mxidlmaxv - Mathbox for Thierry Arnoux

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Theorem mxidlmaxv 32542

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 32539	. . 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 3959	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10	8	eldifbd 3960	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀)
11		nelne1 3040	. . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀)
12	9, 10, 11	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀)
13	12	neneqd 2946	. 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 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3944 ⊆ wss 3947 ‘cfv 6540 Basecbs 17140 Ringcrg 20047 LIdealclidl 20771 MaxIdealcmxidl 32533

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-mxidl 32534

This theorem is referenced by: qsdrngilem 32561

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