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Theorem isprrngo 30609
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 5872 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 isprrng.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2516 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43fveq2d 5876 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
5 isprrng.2 . . . . 5  |-  Z  =  (GId `  G )
64, 5syl6eqr 2516 . . . 4  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
76sneqd 4044 . . 3  |-  ( r  =  R  ->  { (GId
`  ( 1st `  r
) ) }  =  { Z } )
8 fveq2 5872 . . 3  |-  ( r  =  R  ->  ( PrIdl `  r )  =  ( PrIdl `  R )
)
97, 8eleq12d 2539 . 2  |-  ( r  =  R  ->  ( { (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
)  <->  { Z }  e.  ( PrIdl `  R )
) )
10 df-prrngo 30607 . 2  |-  PrRing  =  {
r  e.  RingOps  |  {
(GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
) }
119, 10elrab2 3259 1  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {csn 4032   ` cfv 5594   1stc1st 6797  GIdcgi 25315   RingOpscrngo 25503   PrIdlcpridl 30567   PrRingcprrng 30605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-prrngo 30607
This theorem is referenced by:  prrngorngo  30610  smprngopr  30611  isdmn3  30633
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