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Theorem divrngpr 36726
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 2731 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2731 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2731 . . . 4 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
4 eqid 2731 . . . 4 ran (1st𝑅) = ran (1st𝑅)
51, 2, 3, 4isdrngo1 36629 . . 3 (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ ((2nd𝑅) ↾ ((ran (1st𝑅) ∖ {(GId‘(1st𝑅))}) × (ran (1st𝑅) ∖ {(GId‘(1st𝑅))}))) ∈ GrpOp))
65simplbi 498 . 2 (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps)
7 eqid 2731 . . 3 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
81, 2, 4, 3, 7dvrunz 36627 . 2 (𝑅 ∈ DivRingOps → (GId‘(2nd𝑅)) ≠ (GId‘(1st𝑅)))
91, 2, 4, 3divrngidl 36701 . 2 (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{(GId‘(1st𝑅))}, ran (1st𝑅)})
101, 2, 4, 3, 7smprngopr 36725 . 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 2939  cdif 3941  {csn 4622  {cpr 4624   × cxp 5667  ran crn 5670  cres 5671  cfv 6532  1st c1st 7955  2nd c2nd 7956  GrpOpcgr 29605  GIdcgi 29606  RingOpscrngo 36567  DivRingOpscdrng 36621  Idlcidl 36680  PrRingcprrng 36719
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-1st 7957  df-2nd 7958  df-1o 8448  df-en 8923  df-grpo 29609  df-gid 29610  df-ginv 29611  df-ablo 29661  df-ass 36516  df-exid 36518  df-mgmOLD 36522  df-sgrOLD 36534  df-mndo 36540  df-rngo 36568  df-drngo 36622  df-idl 36683  df-pridl 36684  df-prrngo 36721
This theorem is referenced by:  flddmn  36731
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