Theorem lpirring 21563

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 21560	. 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 1570   ∈ wcel 2145  ‘cfv 6533  Ringcrg 20373  LIdealclidl 21394  LPIdealclpidl 21552  LPIRclpir 21553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-lpir 21555

This theorem is used by:  lpirlnr  43959