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Theorem divrngpr 26655
Description: A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
divrngpr  |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )

Proof of Theorem divrngpr
StepHypRef Expression
1 eqid 2436 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2436 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2436 . . . 4  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
4 eqid 2436 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
51, 2, 3, 4isdrngo1 26564 . . 3  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( ( 2nd `  R
)  |`  ( ( ran  ( 1st `  R
)  \  { (GId `  ( 1st `  R
) ) } )  X.  ( ran  ( 1st `  R )  \  { (GId `  ( 1st `  R ) ) } ) ) )  e. 
GrpOp ) )
65simplbi 447 . 2  |-  ( R  e.  DivRingOps  ->  R  e.  RingOps )
7 eqid 2436 . . 3  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 2, 4, 3, 7dvrunz 22014 . 2  |-  ( R  e.  DivRingOps  ->  (GId `  ( 2nd `  R ) )  =/=  (GId `  ( 1st `  R ) ) )
91, 2, 4, 3divrngidl 26630 . 2  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { {
(GId `  ( 1st `  R ) ) } ,  ran  ( 1st `  R ) } )
101, 2, 4, 3, 7smprngopr 26654 . 2  |-  ( ( R  e.  RingOps  /\  (GId `  ( 2nd `  R
) )  =/=  (GId `  ( 1st `  R
) )  /\  ( Idl `  R )  =  { { (GId `  ( 1st `  R ) ) } ,  ran  ( 1st `  R ) } )  ->  R  e.  PrRing )
116, 8, 9, 10syl3anc 1184 1  |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3310   {csn 3807   {cpr 3808    X. cxp 4869   ran crn 4872    |` cres 4873   ` cfv 5447   1stc1st 6340   2ndc2nd 6341   GrpOpcgr 21767  GIdcgi 21768   RingOpscrngo 21956   DivRingOpscdrng 21986   Idlcidl 26609   PrRingcprrng 26648
This theorem is referenced by:  flddmn  26660
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-1st 6342  df-2nd 6343  df-riota 6542  df-1o 6717  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-grpo 21772  df-gid 21773  df-ginv 21774  df-ablo 21863  df-ass 21894  df-exid 21896  df-mgm 21900  df-sgr 21912  df-mndo 21919  df-rngo 21957  df-drngo 21987  df-idl 26612  df-pridl 26613  df-prrngo 26650
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