Theorem zarclssn 34036

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 20220	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	ad2antrr 727	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ Ring)
3		simplr 769	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ 𝐵)
4		zarclssn.1	. . . . 5 ⊢ 𝐵 = (LIdeal‘𝑅)
5	3, 4	eleqtrdi 2847	. . . 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 4720	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} ≠ ∅)
8	6, 7	eqnetrrd 3001	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) ≠ ∅)
9		simpll 767	. . . . . 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 2737	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	zarcls1 34032	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅)))
13	12	necon3bid 2977	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
14	9, 5, 13	syl2anc 585	. . . . 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 735	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
20	19	mxidlprm 33548	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
21	17, 18, 20	syl2anc 585	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . 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 3948	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
26	25	rabbidv 3397	. . . . . . . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (PrmIdeal‘𝑅) ∈ V
29	28	rabex 5277	. . . . . . . . . . . . . . . . . . 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 6950	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
32	6, 31	eqtr2d 2773	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
33		rabeqsn 4612	. . . . . . . . . . . . . . . 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 735	. . . . . . . . . . . . . 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 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ V
37		eleq1w 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
38		sseq2 3949	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚))
39	37, 38	anbi12d 633	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚)))
40		eqeq1 2741	. . . . . . . . . . . . . . . 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 3548	. . . . . . . . . . . . . 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 713	. . . . . . . . . . . 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 3959	. . . . . . . . . . 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 3940	. . . . . . . . . 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 735	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring)
48		simpllr 776	. . . . . . . . . . 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 2940	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
51	11	ssmxidl 33552	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚)
52	47, 48, 50, 51	syl3anc 1374	. . . . . . . . . 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 3146	. . . . . . . . 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 864	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀))
56	55	orcomd 872	. . . . . 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 3130	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))
59	5, 15, 58	3jca 1129	. . 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 33540	. . . 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 585	. 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 33541	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
66	1, 65	sylan 581	. . . . 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 6950	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
69	1	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑅 ∈ Ring)
70		simplr 769	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅))
71		simprl 771	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅))
72		prmidlidl 33522	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
73	69, 71, 72	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (LIdeal‘𝑅))
74		simprr 773	. . . . . . . . . 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 33543	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
77	69, 70, 75, 76	syl21anc 838	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))
78		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
79	11, 78	prmidlnr 33517	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
80	69, 71, 79	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ≠ (Base‘𝑅))
81	80	neneqd 2938	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → ¬ 𝑗 = (Base‘𝑅))
82	77, 81	olcnd 878	. . . . . . 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 33548	. . . . . . . . . 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 2837	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
87		ssidd 3946	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ⊆ 𝑗)
88	83, 87	eqsstrrd 3958	. . . . . . . 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 801	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀))
91	90	alrimiv 1929	. . . . 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 2773	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
94	93	adantlr 716	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀))
95	62, 94	impbida 801	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  {csn 4568   ↦ cmpt 5167  ‘cfv 6493  Basecbs 17173  ._rcmulr 17215  LSSumclsm 19603  mulGrpcmgp 20115  Ringcrg 20208  CRingccrg 20209  LIdealclidl 21199  PrmIdealcprmidl 33513  MaxIdealcmxidl 33537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-ac2 10379  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-rpss 7671  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9819  df-card 9857  df-ac 10032  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-0g 17398  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-subg 19093  df-cntz 19286  df-lsm 19605  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-subrg 20541  df-lmod 20851  df-lss 20921  df-lsp 20961  df-sra 21163  df-rgmod 21164  df-lidl 21201  df-rsp 21202  df-lpidl 21315  df-prmidl 33514  df-mxidl 33538

This theorem is referenced by:  zarmxt1  34043