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Theorem divrngpr 32825
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 2609 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2609 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2609 . . . 4 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
4 eqid 2609 . . . 4 ran (1st𝑅) = ran (1st𝑅)
51, 2, 3, 4isdrngo1 32728 . . 3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd𝑅) ↾ ((ran (1st𝑅) ∖ {(GId‘(1st𝑅))}) × (ran (1st𝑅) ∖ {(GId‘(1st𝑅))}))) ∈ GrpOp))
65simplbi 474 . 2 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7 eqid 2609 . . 3 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 2, 4, 3, 7dvrunz 32726 . 2 (𝑅 ∈ DivRingOps → (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)))
91, 2, 4, 3divrngidl 32800 . 2 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)})
101, 2, 4, 3, 7smprngopr 32824 . 2 ((𝑅 ∈ RingOps ∧ (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)) ∧ (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)}) → 𝑅 ∈ PrRing)
116, 8, 9, 10syl3anc 1317 1 (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wne 2779  cdif 3536  {csn 4124  {cpr 4126   × cxp 5026  ran crn 5029  cres 5030  cfv 5790  1st c1st 7034  2nd c2nd 7035  GrpOpcgr 26493  GIdcgi 26494  RingOpscrngo 32666  DivRingOpscdrng 32720  Idlcidl 32779  PrRingcprrng 32818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-om 6935  df-1st 7036  df-2nd 7037  df-1o 7424  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-grpo 26497  df-gid 26498  df-ginv 26499  df-ablo 26552  df-ass 32615  df-exid 32617  df-mgmOLD 32621  df-sgrOLD 32633  df-mndo 32639  df-rngo 32667  df-drngo 32721  df-idl 32782  df-pridl 32783  df-prrngo 32820
This theorem is referenced by:  flddmn  32830
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