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Theorem istrg 19738
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 3539 . . 3  |-  ( R  e.  ( TopGrp  i^i  Ring ) 
<->  ( R  e.  TopGrp  /\  R  e.  Ring )
)
21anbi1i 695 . 2  |-  ( ( R  e.  ( TopGrp  i^i 
Ring )  /\  M  e. TopMnd )  <->  ( ( R  e.  TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd
) )
3 fveq2 5691 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . 5  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2493 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
65eleq1d 2509 . . 3  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. TopMnd  <-> 
M  e. TopMnd ) )
7 df-trg 19734 . . 3  |-  TopRing  =  {
r  e.  ( TopGrp  i^i 
Ring )  |  (mulGrp `  r )  e. TopMnd }
86, 7elrab2 3119 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  (
TopGrp  i^i  Ring )  /\  M  e. TopMnd ) )
9 df-3an 967 . 2  |-  ( ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd )  <-> 
( ( R  e. 
TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd )
)
102, 8, 93bitr4i 277 1  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3327   ` cfv 5418  mulGrpcmgp 16591   Ringcrg 16645  TopMndctmd 19641   TopGrpctgp 19642   TopRingctrg 19730
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 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-trg 19734
This theorem is referenced by:  trgtmd  19739  trgtgp  19742  trgrng  19745  nrgtrg  20270
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