mxidlmaxv - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mxidlmaxv		Structured version Visualization version GIF version

Theorem mxidlmaxv 33496

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 33493	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
7	1, 2, 3, 4, 6	syl22anc 839	. 2 ⊢ (𝜑 → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
8		mxidlmaxv.6	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐼 ∖ 𝑀))
9	8	eldifad 3963	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
10	8	eldifbd 3964	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑀)
11		nelne1 3039	. . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝑀) → 𝐼 ≠ 𝑀)
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐼 ≠ 𝑀)
13	12	neneqd 2945	. 2 ⊢ (𝜑 → ¬ 𝐼 = 𝑀)
14	7, 13	orcnd 879	1 ⊢ (𝜑 → 𝐼 = 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ‘cfv 6561 Basecbs 17247 Ringcrg 20230 LIdealclidl 21216 MaxIdealcmxidl 33487

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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-mxidl 33488

This theorem is referenced by: mxidlirredi 33499 qsdrngilem 33522

W3C validator