Theorem lpirlidllpi 33999

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 2763	. . . . 5 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
3		lpirlidllpi.2	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	islpir 21682	. . . 4 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
5	1, 4	sylib 221	. . 3 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
6	5	simpld 500	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		lpirlidllpi.5	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
8	5	simprd 501	. . 3 ⊢ (𝜑 → 𝐼 = (LPIdeal‘𝑅))
9	7, 8	eleqtrd 2866	. 2 ⊢ (𝜑 → 𝐽 ∈ (LPIdeal‘𝑅))
10		lpirlidllpi.3	. . . 4 ⊢ 𝐾 = (RSpan‘𝑅)
11		lpirlidllpi.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
12	2, 10, 11	islpidl 21679	. . 3 ⊢ (𝑅 ∈ Ring → (𝐽 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥})))
13	12	biimpa 482	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))
14	6, 9, 13	syl2anc 596	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {csn 4588  ‘cfv 6558  Basecbs 17416  Ringcrg 20489  LIdealclidl 21514  RSpancrsp 21515  LPIdealclpidl 21674  LPIRclpir 21675

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fv 6566  df-lpidl 21676  df-lpir 21677

This theorem is used by:  pidufd  34145