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Theorem istdrg2 17860
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 2283 . . 3  |-  (Unit `  R )  =  (Unit `  R )
31, 2istdrg 17848 . 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 15516 . . . . . . . 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 5874 . . . . 5  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  ( Ms  (Unit `  R ) )  =  ( Ms  ( B 
\  {  .0.  }
) ) )
109eleq1d 2349 . . . 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 1623    e. wcel 1684    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   0gc0g 13400  mulGrpcmgp 15325   Ringcrg 15337  Unitcui 15421   DivRingcdr 15512   TopGrpctgp 17754   TopRingctrg 17838  TopDRingctdrg 17839
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-ov 5861  df-drng 15514  df-tdrg 17843
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