Theorem lpirring 21467

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

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2		eqid 2769	. . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3	1, 2	islpir 21464	. 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  Ringcrg 20314  LIdealclidl 21307  LPIdealclpidl 21456  LPIRclpir 21457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-lpir 21459

This theorem is referenced by:  lpirlnr  43735