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Theorem divrngpr 33022
 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 2610 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2610 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2610 . . . 4 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
4 eqid 2610 . . . 4 ran (1st𝑅) = ran (1st𝑅)
51, 2, 3, 4isdrngo1 32925 . . 3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd𝑅) ↾ ((ran (1st𝑅) ∖ {(GId‘(1st𝑅))}) × (ran (1st𝑅) ∖ {(GId‘(1st𝑅))}))) ∈ GrpOp))
65simplbi 475 . 2 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7 eqid 2610 . . 3 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 2, 4, 3, 7dvrunz 32923 . 2 (𝑅 ∈ DivRingOps → (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)))
91, 2, 4, 3divrngidl 32997 . 2 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)})
101, 2, 4, 3, 7smprngopr 33021 . 2 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)}) → 𝑅 ∈ PrRing)
116, 8, 9, 10syl3anc 1318 1 (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  {csn 4125  {cpr 4127   × cxp 5036  ran crn 5039   ↾ cres 5040  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058  GrpOpcgr 26727  GIdcgi 26728  RingOpscrngo 32863  DivRingOpscdrng 32917  Idlcidl 32976  PrRingcprrng 33015 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-drngo 32918  df-idl 32979  df-pridl 32980  df-prrngo 33017 This theorem is referenced by:  flddmn  33027
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