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Theorem lpirrng 17467
 Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirrng LPIR

Proof of Theorem lpirrng
StepHypRef Expression
1 eqid 2454 . . 3 LPIdeal LPIdeal
2 eqid 2454 . . 3 LIdeal LIdeal
31, 2islpir 17464 . 2 LPIR LIdeal LPIdeal
43simplbi 460 1 LPIR
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1370   wcel 1758  cfv 5529  crg 16778  LIdealclidl 17384  LPIdealclpidl 17456  LPIRclpir 17457 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-lpir 17459 This theorem is referenced by:  lpirlnr  29644
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