MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istrg Unicode version

Theorem istrg 17846
Description: Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
istrg.1  |-  M  =  (mulGrp `  R )
Assertion
Ref Expression
istrg  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )

Proof of Theorem istrg
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elin 3358 . . 3  |-  ( R  e.  ( TopGrp  i^i  Ring ) 
<->  ( R  e.  TopGrp  /\  R  e.  Ring )
)
21anbi1i 676 . 2  |-  ( ( R  e.  ( TopGrp  i^i 
Ring )  /\  M  e. TopMnd )  <->  ( ( R  e.  TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd
) )
3 fveq2 5525 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . 5  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
65eleq1d 2349 . . 3  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. TopMnd  <-> 
M  e. TopMnd ) )
7 df-trg 17842 . . 3  |-  TopRing  =  {
r  e.  ( TopGrp  i^i 
Ring )  |  (mulGrp `  r )  e. TopMnd }
86, 7elrab2 2925 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  (
TopGrp  i^i  Ring )  /\  M  e. TopMnd ) )
9 df-3an 936 . 2  |-  ( ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd )  <-> 
( ( R  e. 
TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd )
)
102, 8, 93bitr4i 268 1  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151   ` cfv 5255  mulGrpcmgp 15325   Ringcrg 15337  TopMndctmd 17753   TopGrpctgp 17754   TopRingctrg 17838
This theorem is referenced by:  trgtmd  17847  trgtgp  17850  trgrng  17853  nrgtrg  18200  iistmd  23286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-trg 17842
  Copyright terms: Public domain W3C validator