qsssubdrg  16447  drnguc1p  19572  lgseisenlem3  20606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-drng 15530
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