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Theorem tdrgunit 18197
Description: The unit group of a topological division ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istrg.1  |-  M  =  (mulGrp `  R )
istdrg.1  |-  U  =  (Unit `  R )
Assertion
Ref Expression
tdrgunit  |-  ( R  e. TopDRing  ->  ( Ms  U )  e.  TopGrp )

Proof of Theorem tdrgunit
StepHypRef Expression
1 istrg.1 . . 3  |-  M  =  (mulGrp `  R )
2 istdrg.1 . . 3  |-  U  =  (Unit `  R )
31, 2istdrg 18196 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
43simp3bi 975 1  |-  ( R  e. TopDRing  ->  ( Ms  U )  e.  TopGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   ↾s cress 13471  mulGrpcmgp 15649  Unitcui 15745   DivRingcdr 15836   TopGrpctgp 18102   TopRingctrg 18186  TopDRingctdrg 18187
This theorem is referenced by:  invrcn2  18210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-tdrg 18191
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