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Theorem isprrngo 28862
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 5703 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
2 isprrng.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
31, 2syl6eqr 2493 . . . . . 6  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
43fveq2d 5707 . . . . 5  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  (GId
`  G ) )
5 isprrng.2 . . . . 5  |-  Z  =  (GId `  G )
64, 5syl6eqr 2493 . . . 4  |-  ( r  =  R  ->  (GId `  ( 1st `  r
) )  =  Z )
76sneqd 3901 . . 3  |-  ( r  =  R  ->  { (GId
`  ( 1st `  r
) ) }  =  { Z } )
8 fveq2 5703 . . 3  |-  ( r  =  R  ->  ( PrIdl `  r )  =  ( PrIdl `  R )
)
97, 8eleq12d 2511 . 2  |-  ( r  =  R  ->  ( { (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
)  <->  { Z }  e.  ( PrIdl `  R )
) )
10 df-prrngo 28860 . 2  |-  PrRing  =  {
r  e.  RingOps  |  {
(GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r
) }
119, 10elrab2 3131 1  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3889   ` cfv 5430   1stc1st 6587  GIdcgi 23686   RingOpscrngo 23874   PrIdlcpridl 28820   PrRingcprrng 28858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-prrngo 28860
This theorem is referenced by:  prrngorngo  28863  smprngopr  28864  isdmn3  28886
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