Theorem islpir 19610

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 6433	. . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) = (LIdeal‘𝑅))
2		fveq2 6433	. . . 4 ⊢ (𝑟 = 𝑅 → (LPIdeal‘𝑟) = (LPIdeal‘𝑅))
3	1, 2	eqeq12d 2840	. . 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 2839	. . 3 ⊢ (𝑈 = 𝑃 ↔ (LIdeal‘𝑅) = (LPIdeal‘𝑅))
7	3, 6	syl6bbr 281	. 2 ⊢ (𝑟 = 𝑅 → ((LIdeal‘𝑟) = (LPIdeal‘𝑟) ↔ 𝑈 = 𝑃))
8		df-lpir 19605	. 2 ⊢ LPIR = {𝑟 ∈ Ring ∣ (LIdeal‘𝑟) = (LPIdeal‘𝑟)}
9	7, 8	elrab2 3589	1 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ‘cfv 6123  Ringcrg 18901  LIdealclidl 19531  LPIdealclpidl 19602  LPIRclpir 19603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-lpir 19605

This theorem is referenced by:  islpir2  19612  lpirring  19613  lpirlnr  38530