Theorem zart0 34557

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 34554	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		sseq2 3934	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑥))
5		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6		ssidd 3931	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ⊆ 𝑥)
7	4, 5, 6	elrabd 3646	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
8	7	ad2antrr 739	. . . . . . . . 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 3933	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑗))
10	9	rabbidv 3419	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11	10	cbvmptv 5208	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
12		crngring 20477	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	12	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15		prmidlidl 21630	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
16	13, 14, 15	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
17		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (PrmIdeal‘𝑅) ∈ V
18	17	rabex 5299	. . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝑖 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑗))
21	20	rabbidv 3419	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
22	21	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
23	22	adantl 487	. . . . . . . . . . . 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 5946	. . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
26	25	eleq2d 2846	. . . . . . . . . . . 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 2846	. . . . . . . . . . . 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 348	. . . . . . . . . . 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 3568	. . . . . . . . . 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 412	. . . . . . . . 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 235	. . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑦))
33	32	elrab 3644	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥 ⊆ 𝑦))
34	33	simprbi 503	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} → 𝑥 ⊆ 𝑦)
35	31, 34	syl 18	. . . . . . 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 3934	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑦))
37		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38		ssidd 3931	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ⊆ 𝑦)
39	36, 37, 38	elrabd 3646	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
40	39	ad4ant13 764	. . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42		prmidlidl 21630	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
43	13, 41, 42	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
44	17	rabex 5299	. . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑦 → (𝑖 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑗))
47	46	rabbidv 3419	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
48	47	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
49	48	adantl 487	. . . . . . . . . . . 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 5946	. . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
52	51	eleq2d 2846	. . . . . . . . . . . 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 2846	. . . . . . . . . . . 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 348	. . . . . . . . . . 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 3568	. . . . . . . . . 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 412	. . . . . . . . 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 260	. . . . . . . 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 3934	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑥))
59	58	elrab 3644	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ⊆ 𝑥))
60	59	simprbi 503	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} → 𝑦 ⊆ 𝑥)
61	57, 60	syl 18	. . . . . . 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 3925	. . . . . 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 418	. . . . 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 472	. . . 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 3205	. . 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 521	. 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 2760	. . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
68	1, 2, 67	zartopon 34555	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69		toponuni 23237	. . . 4 ⊢ (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = ∪ 𝐽)
70	68, 69	syl 18	. . 3 ⊢ (𝑅 ∈ CRing → (PrmIdeal‘𝑅) = ∪ 𝐽)
71	1, 2, 67, 11	zartopn 34553	. . . 4 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽)))
72	71	simprd 501	. . 3 ⊢ (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽))
73	70, 72	ist0cld 34511	. 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 260	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895  ∪ cuni 4866   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  TopOpenctopn 17597  Ringcrg 20464  CRingccrg 20465  LIdealclidl 21489  PrmIdealcprmidl 21621  Topctop 23216  TopOnctopon 23233  Clsdccld 23339  Kol2ct0 23629  Speccrspec 34540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-ac2 10544  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-oadd 8483  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-dju 9985  df-card 10023  df-ac 10198  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-struct 17330  df-sets 17347  df-slot 17365  df-ndx 17377  df-base 17393  df-ress 17414  df-plusg 17446  df-mulr 17447  df-sca 17449  df-vsca 17450  df-ip 17451  df-tset 17452  df-ple 17453  df-rest 17598  df-topn 17599  df-0g 17617  df-mre 17761  df-mgm 18821  df-sgrp 18913  df-mnd 18929  df-submnd 18984  df-grp 19152  df-minusg 19153  df-sbg 19154  df-subg 19338  df-cntz 19536  df-lsm 19855  df-cmn 20001  df-abl 20002  df-mgp 20366  df-rng 20380  df-ur 20413  df-ring 20466  df-cring 20467  df-subrg 20827  df-lmod 21142  df-lss 21212  df-lsp 21252  df-sra 21453  df-rgmod 21454  df-lidl 21491  df-rsp 21492  df-prmidl 21622  df-lpidl 21651  df-top 23217  df-topon 23234  df-cld 23342  df-t0 23636  df-mxidl 34031  df-idlsrg 34079  df-rspec 34541

This theorem is used by: (None)