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Theorem prrngorngo 26352
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 2387 . . 3  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2387 . . 3  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
31, 2isprrngo 26351 . 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 1717   {csn 3757   ` cfv 5394   1stc1st 6286  GIdcgi 21623   RingOpscrngo 21811   PrIdlcpridl 26309   PrRingcprrng 26347
This theorem is referenced by:  isdmn2  26356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-prrngo 26349
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