Theorem zart0 33869

Description: The Zariski topology is T₀ . Corollary 1.1.8 of [EGA] p. 81. (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)

Assertion

Ref	Expression
zart0	⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)

Proof of Theorem zart0

Dummy variables 𝑖 𝑗 𝑘 𝑑 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zartop.1	. . . 4 ⊢ 𝑆 = (Spec‘𝑅)
2		zartop.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑆)
3	1, 2	zartop 33866	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		sseq2 3973	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑥))
5		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6		ssidd 3970	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ⊆ 𝑥)
7	4, 5, 6	elrabd 3661	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
9		sseq1 3972	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑗))
10	9	rabbidv 3413	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11	10	cbvmptv 5211	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
12		crngring 20154	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	12	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15		prmidlidl 33415	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
16	13, 14, 15	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
17		fvex 6871	. . . . . . . . . . . . . 14 ⊢ (PrmIdeal‘𝑅) ∈ V
18	17	rabex 5294	. . . . . . . . . . . . 13 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ∈ V)
20		sseq1 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝑖 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑗))
21	20	rabbidv 3413	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
22	21	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑥) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
24	11, 16, 19, 23	elrnmptdv 5929	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}))
25		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
26	25	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → (𝑥 ∈ 𝑑 ↔ 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}))
27	25	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → (𝑦 ∈ 𝑑 ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}))
28	26, 27	bibi12d 345	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → ((𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) ↔ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})))
29	24, 28	rspcdv 3580	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})))
30	29	imp 406	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}))
31	8, 30	mpbid 232	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
32		sseq2 3973	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑦))
33	32	elrab 3659	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥 ⊆ 𝑦))
34	33	simprbi 496	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} → 𝑥 ⊆ 𝑦)
35	31, 34	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑥 ⊆ 𝑦)
36		sseq2 3973	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑦))
37		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38		ssidd 3970	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ⊆ 𝑦)
39	36, 37, 38	elrabd 3661	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
40	39	ad4ant13 751	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
41		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42		prmidlidl 33415	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
43	13, 41, 42	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
44	17	rabex 5294	. . . . . . . . . . . . 13 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ∈ V
45	44	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ∈ V)
46		sseq1 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑦 → (𝑖 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑗))
47	46	rabbidv 3413	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
48	47	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
49	48	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑦) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
50	11, 43, 45, 49	elrnmptdv 5929	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}))
51		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
52	51	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → (𝑥 ∈ 𝑑 ↔ 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}))
53	51	eleq2d 2814	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → (𝑦 ∈ 𝑑 ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}))
54	52, 53	bibi12d 345	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → ((𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) ↔ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})))
55	50, 54	rspcdv 3580	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})))
56	55	imp 406	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}))
57	40, 56	mpbird 257	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
58		sseq2 3973	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑥))
59	58	elrab 3659	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ⊆ 𝑥))
60	59	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} → 𝑦 ⊆ 𝑥)
61	57, 60	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑦 ⊆ 𝑥)
62	35, 61	eqssd 3964	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) → 𝑥 = 𝑦)
63	62	ex 412	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))
64	63	anasss 466	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ∈ (PrmIdeal‘𝑅))) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))
65	64	ralrimivva 3180	. . 3 ⊢ (𝑅 ∈ CRing → ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))
66	3, 65	jca 511	. 2 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ Top ∧ ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦)))
67		eqid 2729	. . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
68	1, 2, 67	zartopon 33867	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69		toponuni 22801	. . . 4 ⊢ (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = ∪ 𝐽)
70	68, 69	syl 17	. . 3 ⊢ (𝑅 ∈ CRing → (PrmIdeal‘𝑅) = ∪ 𝐽)
71	1, 2, 67, 11	zartopn 33865	. . . 4 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽)))
72	71	simprd 495	. . 3 ⊢ (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽))
73	70, 72	ist0cld 33823	. 2 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗})(𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))))
74	66, 73	mpbird 257	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ⊆ wss 3914  ∪ cuni 4871   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  TopOpenctopn 17384  Ringcrg 20142  CRingccrg 20143  LIdealclidl 21116  Topctop 22780  TopOnctopon 22797  Clsdccld 22903  Kol2ct0 23193  PrmIdealcprmidl 33406  Speccrspec 33852

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-ac2 10416  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-rpss 7699  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-ac 10069  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  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-tset 17239  df-ple 17240  df-rest 17385  df-topn 17386  df-0g 17404  df-mre 17547  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-cntz 19249  df-lsm 19566  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-subrg 20479  df-lmod 20768  df-lss 20838  df-lsp 20878  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-rsp 21119  df-lpidl 21232  df-top 22781  df-topon 22798  df-cld 22906  df-t0 23200  df-prmidl 33407  df-mxidl 33431  df-idlsrg 33472  df-rspec 33853

This theorem is referenced by: (None)