MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drngui Unicode version

Theorem drngui 15534
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 2296 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
4 drngui.z . . . . 5  |-  .0.  =  ( 0g `  R )
52, 3, 4isdrng 15532 . . . 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 2300 1  |-  ( B 
\  {  .0.  }
)  =  (Unit `  R )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162   {csn 3653   ` cfv 5271   Basecbs 13164   0gc0g 13416   Ringcrg 15353  Unitcui 15437   DivRingcdr 15528
This theorem is referenced by:  cnflddiv  16420  cnfldinv  16421  cnsubdrglem  16438  cnmgpabl  16449  cnmsubglem  16450  gzrngunit  16453  zrngunit  16454  expghm  16466  amgmlem  20300  dchrghm  20511  dchrabs  20515  sum2dchr  20529  lgseisenlem4  20607  qrngdiv  20789  proot1ex  27623
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-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
  Copyright terms: Public domain W3C validator