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Theorem lpirring 19649
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 2777 . . 3 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2 eqid 2777 . . 3 (LIdeal‘𝑅) = (LIdeal‘𝑅)
31, 2islpir 19646 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
43simplbi 493 1 (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  cfv 6135  Ringcrg 18934  LIdealclidl 19567  LPIdealclpidl 19638  LPIRclpir 19639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-lpir 19641
This theorem is referenced by:  lpirlnr  38638
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