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Theorem istdrg 20753
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 3601 . . 3  |-  ( R  e.  ( TopRing  i^i  DivRing )  <->  ( R  e.  TopRing  /\  R  e.  DivRing ) )
21anbi1i 693 . 2  |-  ( ( R  e.  ( TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp )  <->  ( ( R  e.  TopRing  /\  R  e.  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
3 fveq2 5774 . . . . . 6  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . . 6  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2441 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
6 fveq2 5774 . . . . . 6  |-  ( r  =  R  ->  (Unit `  r )  =  (Unit `  R ) )
7 istdrg.1 . . . . . 6  |-  U  =  (Unit `  R )
86, 7syl6eqr 2441 . . . . 5  |-  ( r  =  R  ->  (Unit `  r )  =  U )
95, 8oveq12d 6214 . . . 4  |-  ( r  =  R  ->  (
(mulGrp `  r )s  (Unit `  r ) )  =  ( Ms  U ) )
109eleq1d 2451 . . 3  |-  ( r  =  R  ->  (
( (mulGrp `  r
)s  (Unit `  r )
)  e.  TopGrp  <->  ( Ms  U
)  e.  TopGrp ) )
11 df-tdrg 20748 . . 3  |- TopDRing  =  {
r  e.  ( TopRing  i^i  DivRing )  |  ( (mulGrp `  r )s  (Unit `  r )
)  e.  TopGrp }
1210, 11elrab2 3184 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  (
TopRing  i^i  DivRing )  /\  ( Ms  U )  e.  TopGrp ) )
13 df-3an 973 . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    i^i cin 3388   ` cfv 5496  (class class class)co 6196   ↾s cress 14635  mulGrpcmgp 17254  Unitcui 17401   DivRingcdr 17509   TopGrpctgp 20655   TopRingctrg 20743  TopDRingctdrg 20744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-iota 5460  df-fv 5504  df-ov 6199  df-tdrg 20748
This theorem is referenced by:  tdrgunit  20754  tdrgtrg  20760  tdrgdrng  20761  istdrg2  20765  nrgtdrg  21286
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