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Theorem isprrngo 26778
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 5541 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 isprrng.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2346 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43fveq2d 5545 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
5 isprrng.2 . . . . 5  |-  Z  =  (GId `  G )
64, 5syl6eqr 2346 . . . 4  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
76sneqd 3666 . . 3  |-  ( r  =  R  ->  { (GId
`  ( 1st `  r
) ) }  =  { Z } )
8 fveq2 5541 . . 3  |-  ( r  =  R  ->  ( PrIdl `  r )  =  ( PrIdl `  R )
)
97, 8eleq12d 2364 . 2  |-  ( r  =  R  ->  ( { (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
)  <->  { Z }  e.  ( PrIdl `  R )
) )
10 df-prrngo 26776 . 2  |-  PrRing  =  {
r  e.  RingOps  |  {
(GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
) }
119, 10elrab2 2938 1  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   ` cfv 5271   1stc1st 6136  GIdcgi 20870   RingOpscrngo 21058   PrIdlcpridl 26736   PrRingcprrng 26774
This theorem is referenced by:  prrngorngo  26779  smprngopr  26780  isdmn3  26802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-prrngo 26776
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