Theorem zart0 33840

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 33837	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		sseq2 4022	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑥))
5		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6		ssidd 4019	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ⊆ 𝑥)
7	4, 5, 6	elrabd 3697	. . . . . . . . . 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 4021	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑗))
10	9	rabbidv 3441	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11	10	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
12		crngring 20263	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	12	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15		prmidlidl 33452	. . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . 14 ⊢ (PrmIdeal‘𝑅) ∈ V
18	17	rabex 5345	. . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝑖 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑗))
21	20	rabbidv 3441	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
22	21	eqcomd 2741	. . . . . . . . . . . . 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 5979	. . . . . . . . . . 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 2825	. . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 3614	. . . . . . . . . 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 4022	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑦))
33	32	elrab 3695	. . . . . . . . 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 4022	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑦))
37		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38		ssidd 4019	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ⊆ 𝑦)
39	36, 37, 38	elrabd 3697	. . . . . . . . . 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 33452	. . . . . . . . . . . . 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 5345	. . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑦 → (𝑖 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑗))
47	46	rabbidv 3441	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
48	47	eqcomd 2741	. . . . . . . . . . . . 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 5979	. . . . . . . . . . 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 2825	. . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 3614	. . . . . . . . . 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 4022	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑥))
59	58	elrab 3695	. . . . . . . . 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 4013	. . . . . 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 3200	. . 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 2735	. . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
68	1, 2, 67	zartopon 33838	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69		toponuni 22936	. . . 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 33836	. . . 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 33794	. 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 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ⊆ wss 3963  ∪ cuni 4912   ↦ cmpt 5231  ran crn 5690  ‘cfv 6563  TopOpenctopn 17468  Ringcrg 20251  CRingccrg 20252  LIdealclidl 21234  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  Kol2ct0 23330  PrmIdealcprmidl 33443  Speccrspec 33823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-rest 17469  df-topn 17470  df-0g 17488  df-mre 17631  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-lpidl 21350  df-top 22916  df-topon 22933  df-cld 23043  df-t0 23337  df-prmidl 33444  df-mxidl 33468  df-idlsrg 33509  df-rspec 33824

This theorem is referenced by: (None)