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Theorem drngunit 15517
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 15516 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  U  =  ( B 
\  {  .0.  }
) ) )
54simprbi 450 . . 3  |-  ( R  e.  DivRing  ->  U  =  ( B  \  {  .0.  } ) )
65eleq2d 2350 . 2  |-  ( R  e.  DivRing  ->  ( X  e.  U  <->  X  e.  ( B  \  {  .0.  }
) ) )
7 eldifsn 3749 . 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   Ringcrg 15337  Unitcui 15421   DivRingcdr 15512
This theorem is referenced by:  drngunz  15527  drnginvrcl  15529  drnginvrn0  15530  drnginvrl  15531  drnginvrr  15532  issubdrg  15570  abvdiv  15602  qsssubdrg  16431  drnguc1p  19556  lgseisenlem3  20590
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-ne 2448  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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