Theorem lpirlidllpi 33186

Description: In a principal ideal ring, ideals are principal. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypotheses

Ref	Expression
lpirlidllpi.1	⊢ 𝐵 = (Base‘𝑅)
lpirlidllpi.2	⊢ 𝐼 = (LIdeal‘𝑅)
lpirlidllpi.3	⊢ 𝐾 = (RSpan‘𝑅)
lpirlidllpi.4	⊢ (𝜑 → 𝑅 ∈ LPIR)
lpirlidllpi.5	⊢ (𝜑 → 𝐽 ∈ 𝐼)

Assertion

Ref	Expression
lpirlidllpi	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐽   𝑥,𝐾   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝐼(𝑥)

Proof of Theorem lpirlidllpi

Step	Hyp	Ref	Expression
1		lpirlidllpi.4	. . . 4 ⊢ (𝜑 → 𝑅 ∈ LPIR)
2		eqid 2725	. . . . 5 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
3		lpirlidllpi.2	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	islpir 21235	. . . 4 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
5	1, 4	sylib 217	. . 3 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
6	5	simpld 493	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		lpirlidllpi.5	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
8	5	simprd 494	. . 3 ⊢ (𝜑 → 𝐼 = (LPIdeal‘𝑅))
9	7, 8	eleqtrd 2827	. 2 ⊢ (𝜑 → 𝐽 ∈ (LPIdeal‘𝑅))
10		lpirlidllpi.3	. . . 4 ⊢ 𝐾 = (RSpan‘𝑅)
11		lpirlidllpi.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
12	2, 10, 11	islpidl 21232	. . 3 ⊢ (𝑅 ∈ Ring → (𝐽 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥})))
13	12	biimpa 475	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))
14	6, 9, 13	syl2anc 582	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  {csn 4630  ‘cfv 6549  Basecbs 17183  Ringcrg 20185  LIdealclidl 21114  RSpancrsp 21115  LPIdealclpidl 21227  LPIRclpir 21228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-lpidl 21229  df-lpir 21230

This theorem is referenced by:  pidufd  33358