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Theorem prrngorngo 26779
Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
prrngorngo  |-  ( R  e.  PrRing  ->  R  e.  RingOps )

Proof of Theorem prrngorngo
StepHypRef Expression
1 eqid 2296 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2296 . . 3  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
31, 2isprrngo 26778 . 2  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { (GId `  ( 1st `  R ) ) }  e.  ( PrIdl `  R
) ) )
43simplbi 446 1  |-  ( R  e.  PrRing  ->  R  e.  RingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   {csn 3653   ` cfv 5271   1stc1st 6136  GIdcgi 20870   RingOpscrngo 21058   PrIdlcpridl 26736   PrRingcprrng 26774
This theorem is referenced by:  isdmn2  26783
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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