Theorem lpirring 21499

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 2763	. . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2		eqid 2763	. . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3	1, 2	islpir 21496	. 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  Ringcrg 20310  LIdealclidl 21330  LPIdealclpidl 21488  LPIRclpir 21489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-lpir 21491

This theorem is referenced by:  lpirlnr  43844