Theorem islpir 20433

Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)

Hypotheses

Ref	Expression
lpival.p	⊢ 𝑃 = (LPIdeal‘𝑅)
lpiss.u	⊢ 𝑈 = (LIdeal‘𝑅)

Assertion

Ref	Expression
islpir	⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Proof of Theorem islpir

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6756	. . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2		fveq2 6756	. . . 4 ⊢ (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅))
3	1, 2	eqeq12d 2754	. . 3 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅)))
4		lpiss.u	. . . 4 ⊢ 𝑈 = (LIdeal‘𝑅)
5		lpival.p	. . . 4 ⊢ 𝑃 = (LPIdeal‘𝑅)
6	4, 5	eqeq12i 2756	. . 3 ⊢ (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))
7	3, 6	bitr4di 288	. 2 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃))
8		df-lpir 20428	. 2 ⊢ LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)}
9	7, 8	elrab2 3620	1 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  Ringcrg 19698  LIdealclidl 20347  LPIdealclpidl 20425  LPIRclpir 20426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-lpir 20428

This theorem is referenced by:  islpir2  20435  lpirring  20436  lpirlnr  40858