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Theorem zart0 31837
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.
StepHypRef Expression
1 zartop.1 . . . 4 𝑆 = (Spec‘𝑅)
2 zartop.2 . . . 4 𝐽 = (TopOpen‘𝑆)
31, 2zartop 31834 . . 3 (𝑅 ∈ CRing → 𝐽 ∈ Top)
4 sseq2 3946 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑥𝑗𝑥𝑥))
5 simpr 485 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6 ssidd 3943 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥𝑥)
74, 5, 6elrabd 3625 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
87ad2antrr 723 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
9 sseq1 3945 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109rabbidv 3411 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
1110cbvmptv 5186 . . . . . . . . . . . 12 (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
12 crngring 19805 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1312ad2antrr 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 31627 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
1613, 14, 15syl2anc 584 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
17 fvex 6779 . . . . . . . . . . . . . 14 (PrmIdeal‘𝑅) ∈ V
1817rabex 5254 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V
1918a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V)
20 sseq1 3945 . . . . . . . . . . . . . . 15 (𝑖 = 𝑥 → (𝑖𝑗𝑥𝑗))
2120rabbidv 3411 . . . . . . . . . . . . . 14 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2221eqcomd 2744 . . . . . . . . . . . . 13 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2322adantl 482 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑥) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2411, 16, 19, 23elrnmptdv 5864 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
25 simpr 485 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2625eleq2d 2824 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2725eleq2d 2824 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2826, 27bibi12d 346 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → ((𝑥𝑑𝑦𝑑) ↔ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})))
2924, 28rspcdv 3550 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})))
3029imp 407 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
318, 30mpbid 231 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
32 sseq2 3946 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝑥𝑗𝑥𝑦))
3332elrab 3623 . . . . . . . . 9 (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥𝑦))
3433simprbi 497 . . . . . . . 8 (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} → 𝑥𝑦)
3531, 34syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥𝑦)
36 sseq2 3946 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑦𝑗𝑦𝑦))
37 simpr 485 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38 ssidd 3943 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦𝑦)
3936, 37, 38elrabd 3625 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4039ad4ant13 748 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
41 simpr 485 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42 prmidlidl 31627 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
4313, 41, 42syl2anc 584 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
4417rabex 5254 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V
4544a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V)
46 sseq1 3945 . . . . . . . . . . . . . . 15 (𝑖 = 𝑦 → (𝑖𝑗𝑦𝑗))
4746rabbidv 3411 . . . . . . . . . . . . . 14 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4847eqcomd 2744 . . . . . . . . . . . . 13 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
4948adantl 482 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑦) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
5011, 43, 45, 49elrnmptdv 5864 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
51 simpr 485 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
5251eleq2d 2824 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5351eleq2d 2824 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5452, 53bibi12d 346 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → ((𝑥𝑑𝑦𝑑) ↔ (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})))
5550, 54rspcdv 3550 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})))
5655imp 407 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5740, 56mpbird 256 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
58 sseq2 3946 . . . . . . . . . 10 (𝑗 = 𝑥 → (𝑦𝑗𝑦𝑥))
5958elrab 3623 . . . . . . . . 9 (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦𝑥))
6059simprbi 497 . . . . . . . 8 (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} → 𝑦𝑥)
6157, 60syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑦𝑥)
6235, 61eqssd 3937 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥 = 𝑦)
6362ex 413 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6463anasss 467 . . . 4 ((𝑅 ∈ CRing ∧ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ∈ (PrmIdeal‘𝑅))) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6564ralrimivva 3115 . . 3 (𝑅 ∈ CRing → ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
663, 65jca 512 . 2 (𝑅 ∈ CRing → (𝐽 ∈ Top ∧ ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦)))
67 eqid 2738 . . . . 5 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
681, 2, 67zartopon 31835 . . . 4 (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69 toponuni 22073 . . . 4 (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = 𝐽)
7068, 69syl 17 . . 3 (𝑅 ∈ CRing → (PrmIdeal‘𝑅) = 𝐽)
711, 2, 67, 11zartopn 31833 . . . 4 (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽)))
7271simprd 496 . . 3 (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽))
7370, 72ist0cld 31791 . 2 (𝑅 ∈ CRing → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))))
7466, 73mpbird 256 1 (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3429  wss 3886   cuni 4839  cmpt 5156  ran crn 5585  cfv 6426  TopOpenctopn 17142  Ringcrg 19793  CRingccrg 19794  LIdealclidl 20442  Topctop 22052  TopOnctopon 22069  Clsdccld 22177  Kol2ct0 22467  PrmIdealcprmidl 31618  Speccrspec 31820
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 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-ac2 10229  ax-cnex 10937  ax-resscn 10938  ax-1cn 10939  ax-icn 10940  ax-addcl 10941  ax-addrcl 10942  ax-mulcl 10943  ax-mulrcl 10944  ax-mulcom 10945  ax-addass 10946  ax-mulass 10947  ax-distr 10948  ax-i2m1 10949  ax-1ne0 10950  ax-1rid 10951  ax-rnegex 10952  ax-rrecex 10953  ax-cnre 10954  ax-pre-lttri 10955  ax-pre-lttrn 10956  ax-pre-ltadd 10957  ax-pre-mulgt0 10958
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-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  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 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-se 5540  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-isom 6435  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-rpss 7566  df-om 7703  df-1st 7820  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-1o 8284  df-oadd 8288  df-er 8485  df-en 8721  df-dom 8722  df-sdom 8723  df-fin 8724  df-dju 9669  df-card 9707  df-ac 9882  df-pnf 11021  df-mnf 11022  df-xr 11023  df-ltxr 11024  df-le 11025  df-sub 11217  df-neg 11218  df-nn 11984  df-2 12046  df-3 12047  df-4 12048  df-5 12049  df-6 12050  df-7 12051  df-8 12052  df-9 12053  df-n0 12244  df-z 12330  df-dec 12448  df-uz 12593  df-fz 13250  df-struct 16858  df-sets 16875  df-slot 16893  df-ndx 16905  df-base 16923  df-ress 16952  df-plusg 16985  df-mulr 16986  df-sca 16988  df-vsca 16989  df-ip 16990  df-tset 16991  df-ple 16992  df-rest 17143  df-topn 17144  df-0g 17162  df-mre 17305  df-mgm 18336  df-sgrp 18385  df-mnd 18396  df-submnd 18441  df-grp 18590  df-minusg 18591  df-sbg 18592  df-subg 18762  df-cntz 18933  df-lsm 19251  df-cmn 19398  df-abl 19399  df-mgp 19731  df-ur 19748  df-ring 19795  df-cring 19796  df-subrg 20032  df-lmod 20135  df-lss 20204  df-lsp 20244  df-sra 20444  df-rgmod 20445  df-lidl 20446  df-rsp 20447  df-lpidl 20524  df-top 22053  df-topon 22070  df-cld 22180  df-t0 22474  df-prmidl 31619  df-mxidl 31640  df-idlsrg 31654  df-rspec 31821
This theorem is referenced by: (None)
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