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Theorem istdrg 17864
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 3371 . . 3  |-  ( R  e.  ( TopRing  i^i  DivRing )  <->  ( R  e.  TopRing  /\  R  e.  DivRing ) )
21anbi1i 676 . 2  |-  ( ( R  e.  ( TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
3 fveq2 5541 . . . . . 6  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . . 6  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
6 fveq2 5541 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
7 istdrg.1 . . . . . 6  |-  U  =  (Unit `  R )
86, 7syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
95, 8oveq12d 5892 . . . 4  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( Ms  U ) )
109eleq1d 2362 . . 3  |-  ( r  =  R  ->  (
( (mulGrp `  r
)s  (Unit `  r )
)  e.  TopGrp  <->  ( Ms  U
)  e.  TopGrp ) )
11 df-tdrg 17859 . . 3  |- TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }
1210, 11elrab2 2938 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  (
TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
13 df-3an 936 . 2  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U
)  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
142, 12, 133bitr4i 268 1  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  (class class class)co 5874   ↾s cress 13165  mulGrpcmgp 15341  Unitcui 15437   DivRingcdr 15528   TopGrpctgp 17770   TopRingctrg 17854  TopDRingctdrg 17855
This theorem is referenced by:  tdrgunit  17865  tdrgtrg  17871  tdrgdrng  17872  istdrg2  17876  nrgtdrg  18219
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-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-tdrg 17859
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