Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  divrngpr Structured version   Visualization version   GIF version

Theorem divrngpr 35793
 Description: A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
divrngpr (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)

Proof of Theorem divrngpr
StepHypRef Expression
1 eqid 2758 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2758 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2758 . . . 4 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
4 eqid 2758 . . . 4 ran (1st𝑅) = ran (1st𝑅)
51, 2, 3, 4isdrngo1 35696 . . 3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd𝑅) ↾ ((ran (1st𝑅) ∖ {(GId‘(1st𝑅))}) × (ran (1st𝑅) ∖ {(GId‘(1st𝑅))}))) ∈ GrpOp))
65simplbi 501 . 2 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7 eqid 2758 . . 3 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 2, 4, 3, 7dvrunz 35694 . 2 (𝑅 ∈ DivRingOps → (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)))
91, 2, 4, 3divrngidl 35768 . 2 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)})
101, 2, 4, 3, 7smprngopr 35792 . 2 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)}) → 𝑅 ∈ PrRing)
116, 8, 9, 10syl3anc 1368 1 (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∖ cdif 3855  {csn 4522  {cpr 4524   × cxp 5522  ran crn 5525   ↾ cres 5526  ‘cfv 6335  1st c1st 7691  2nd c2nd 7692  GrpOpcgr 28371  GIdcgi 28372  RingOpscrngo 35634  DivRingOpscdrng 35688  Idlcidl 35747  PrRingcprrng 35786 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-om 7580  df-1st 7693  df-2nd 7694  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-grpo 28375  df-gid 28376  df-ginv 28377  df-ablo 28427  df-ass 35583  df-exid 35585  df-mgmOLD 35589  df-sgrOLD 35601  df-mndo 35607  df-rngo 35635  df-drngo 35689  df-idl 35750  df-pridl 35751  df-prrngo 35788 This theorem is referenced by:  flddmn  35798
 Copyright terms: Public domain W3C validator