mxidlmaxv - Mathbox for Thierry Arnoux

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Theorem mxidlmaxv 33483

Description: An ideal 𝐼 strictly containing a maximal ideal 𝑀 is the whole ring 𝐵. (Contributed by Thierry Arnoux, 9-Mar-2025.)

**Hypotheses**
Ref	Expression
mxidlmaxv.1	⊢ 𝐵 = (Base‘𝑅)
mxidlmaxv.2	⊢ (𝜑 → 𝑅 ∈ Ring)
mxidlmaxv.3	⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
mxidlmaxv.4	⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
mxidlmaxv.5	⊢ (𝜑 → 𝑀 ⊆ 𝐼)
mxidlmaxv.6	⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀))

**Assertion**
Ref	Expression
mxidlmaxv	⊢ (𝜑 → 𝐼 = 𝐵)

**Proof of Theorem mxidlmaxv**
Step	Hyp	Ref	Expression
1		mxidlmaxv.2	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		mxidlmaxv.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅))
3		mxidlmaxv.4	. . 3 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
4		mxidlmaxv.5	. . 3 ⊢ (𝜑 → 𝑀 ⊆ 𝐼)
5		mxidlmaxv.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
6	5	mxidlmax 33480	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
7	1, 2, 3, 4, 6	syl22anc 838	. 2 ⊢ (𝜑 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
8		mxidlmaxv.6	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀))
9	8	eldifad 3938	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10	8	eldifbd 3939	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀)
11		nelne1 3029	. . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀)
13	12	neneqd 2937	. 2 ⊢ (𝜑 → ¬ 𝐼 = 𝑀)
14	7, 13	orcnd 878	1 ⊢ (𝜑 → 𝐼 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 ⊆ wss 3926 ‘cfv 6531 Basecbs 17228 Ringcrg 20193 LIdealclidl 21167 MaxIdealcmxidl 33474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-iota 6484 df-fun 6533 df-fv 6539 df-mxidl 33475

This theorem is referenced by: mxidlirredi 33486 qsdrngilem 33509

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