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Theorem isprrngo 26662
Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isprrng.1  |-  G  =  ( 1st `  R
)
isprrng.2  |-  Z  =  (GId `  G )
Assertion
Ref Expression
isprrngo  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )

Proof of Theorem isprrngo
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 isprrng.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2488 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43fveq2d 5734 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
5 isprrng.2 . . . . 5  |-  Z  =  (GId `  G )
64, 5syl6eqr 2488 . . . 4  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
76sneqd 3829 . . 3  |-  ( r  =  R  ->  { (GId
`  ( 1st `  r
) ) }  =  { Z } )
8 fveq2 5730 . . 3  |-  ( r  =  R  ->  ( PrIdl `  r )  =  ( PrIdl `  R )
)
97, 8eleq12d 2506 . 2  |-  ( r  =  R  ->  ( { (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
)  <->  { Z }  e.  ( PrIdl `  R )
) )
10 df-prrngo 26660 . 2  |-  PrRing  =  {
r  e.  RingOps  |  {
(GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
) }
119, 10elrab2 3096 1  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {csn 3816   ` cfv 5456   1stc1st 6349  GIdcgi 21777   RingOpscrngo 21965   PrIdlcpridl 26620   PrRingcprrng 26658
This theorem is referenced by:  prrngorngo  26663  smprngopr  26664  isdmn3  26686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-prrngo 26660
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