Theorem zarclssn 33904

Description: The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zarclsx.1	⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
zarclssn.1	⊢ 𝐵 = (LIdeal‘𝑅)

Assertion

Ref	Expression
zarclssn	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))

Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝑉   𝐵,𝑖,𝑗   𝑖,𝑀,𝑗   𝑗,𝑉

Proof of Theorem zarclssn

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20205	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad2antrr 726	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ Ring)
3		simplr 768	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ 𝐵)
4		zarclssn.1	. . . . 5 ⊢ 𝐵 = (LIdeal‘𝑅)
5	3, 4	eleqtrdi 2844	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ (LIdeal‘𝑅))
6		simpr 484	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} = (𝑉‘𝑀))
7	3	snn0d 4751	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} ≠ ∅)
8	6, 7	eqnetrrd 3000	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) ≠ ∅)
9		simpll 766	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ CRing)
10		zarclsx.1	. . . . . . . 8 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	zarcls1 33900	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅)))
13	12	necon3bid 2976	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
14	9, 5, 13	syl2anc 584	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
15	8, 14	mpbid 232	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ≠ (Base‘𝑅))
16		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 ⊆ 𝑚)
17	9	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑅 ∈ CRing)
18		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
19		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
20	19	mxidlprm 33485	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
21	17, 18, 20	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 ∈ (PrmIdeal‘𝑅))
22		simp-4r 783	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑀 ⊆ 𝑗)
23	22, 16	sstrd 3969	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑀 ⊆ 𝑚)
24	10	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
25		sseq1 3984	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
26	25	rabbidv 3423	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑀 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
27	26	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
28		fvex 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (PrmIdeal‘𝑅) ∈ V
29	28	rabex 5309	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V
30	29	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V)
31	24, 27, 5, 30	fvmptd 6993	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
32	6, 31	eqtr2d 2771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
33		rabeqsn 4643	. . . . . . . . . . . . . . . 16 ⊢ ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀} ↔ ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
34	32, 33	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
35	34	ad5antr 734	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
36		vex 3463	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ V
37		eleq1w 2817	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
38		sseq2 3985	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚))
39	37, 38	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚)))
40		eqeq1 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → (𝑗 = 𝑀 ↔ 𝑚 = 𝑀))
41	39, 40	bibi12d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) ↔ ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀)))
42	36, 41	spcv 3584	. . . . . . . . . . . . . 14 ⊢ (∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀))
43	35, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀))
44	21, 23, 43	mpbi2and 712	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 = 𝑀)
45	16, 44	sseqtrd 3995	. . . . . . . . . . 11 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 ⊆ 𝑀)
46	45, 22	eqssd 3976	. . . . . . . . . 10 ⊢ ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 = 𝑀)
47	1	ad5antr 734	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring)
48		simpllr 775	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
49		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ¬ 𝑗 = (Base‘𝑅))
50	49	neqned 2939	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
51	11	ssmxidl 33489	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚)
52	47, 48, 50, 51	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚)
53	46, 52	r19.29a 3148	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 = 𝑀)
54	53	ex 412	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = (Base‘𝑅) → 𝑗 = 𝑀))
55	54	orrd 863	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀))
56	55	orcomd 871	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
57	56	ex 412	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
58	57	ralrimiva 3132	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
59	5, 15, 58	3jca 1128	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))))
60	11	ismxidl 33477	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))))
61	60	biimpar 477	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅))
62	2, 59, 61	syl2anc 584	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
63	10	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
64	26	adantl 481	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
65	11	mxidlidl 33478	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
66	1, 65	sylan 580	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
67	29	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V)
68	63, 64, 66, 67	fvmptd 6993	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
69	1	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑅 ∈ Ring)
70		simplr 768	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅))
71		simprl 770	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅))
72		prmidlidl 33459	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
73	69, 71, 72	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (LIdeal‘𝑅))
74		simprr 772	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ⊆ 𝑗)
75	73, 74	jca 511	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗))
76	11	mxidlmax 33480	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
77	69, 70, 75, 76	syl21anc 837	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
78		eqid 2735	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
79	11, 78	prmidlnr 33454	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
80	69, 71, 79	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ≠ (Base‘𝑅))
81	80	neneqd 2937	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → ¬ 𝑗 = (Base‘𝑅))
82	77, 81	olcnd 877	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 = 𝑀)
83		simpr 484	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 = 𝑀)
84	19	mxidlprm 33485	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
85	84	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ∈ (PrmIdeal‘𝑅))
86	83, 85	eqeltrd 2834	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
87		ssidd 3982	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ⊆ 𝑗)
88	83, 87	eqsstrrd 3994	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
89	86, 88	jca 511	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗))
90	82, 89	impbida 800	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
91	90	alrimiv 1927	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
92	91, 33	sylibr 234	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
93	68, 92	eqtr2d 2771	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
94	93	adantlr 715	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
95	62, 94	impbida 800	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  {csn 4601   ↦ cmpt 5201  ‘cfv 6531  Basecbs 17228  ._rcmulr 17272  LSSumclsm 19615  mulGrpcmgp 20100  Ringcrg 20193  CRingccrg 20194  LIdealclidl 21167  PrmIdealcprmidl 33450  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-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-ac2 10477  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rpss 7717  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-ac 10130  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-subg 19106  df-cntz 19300  df-lsm 19617  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-subrg 20530  df-lmod 20819  df-lss 20889  df-lsp 20929  df-sra 21131  df-rgmod 21132  df-lidl 21169  df-rsp 21170  df-lpidl 21283  df-prmidl 33451  df-mxidl 33475

This theorem is referenced by:  zarmxt1  33911