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Theorem islpir 17455
 Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p LPIdeal
lpiss.u LIdeal
Assertion
Ref Expression
islpir LPIR

Proof of Theorem islpir
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5800 . . . 4 LIdeal LIdeal
2 fveq2 5800 . . . 4 LPIdeal LPIdeal
31, 2eqeq12d 2476 . . 3 LIdeal LPIdeal LIdeal LPIdeal
4 lpiss.u . . . 4 LIdeal
5 lpival.p . . . 4 LPIdeal
64, 5eqeq12i 2474 . . 3 LIdeal LPIdeal
73, 6syl6bbr 263 . 2 LIdeal LPIdeal
8 df-lpir 17450 . 2 LPIR LIdeal LPIdeal
97, 8elrab2 3226 1 LPIR
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1370   wcel 1758  cfv 5527  crg 16769  LIdealclidl 17375  LPIdealclpidl 17447  LPIRclpir 17448 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 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-lpir 17450 This theorem is referenced by:  islpir2  17457  lpirrng  17458  lpirlnr  29622
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