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Theorem isprrngo 26675
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 5525 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 isprrng.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2333 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43fveq2d 5529 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
5 isprrng.2 . . . . 5  |-  Z  =  (GId `  G )
64, 5syl6eqr 2333 . . . 4  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
76sneqd 3653 . . 3  |-  ( r  =  R  ->  { (GId
`  ( 1st `  r
) ) }  =  { Z } )
8 fveq2 5525 . . 3  |-  ( r  =  R  ->  ( PrIdl `  r )  =  ( PrIdl `  R )
)
97, 8eleq12d 2351 . 2  |-  ( r  =  R  ->  ( { (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
)  <->  { Z }  e.  ( PrIdl `  R )
) )
10 df-prrngo 26673 . 2  |-  PrRing  =  {
r  e.  RingOps  |  {
(GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
) }
119, 10elrab2 2925 1  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   ` cfv 5255   1stc1st 6120  GIdcgi 20854   RingOpscrngo 21042   PrIdlcpridl 26633   PrRingcprrng 26671
This theorem is referenced by:  prrngorngo  26676  smprngopr  26677  isdmn3  26699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-prrngo 26673
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