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Theorem lpirring 19073
Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirring (𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Proof of Theorem lpirring
StepHypRef Expression
1 eqid 2610 . . 3 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2 eqid 2610 . . 3 (LIdeal‘𝑅) = (LIdeal‘𝑅)
31, 2islpir 19070 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
43simplbi 475 1 (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  cfv 5804  Ringcrg 18370  LIdealclidl 18991  LPIdealclpidl 19062  LPIRclpir 19063
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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-lpir 19065
This theorem is referenced by:  lpirlnr  36706
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