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Theorem lpirring 20157
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 2739 . . 3 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2 eqid 2739 . . 3 (LIdeal‘𝑅) = (LIdeal‘𝑅)
31, 2islpir 20154 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
43simplbi 501 1 (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cfv 6350  Ringcrg 19429  LIdealclidl 20074  LPIdealclpidl 20146  LPIRclpir 20147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-un 3858  df-in 3860  df-ss 3870  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-lpir 20149
This theorem is referenced by:  lpirlnr  40555
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