lpirlidllpi - Mathbox for Thierry Arnoux

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Theorem lpirlidllpi 33461

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 2741	. . . . 5 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
3		lpirlidllpi.2	. . . . 5 ⊢ 𝐼 = (LIdeal‘𝑅)
4	2, 3	islpir 21325	. . . 4 ⊢ (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
5	1, 4	sylib 220	. . 3 ⊢ (𝜑 → (𝑅 ∈ Ring ∧ 𝐼 = (LPIdeal‘𝑅)))
6	5	simpld 496	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		lpirlidllpi.5	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
8	5	simprd 497	. . 3 ⊢ (𝜑 → 𝐼 = (LPIdeal‘𝑅))
9	7, 8	eleqtrd 2843	. 2 ⊢ (𝜑 → 𝐽 ∈ (LPIdeal‘𝑅))
10		lpirlidllpi.3	. . . 4 ⊢ 𝐾 = (RSpan‘𝑅)
11		lpirlidllpi.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
12	2, 10, 11	islpidl 21322	. . 3 ⊢ (𝑅 ∈ Ring → (𝐽 ∈ (LPIdeal‘𝑅) ↔ ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥})))
13	12	biimpa 478	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ (LPIdeal‘𝑅)) → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))
14	6, 9, 13	syl2anc 591	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ∃wrex 3065 {csn 4558 ‘cfv 6489 Basecbs 17174 Ringcrg 20209 LIdealclidl 21203 RSpancrsp 21204 LPIdealclpidl 21317 LPIRclpir 21318

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-iota 6445 df-fun 6491 df-fv 6497 df-lpidl 21319 df-lpir 21320

This theorem is referenced by: pidufd 33638

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