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Theorem prrngorngo 26653
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 2436 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2436 . . 3  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
31, 2isprrngo 26652 . 2  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { (GId `  ( 1st `  R ) ) }  e.  ( PrIdl `  R
) ) )
43simplbi 447 1  |-  ( R  e.  PrRing  ->  R  e.  RingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   {csn 3807   ` cfv 5447   1stc1st 6340  GIdcgi 21768   RingOpscrngo 21956   PrIdlcpridl 26610   PrRingcprrng 26648
This theorem is referenced by:  isdmn2  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-iota 5411  df-fv 5455  df-prrngo 26650
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