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Theorem istdrg 19762
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 3560 . . 3  |-  ( R  e.  ( TopRing  i^i  DivRing )  <->  ( R  e.  TopRing  /\  R  e.  DivRing ) )
21anbi1i 695 . 2  |-  ( ( R  e.  ( TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
3 fveq2 5712 . . . . . 6  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . . 6  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
6 fveq2 5712 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
7 istdrg.1 . . . . . 6  |-  U  =  (Unit `  R )
86, 7syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
95, 8oveq12d 6130 . . . 4  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( Ms  U ) )
109eleq1d 2509 . . 3  |-  ( r  =  R  ->  (
( (mulGrp `  r
)s  (Unit `  r )
)  e.  TopGrp  <->  ( Ms  U
)  e.  TopGrp ) )
11 df-tdrg 19757 . . 3  |- TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }
1210, 11elrab2 3140 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  (
TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
13 df-3an 967 . 2  |-  ( ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U
)  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
142, 12, 133bitr4i 277 1  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  ( Ms  U )  e.  TopGrp ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3348   ` cfv 5439  (class class class)co 6112   ↾s cress 14196  mulGrpcmgp 16613  Unitcui 16753   DivRingcdr 16854   TopGrpctgp 19664   TopRingctrg 19752  TopDRingctdrg 19753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115  df-tdrg 19757
This theorem is referenced by:  tdrgunit  19763  tdrgtrg  19769  tdrgdrng  19770  istdrg2  19774  nrgtdrg  20295
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