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Theorem istrg 17862
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 3371 . . 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 5541 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . 5  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2346 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
65eleq1d 2362 . . 3  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. TopMnd  <-> 
M  e. TopMnd ) )
7 df-trg 17858 . . 3  |-  TopRing  =  {
r  e.  ( TopGrp  i^i 
Ring )  |  (mulGrp `  r )  e. TopMnd }
86, 7elrab2 2938 . 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 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  mulGrpcmgp 15341   Ringcrg 15353  TopMndctmd 17769   TopGrpctgp 17770   TopRingctrg 17854
This theorem is referenced by:  trgtmd  17863  trgtgp  17866  trgrng  17869  nrgtrg  18216  iistmd  23301
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-trg 17858
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