Theorem zarclssn 34017

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 20226	. . . 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 2846	. . . 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 4719	. . . . . 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 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 2736	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 11	zarcls1 34013	. . . . . . 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 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 2736	. . . . . . . . . . . . . . 15 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
20	19	mxidlprm 33530	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗))
26	25	rabbidv 3396	. . . . . . . . . . . . . . . . . . 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 6853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (PrmIdeal‘𝑅) ∈ V
29	28	rabex 5280	. . . . . . . . . . . . . . . . . . 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 6955	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗})
32	6, 31	eqtr2d 2772	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀})
33		rabeqsn 4611	. . . . . . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑚 ∈ V
37		eleq1w 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
38		sseq2 3948	. . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 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 3547	. . . . . . . . . . . . . 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 3958	. . . . . . . . . . 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 3939	. . . . . . . . . 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 2939	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
51	11	ssmxidl 33534	. . . . . . . . . . 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 3145	. . . . . . . . 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 3129	. . . 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 33522	. . . 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 33523	. . . . . 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 6955	. . . 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 33504	. . . . . . . . . . 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 33525	. . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
79	11, 78	prmidlnr 33499	. . . . . . . . . 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 2937	. . . . . . . 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 33530	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
87		ssidd 3945	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ⊆ 𝑗)
88	83, 87	eqsstrrd 3957	. . . . . . . 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 2772	. . 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 2932  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  {csn 4567   ↦ cmpt 5166  ‘cfv 6498  Basecbs 17179  ._rcmulr 17221  LSSumclsm 19609  mulGrpcmgp 20121  Ringcrg 20214  CRingccrg 20215  LIdealclidl 21204  PrmIdealcprmidl 33495  MaxIdealcmxidl 33519

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-cntz 19292  df-lsm 19611  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-sra 21168  df-rgmod 21169  df-lidl 21206  df-rsp 21207  df-lpidl 21320  df-prmidl 33496  df-mxidl 33520

This theorem is referenced by:  zarmxt1  34024