Theorem zart0 31829

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 31826	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		sseq2 3947	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑥 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑥))
5		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6		ssidd 3944	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ⊆ 𝑥)
7	4, 5, 6	elrabd 3626	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
8	7	ad2antrr 723	. . . . . . . . 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 3946	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑗))
10	9	rabbidv 3414	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11	10	cbvmptv 5187	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
12		crngring 19795	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
13	12	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15		prmidlidl 31619	. . . . . . . . . . . . 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 6787	. . . . . . . . . . . . . 14 ⊢ (PrmIdeal‘𝑅) ∈ V
18	17	rabex 5256	. . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝑖 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑗))
21	20	rabbidv 3414	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
22	21	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
23	22	adantl 482	. . . . . . . . . . . 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 5871	. . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗})
26	25	eleq2d 2824	. . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 346	. . . . . . . . . . 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 3553	. . . . . . . . . 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 407	. . . . . . . . 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 231	. . . . . . . 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 3947	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑥 ⊆ 𝑗 ↔ 𝑥 ⊆ 𝑦))
33	32	elrab 3624	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥 ⊆ 𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥 ⊆ 𝑦))
34	33	simprbi 497	. . . . . . . 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 3947	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑦 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑦))
37		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38		ssidd 3944	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ⊆ 𝑦)
39	36, 37, 38	elrabd 3626	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
40	39	ad4ant13 748	. . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42		prmidlidl 31619	. . . . . . . . . . . . 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 5256	. . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑦 → (𝑖 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑗))
47	46	rabbidv 3414	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
48	47	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
49	48	adantl 482	. . . . . . . . . . . 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 5871	. . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗})
52	51	eleq2d 2824	. . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 346	. . . . . . . . . . 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 3553	. . . . . . . . . 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 407	. . . . . . . . 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 256	. . . . . . . 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 3947	. . . . . . . . . 10 ⊢ (𝑗 = 𝑥 → (𝑦 ⊆ 𝑗 ↔ 𝑦 ⊆ 𝑥))
59	58	elrab 3624	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦 ⊆ 𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ⊆ 𝑥))
60	59	simprbi 497	. . . . . . . 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 3938	. . . . . 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 413	. . . . 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 467	. . . 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 3123	. . 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 512	. 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 2738	. . . . 5 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
68	1, 2, 67	zartopon 31827	. . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69		toponuni 22063	. . . 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 31825	. . . 4 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽)))
72	71	simprd 496	. . 3 ⊢ (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑗}) = (Clsd‘𝐽))
73	70, 72	ist0cld 31783	. 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 256	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839   ↦ cmpt 5157  ran crn 5590  ‘cfv 6433  TopOpenctopn 17132  Ringcrg 19783  CRingccrg 19784  LIdealclidl 20432  Topctop 22042  TopOnctopon 22059  Clsdccld 22167  Kol2ct0 22457  PrmIdealcprmidl 31610  Speccrspec 31812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-rpss 7576  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-rest 17133  df-topn 17134  df-0g 17152  df-mre 17295  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-subrg 20022  df-lmod 20125  df-lss 20194  df-lsp 20234  df-sra 20434  df-rgmod 20435  df-lidl 20436  df-rsp 20437  df-lpidl 20514  df-top 22043  df-topon 22060  df-cld 22170  df-t0 22464  df-prmidl 31611  df-mxidl 31632  df-idlsrg 31646  df-rspec 31813

This theorem is referenced by: (None)