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

Theorem istrg 18185
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 3522 . . 3  |-  ( R  e.  ( TopGrp  i^i  Ring ) 
<->  ( R  e.  TopGrp  /\  R  e.  Ring )
)
21anbi1i 677 . 2  |-  ( ( R  e.  ( TopGrp  i^i 
Ring )  /\  M  e. TopMnd )  <->  ( ( R  e.  TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd
) )
3 fveq2 5720 . . . . 5  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
4 istrg.1 . . . . 5  |-  M  =  (mulGrp `  R )
53, 4syl6eqr 2485 . . . 4  |-  ( r  =  R  ->  (mulGrp `  r )  =  M )
65eleq1d 2501 . . 3  |-  ( r  =  R  ->  (
(mulGrp `  r )  e. TopMnd  <-> 
M  e. TopMnd ) )
7 df-trg 18181 . . 3  |-  TopRing  =  {
r  e.  ( TopGrp  i^i 
Ring )  |  (mulGrp `  r )  e. TopMnd }
86, 7elrab2 3086 . 2  |-  ( R  e.  TopRing 
<->  ( R  e.  (
TopGrp  i^i  Ring )  /\  M  e. TopMnd ) )
9 df-3an 938 . 2  |-  ( ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd )  <-> 
( ( R  e. 
TopGrp  /\  R  e.  Ring )  /\  M  e. TopMnd )
)
102, 8, 93bitr4i 269 1  |-  ( R  e.  TopRing 
<->  ( R  e.  TopGrp  /\  R  e.  Ring  /\  M  e. TopMnd ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311   ` cfv 5446  mulGrpcmgp 15640   Ringcrg 15652  TopMndctmd 18092   TopGrpctgp 18093   TopRingctrg 18177
This theorem is referenced by:  trgtmd  18186  trgtgp  18189  trgrng  18192  nrgtrg  18717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-trg 18181
  Copyright terms: Public domain W3C validator