Theorem lpirring 21364

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 2740	. . 3 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
2		eqid 2740	. . 3 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
3	1, 2	islpir 21361	. 2 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4	3	simplbi 497	1 ⊢ (𝑅 ∈ LPIR → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  Ringcrg 20260  LIdealclidl 21239  LPIdealclpidl 21353  LPIRclpir 21354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-lpir 21356

This theorem is referenced by:  lpirlnr  43074