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Theorem divrngpr 37007
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 2732 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2732 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2732 . . . 4 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
4 eqid 2732 . . . 4 ran (1st𝑅) = ran (1st𝑅)
51, 2, 3, 4isdrngo1 36910 . . 3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd𝑅) ↾ ((ran (1st𝑅) ∖ {(GId‘(1st𝑅))}) × (ran (1st𝑅) ∖ {(GId‘(1st𝑅))}))) ∈ GrpOp))
65simplbi 498 . 2 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7 eqid 2732 . . 3 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 2, 4, 3, 7dvrunz 36908 . 2 (𝑅 ∈ DivRingOps → (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)))
91, 2, 4, 3divrngidl 36982 . 2 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)})
101, 2, 4, 3, 7smprngopr 37006 . 2 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)}) → 𝑅 ∈ PrRing)
116, 8, 9, 10syl3anc 1371 1 (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wne 2940  cdif 3945  {csn 4628  {cpr 4630   × cxp 5674  ran crn 5677  cres 5678  cfv 6543  1st c1st 7975  2nd c2nd 7976  GrpOpcgr 29780  GIdcgi 29781  RingOpscrngo 36848  DivRingOpscdrng 36902  Idlcidl 36961  PrRingcprrng 37000
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-1o 8468  df-en 8942  df-grpo 29784  df-gid 29785  df-ginv 29786  df-ablo 29836  df-ass 36797  df-exid 36799  df-mgmOLD 36803  df-sgrOLD 36815  df-mndo 36821  df-rngo 36849  df-drngo 36903  df-idl 36964  df-pridl 36965  df-prrngo 37002
This theorem is referenced by:  flddmn  37012
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