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Theorem divrngpr 26678
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 2283 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2283 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2283 . . . 4  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
4 eqid 2283 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
51, 2, 3, 4isdrngo1 26587 . . 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 2283 . . 3  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 2, 4, 3, 7dvrunz 21100 . 2  |-  ( R  e.  DivRingOps  ->  (GId `  ( 2nd `  R ) )  =/=  (GId `  ( 1st `  R ) ) )
91, 2, 4, 3divrngidl 26653 . 2  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { {
(GId `  ( 1st `  R ) ) } ,  ran  ( 1st `  R ) } )
101, 2, 4, 3, 7smprngopr 26677 . 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   {cpr 3641    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072   Idlcidl 26632   PrRingcprrng 26671
This theorem is referenced by:  flddmn  26683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-drngo 21073  df-idl 26635  df-pridl 26636  df-prrngo 26673
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