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Theorem drngui 15518
Description: The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
drngui.b  |-  B  =  ( Base `  R
)
drngui.z  |-  .0.  =  ( 0g `  R )
drngui.r  |-  R  e.  DivRing
Assertion
Ref Expression
drngui  |-  ( B 
\  {  .0.  }
)  =  (Unit `  R )

Proof of Theorem drngui
StepHypRef Expression
1 drngui.r . . . 4  |-  R  e.  DivRing
2 drngui.b . . . . 5  |-  B  =  ( Base `  R
)
3 eqid 2283 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
4 drngui.z . . . . 5  |-  .0.  =  ( 0g `  R )
52, 3, 4isdrng 15516 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
61, 5mpbi 199 . . 3  |-  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) )
76simpri 448 . 2  |-  (Unit `  R )  =  ( B  \  {  .0.  } )
87eqcomi 2287 1  |-  ( B 
\  {  .0.  }
)  =  (Unit `  R )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   Ringcrg 15337  Unitcui 15421   DivRingcdr 15512
This theorem is referenced by:  cnflddiv  16404  cnfldinv  16405  cnsubdrglem  16422  cnmgpabl  16433  cnmsubglem  16434  gzrngunit  16437  zrngunit  16438  expghm  16450  amgmlem  20284  dchrghm  20495  dchrabs  20499  sum2dchr  20513  lgseisenlem4  20591  qrngdiv  20773  proot1ex  27520
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-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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