Theorem zart0 34137

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 34134	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑥))
5		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6		ssidd 3959	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ⊆ 𝑥)
7	4, 5, 6	elrabd 3652	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
8	7	ad2antrr 736	. . . . . . . . 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 3961	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑗))
10	9	rabbidv 3420	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11	10	cbvmptv 5203	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
12		crngring 20274	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	12	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14		simplr 778	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15		prmidlidl 33591	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
16	13, 14, 15	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
17		fvex 6876	. . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝑖 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑗))
21	20	rabbidv 3420	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
22	21	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
23	22	adantl 485	. . . . . . . . . . . 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 5939	. . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
26	25	eleq2d 2847	. . . . . . . . . . . 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 2847	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3573	. . . . . . . . . 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 410	. . . . . . . . 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 234	. . . . . . . 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 3962	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑦))
33	32	elrab 3650	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥 ⊆ 𝑦))
34	33	simprbi 501	. . . . . . . 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 3962	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑦))
37		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38		ssidd 3959	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ⊆ 𝑦)
39	36, 37, 38	elrabd 3652	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
40	39	ad4ant13 761	. . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42		prmidlidl 33591	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
43	13, 41, 42	syl2anc 593	. . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑦 → (𝑖 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑗))
47	46	rabbidv 3420	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
48	47	eqcomd 2767	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
49	48	adantl 485	. . . . . . . . . . . 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 5939	. . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
52	51	eleq2d 2847	. . . . . . . . . . . 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 2847	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3573	. . . . . . . . . 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 410	. . . . . . . . 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 259	. . . . . . . 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 3962	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑥))
59	58	elrab 3650	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ⊆ 𝑥))
60	59	simprbi 501	. . . . . . . 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 3953	. . . . . 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 416	. . . . 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 470	. . . 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 3204	. . 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 519	. 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 2761	. . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
68	1, 2, 67	zartopon 34135	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69		toponuni 22954	. . . 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 34133	. . . 4 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽)))
72	71	simprd 499	. . 3 ⊢ (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽))
73	70, 72	ist0cld 34091	. 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 259	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3904  ∪ cuni 4864   ↦ cmpt 5180  ran crn 5646  ‘cfv 6517  TopOpenctopn 17433  Ringcrg 20262  CRingccrg 20263  LIdealclidl 21256  Topctop 22933  TopOnctopon 22950  Clsdccld 23056  Kol2ct0 23346  PrmIdealcprmidl 33582  Speccrspec 34120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-ac2 10417  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-rpss 7702  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-oadd 8436  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-dju 9856  df-card 9894  df-ac 10069  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-rest 17434  df-topn 17435  df-0g 17453  df-mre 17597  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-submnd 18801  df-grp 18961  df-minusg 18962  df-sbg 18963  df-subg 19148  df-cntz 19340  df-lsm 19659  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-ring 20264  df-cring 20265  df-subrg 20599  df-lmod 20909  df-lss 20979  df-lsp 21019  df-sra 21220  df-rgmod 21221  df-lidl 21258  df-rsp 21259  df-lpidl 21372  df-top 22934  df-topon 22951  df-cld 23059  df-t0 23353  df-prmidl 33583  df-mxidl 33609  df-idlsrg 33658  df-rspec 34121

This theorem is referenced by: (None)