Theorem zarclssn 34384

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 20385	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad2antrr 739	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ Ring)
3		simplr 781	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ 𝐵)
4		zarclssn.1	. . . . 5 ⊢ 𝐵 = (LIdeal‘𝑅)
5	3, 4	eleqtrdi 2870	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ (LIdeal‘𝑅))
6		simpr 490	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} = (𝑉‘𝑀))
7	3	snn0d 4736	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} ≠ ∅)
8	6, 7	eqnetrrd 3023	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) ≠ ∅)
9		simpll 779	. . . . . 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 2760	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	zarcls1 34380	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅)))
13	12	necon3bid 2999	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
14	9, 5, 13	syl2anc 596	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
15	8, 14	mpbid 235	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ≠ (Base‘𝑅))
16		simpr 490	. . . . . . . . . . . 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 747	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
20	19	mxidlprm 33874	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
21	17, 18, 20	syl2anc 596	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
26	25	rabbidv 3419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑀 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
27	26	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
28		fvex 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (PrmIdeal‘𝑅) ∈ V
29	28	rabex 5303	. . . . . . . . . . . . . . . . . . 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 6995	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
32	6, 31	eqtr2d 2796	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
33		rabeqsn 4628	. . . . . . . . . . . . . . . 16 ⊢ ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀} ↔ ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
34	32, 33	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
35	34	ad5antr 747	. . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ V
37		eleq1w 2843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
38		sseq2 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚))
39	37, 38	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚)))
40		eqeq1 2764	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → (𝑗 = 𝑀 ↔ 𝑚 = 𝑀))
41	39, 40	bibi12d 348	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑚 → (((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) ↔ ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀)))
42	36, 41	spcv 3559	. . . . . . . . . . . . . 14 ⊢ (∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀))
43	35, 42	syl 18	. . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 3967	. . . . . . . . . . 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 3948	. . . . . . . . . 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 747	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring)
48		simpllr 788	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
49		simpr 490	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ¬ 𝑗 = (Base‘𝑅))
50	49	neqned 2962	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
51	11	ssmxidl 33878	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚)
52	47, 48, 50, 51	syl3anc 1398	. . . . . . . . . 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 3170	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 = 𝑀)
54	53	ex 418	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = (Base‘𝑅) → 𝑗 = 𝑀))
55	54	orrd 877	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀))
56	55	orcomd 885	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
57	56	ex 418	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
58	57	ralrimiva 3154	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
59	5, 15, 58	3jca 1146	. . 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 33866	. . . 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 483	. . 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 596	. 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 487	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
65	11	mxidlidl 33867	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
66	1, 65	sylan 592	. . . . 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 6995	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
69	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑅 ∈ Ring)
70		simplr 781	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅))
71		simprl 783	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅))
72		prmidlidl 21533	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
73	69, 71, 72	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (LIdeal‘𝑅))
74		simprr 785	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ⊆ 𝑗)
75	73, 74	jca 521	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗))
76	11	mxidlmax 33869	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
77	69, 70, 75, 76	syl21anc 851	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
78		eqid 2760	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
79	11, 78	prmidlnr 21528	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
80	69, 71, 79	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ≠ (Base‘𝑅))
81	80	neneqd 2960	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → ¬ 𝑗 = (Base‘𝑅))
82	77, 81	olcnd 891	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 = 𝑀)
83		simpr 490	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 = 𝑀)
84	19	mxidlprm 33874	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
85	84	adantr 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ∈ (PrmIdeal‘𝑅))
86	83, 85	eqeltrd 2860	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
87		ssidd 3954	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ⊆ 𝑗)
88	83, 87	eqsstrrd 3966	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗)
89	86, 88	jca 521	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗))
90	82, 89	impbida 813	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
91	90	alrimiv 1960	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
92	91, 33	sylibr 237	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
93	68, 92	eqtr2d 2796	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
94	93	adantlr 728	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
95	62, 94	impbida 813	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  {csn 4584   ↦ cmpt 5186  ‘cfv 6533  Basecbs 17302  ._rcmulr 17344  LSSumclsm 19762  mulGrpcmgp 20274  Ringcrg 20373  CRingccrg 20374  LIdealclidl 21394  PrmIdealcprmidl 21524  MaxIdealcmxidl 33863

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-ac2 10466  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-rpss 7725  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-oadd 8460  df-er 8697  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9907  df-card 9945  df-ac 10120  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-5 12331  df-6 12332  df-7 12333  df-8 12334  df-sets 17257  df-slot 17275  df-ndx 17287  df-base 17303  df-ress 17324  df-plusg 17356  df-mulr 17357  df-sca 17359  df-vsca 17360  df-ip 17361  df-0g 17527  df-mgm 18731  df-sgrp 18822  df-mnd 18838  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-subg 19247  df-cntz 19445  df-lsm 19764  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-cring 20376  df-subrg 20733  df-lmod 21047  df-lss 21117  df-lsp 21157  df-sra 21358  df-rgmod 21359  df-lidl 21396  df-rsp 21397  df-prmidl 21525  df-lpidl 21554  df-mxidl 33864

This theorem is used by:  zarmxt1  34391