Theorem divrngpr 33852

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( 1st ` R ) = ( 1st ` R )
2		eqid 2622	. . . 4 |- ( 2nd ` R ) = ( 2nd ` R )
3		eqid 2622	. . . 4 |- (GId ` ( 1st ` R ) ) = (GId ` ( 1st ` R ) )
4		eqid 2622	. . . 4 |- ran ( 1st ` R ) = ran ( 1st ` R )
5	1, 2, 3, 4	isdrngo1 33755	. . 3 |- ( R e. DivRingOps <-> ( R e. RingOps /\ ( ( 2nd ` R ) |` ( ( ran ( 1st ` R ) \ { (GId ` ( 1st ` R ) ) } ) X. ( ran ( 1st ` R ) \ { (GId ` ( 1st ` R ) ) } ) ) ) e. GrpOp ) )
6	5	simplbi 476	. 2 |- ( R e. DivRingOps -> R e. RingOps )
7		eqid 2622	. . 3 |- (GId ` ( 2nd ` R ) ) = (GId ` ( 2nd ` R ) )
8	1, 2, 4, 3, 7	dvrunz 33753	. 2 |- ( R e. DivRingOps -> (GId ` ( 2nd ` R ) ) =/= (GId ` ( 1st ` R ) ) )
9	1, 2, 4, 3	divrngidl 33827	. 2 |- ( R e. DivRingOps -> ( Idl ` R ) = { { (GId ` ( 1st ` R ) ) } , ran ( 1st ` R ) } )
10	1, 2, 4, 3, 7	smprngopr 33851	. 2 |- ( ( R e. RingOps /\ (GId ` ( 2nd ` R ) ) =/= (GId ` ( 1st ` R ) ) /\ ( Idl ` R ) = { { (GId ` ( 1st ` R ) ) } , ran ( 1st ` R ) } ) -> R e. PrRing )
11	6, 8, 9, 10	syl3anc 1326	1 |- ( R e. DivRingOps -> R e. PrRing )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   {cpr 4179   X. cxp 5112   ran crn 5115   |` cres 5116   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   RingOpscrngo 33693   DivRingOpscdrng 33747   Idlcidl 33806   PrRingcprrng 33845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-drngo 33748  df-idl 33809  df-pridl 33810  df-prrngo 33847

This theorem is referenced by:  flddmn  33857