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Theorem istdrg2 18207
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 2436 . . 3  |-  (Unit `  R )  =  (Unit `  R )
31, 2istdrg 18195 . 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 15839 . . . . . . . 8  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  {  .0.  } ) ) )
76simprbi 451 . . . . . . 7  |-  ( R  e.  DivRing  ->  (Unit `  R
)  =  ( B 
\  {  .0.  }
) )
87adantl 453 . . . . . 6  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  (Unit `  R )  =  ( B  \  {  .0.  } ) )
98oveq2d 6097 . . . . 5  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  ( Ms  (Unit `  R ) )  =  ( Ms  ( B 
\  {  .0.  }
) ) )
109eleq1d 2502 . . . 4  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing )  ->  (
( Ms  (Unit `  R )
)  e.  TopGrp  <->  ( Ms  ( B  \  {  .0.  }
) )  e.  TopGrp ) )
1110pm5.32i 619 . . 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 938 . . 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 938 . . 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 269 . 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 241 1  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  ( B  \  {  .0.  } ) )  e.  TopGrp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3317   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   0gc0g 13723  mulGrpcmgp 15648   Ringcrg 15660  Unitcui 15744   DivRingcdr 15835   TopGrpctgp 18101   TopRingctrg 18185  TopDRingctdrg 18186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-drng 15837  df-tdrg 18190
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