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Theorem lpirring 21359
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 2735 . . 3 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2 eqid 2735 . . 3 (LIdeal‘𝑅) = (LIdeal‘𝑅)
31, 2islpir 21356 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
43simplbi 497 1 (𝑅 ∈ LPIR → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Ringcrg 20251  LIdealclidl 21234  LPIdealclpidl 21348  LPIRclpir 21349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-lpir 21351
This theorem is referenced by:  lpirlnr  43106
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