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Theorem istrg 20960
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 3628 . . 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 5851 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . 5  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2463 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
65eleq1d 2473 . . 3  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. TopMnd  <-> 
M  e. TopMnd ) )
7 df-trg 20956 . . 3  |-  TopRing  =  {
r  e.  ( TopGrp  i^i 
Ring )  |  (mulGrp `  r )  e. TopMnd }
86, 7elrab2 3211 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  (
TopGrp  i^i  Ring )  /\  M  e. TopMnd ) )
9 df-3an 978 . 2  |-  ( ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd )  <-> 
( ( R  e. 
TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd )
)
102, 8, 93bitr4i 279 1  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    i^i cin 3415   ` cfv 5571  mulGrpcmgp 17463   Ringcrg 17520  TopMndctmd 20863   TopGrpctgp 20864   TopRingctrg 20952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-trg 20956
This theorem is referenced by:  trgtmd  20961  trgtgp  20964  trgring  20967  nrgtrg  21492
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