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Theorem istdrg 18196
Description: Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istrg.1  |-  M  =  (mulGrp `  R )
istdrg.1  |-  U  =  (Unit `  R )
Assertion
Ref Expression
istdrg  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )

Proof of Theorem istdrg
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elin 3531 . . 3  |-  ( R  e.  ( TopRing  i^i  DivRing )  <->  ( R  e.  TopRing  /\  R  e.  DivRing ) )
21anbi1i 678 . 2  |-  ( ( R  e.  ( TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
3 fveq2 5729 . . . . . 6  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . . 6  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2487 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
6 fveq2 5729 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
7 istdrg.1 . . . . . 6  |-  U  =  (Unit `  R )
86, 7syl6eqr 2487 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
95, 8oveq12d 6100 . . . 4  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( Ms  U ) )
109eleq1d 2503 . . 3  |-  ( r  =  R  ->  (
( (mulGrp `  r
)s  (Unit `  r )
)  e.  TopGrp  <->  ( Ms  U
)  e.  TopGrp ) )
11 df-tdrg 18191 . . 3  |- TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }
1210, 11elrab2 3095 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  (
TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
13 df-3an 939 . 2  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U
)  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
142, 12, 133bitr4i 270 1  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320   ` 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:  tdrgunit  18197  tdrgtrg  18203  tdrgdrng  18204  istdrg2  18208  nrgtdrg  18729
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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