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Theorem istdrg2 17876
Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istdrg2.m  |-  M  =  (mulGrp `  R )
istdrg2.b  |-  B  =  ( Base `  R
)
istdrg2.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
istdrg2  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )

Proof of Theorem istdrg2
StepHypRef Expression
1 istdrg2.m . . 3  |-  M  =  (mulGrp `  R )
2 eqid 2296 . . 3  |-  (Unit `  R )  =  (Unit `  R )
31, 2istdrg 17864 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  (Unit `  R ) )  e.  TopGrp ) )
4 istdrg2.b . . . . . . . . 9  |-  B  =  ( Base `  R
)
5 istdrg2.z . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
64, 2, 5isdrng 15532 . . . . . . . 8  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
76simprbi 450 . . . . . . 7  |-  ( R  e.  DivRing  ->  (Unit `  R
)  =  ( B 
\  {  .0.  }
) )
87adantl 452 . . . . . 6  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
98oveq2d 5890 . . . . 5  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  ( Ms  (Unit `  R ) )  =  ( Ms  ( B 
\  {  .0.  }
) ) )
109eleq1d 2362 . . . 4  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  (
( Ms  (Unit `  R )
)  e.  TopGrp  <->  ( Ms  ( B  \  {  .0.  }
) )  e.  TopGrp ) )
1110pm5.32i 618 . . 3  |-  ( ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  (Unit `  R )
)  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
12 df-3an 936 . . 3  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  (Unit `  R ) )  e. 
TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  (Unit `  R ) )  e. 
TopGrp ) )
13 df-3an 936 . . 3  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  }
) )  e.  TopGrp )  <-> 
( ( R  e.  TopRing 
/\  R  e.  DivRing )  /\  ( Ms  ( B 
\  {  .0.  }
) )  e.  TopGrp ) )
1411, 12, 133bitr4i 268 . 2  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  (Unit `  R ) )  e. 
TopGrp )  <->  ( R  e.  TopRing 
/\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
153, 14bitri 240 1  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   0gc0g 13416  mulGrpcmgp 15341   Ringcrg 15353  Unitcui 15437   DivRingcdr 15528   TopGrpctgp 17770   TopRingctrg 17854  TopDRingctdrg 17855
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-ov 5877  df-drng 15530  df-tdrg 17859
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