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Theorem lpirring 21184
Description: Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirring (𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)

Proof of Theorem lpirring
StepHypRef Expression
1 eqid 2726 . . 3 (LPIdealβ€˜π‘…) = (LPIdealβ€˜π‘…)
2 eqid 2726 . . 3 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
31, 2islpir 21181 . 2 (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdealβ€˜π‘…) = (LPIdealβ€˜π‘…)))
43simplbi 497 1 (𝑅 ∈ LPIR β†’ 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6537  Ringcrg 20138  LIdealclidl 21065  LPIdealclpidl 21173  LPIRclpir 21174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-lpir 21176
This theorem is referenced by:  lpirlnr  42434
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