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Theorem divrngpr 26347
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 2380 . . . 4  |-  ( 1st `  R )  =  ( 1st `  R )
2 eqid 2380 . . . 4  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2380 . . . 4  |-  (GId `  ( 1st `  R ) )  =  (GId `  ( 1st `  R ) )
4 eqid 2380 . . . 4  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
51, 2, 3, 4isdrngo1 26256 . . 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 2380 . . 3  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 2, 4, 3, 7dvrunz 21862 . 2  |-  ( R  e.  DivRingOps  ->  (GId `  ( 2nd `  R ) )  =/=  (GId `  ( 1st `  R ) ) )
91, 2, 4, 3divrngidl 26322 . 2  |-  ( R  e.  DivRingOps  ->  ( Idl `  R
)  =  { {
(GId `  ( 1st `  R ) ) } ,  ran  ( 1st `  R ) } )
101, 2, 4, 3, 7smprngopr 26346 . 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 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253   {csn 3750   {cpr 3751    X. cxp 4809   ran crn 4812    |` cres 4813   ` cfv 5387   1stc1st 6279   2ndc2nd 6280   GrpOpcgr 21615  GIdcgi 21616   RingOpscrngo 21804   DivRingOpscdrng 21834   Idlcidl 26301   PrRingcprrng 26340
This theorem is referenced by:  flddmn  26352
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-1st 6281  df-2nd 6282  df-riota 6478  df-1o 6653  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-grpo 21620  df-gid 21621  df-ginv 21622  df-ablo 21711  df-ass 21742  df-exid 21744  df-mgm 21748  df-sgr 21760  df-mndo 21767  df-rngo 21805  df-drngo 21835  df-idl 26304  df-pridl 26305  df-prrngo 26342
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