mxidlmaxv - Mathbox for Thierry Arnoux

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Theorem mxidlmaxv 33439

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 33436	. . 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 3926	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10	8	eldifbd 3927	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀)
11		nelne1 3022	. . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀)
13	12	neneqd 2930	. 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 2109 ≠ wne 2925 ∖ cdif 3911 ⊆ wss 3914 ‘cfv 6511 Basecbs 17179 Ringcrg 20142 LIdealclidl 21116 MaxIdealcmxidl 33430

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6464 df-fun 6513 df-fv 6519 df-mxidl 33431

This theorem is referenced by: mxidlirredi 33442 qsdrngilem 33465

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