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Theorem divrngpr 26781
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 2296 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2296 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2296 . . . 4  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
4 eqid 2296 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
51, 2, 3, 4isdrngo1 26690 . . 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 446 . 2  |-  ( R  e.  DivRingOps  ->  R  e.  RingOps )
7 eqid 2296 . . 3  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 2, 4, 3, 7dvrunz 21116 . 2  |-  ( R  e.  DivRingOps  ->  (GId `  ( 2nd `  R ) )  =/=  (GId `  ( 1st `  R ) ) )
91, 2, 4, 3divrngidl 26756 . 2  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { {
(GId `  ( 1st `  R ) ) } ,  ran  ( 1st `  R ) } )
101, 2, 4, 3, 7smprngopr 26780 . 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 1182 1  |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   {cpr 3654    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   RingOpscrngo 21058   DivRingOpscdrng 21088   Idlcidl 26735   PrRingcprrng 26774
This theorem is referenced by:  flddmn  26786
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-drngo 21089  df-idl 26738  df-pridl 26739  df-prrngo 26776
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