Theorem lpirring 21664

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 2760	. . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2		eqid 2760	. . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3	1, 2	islpir 21661	. 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4	3	simplbi 502	1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  Ringcrg 20468  LIdealclidl 21493  LPIdealclpidl 21653  LPIRclpir 21654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-lpir 21656

This theorem is used by:  lpirlnr  44256