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Theorem zart0 31232
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 31229 . . 3 (𝑅 ∈ CRing → 𝐽 ∈ Top)
4 sseq2 3944 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑥𝑗𝑥𝑥))
5 simpr 488 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6 ssidd 3941 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥𝑥)
74, 5, 6elrabd 3633 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
87ad2antrr 725 . . . . . . . . 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 3943 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109rabbidv 3430 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
1110cbvmptv 5136 . . . . . . . . . . . 12 (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
12 crngring 19305 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1312ad2antrr 725 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14 simplr 768 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15 prmidlidl 31027 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
1613, 14, 15syl2anc 587 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (LIdeal‘𝑅))
17 fvex 6662 . . . . . . . . . . . . . 14 (PrmIdeal‘𝑅) ∈ V
1817rabex 5202 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V
1918a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V)
20 sseq1 3943 . . . . . . . . . . . . . . 15 (𝑖 = 𝑥 → (𝑖𝑗𝑥𝑗))
2120rabbidv 3430 . . . . . . . . . . . . . 14 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2221eqcomd 2807 . . . . . . . . . . . . 13 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2322adantl 485 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑥) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2411, 16, 19, 23elrnmptdv 5802 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
25 simpr 488 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2625eleq2d 2878 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2725eleq2d 2878 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2826, 27bibi12d 349 . . . . . . . . . . 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 3566 . . . . . . . . . 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 410 . . . . . . . . 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 235 . . . . . . . 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 3944 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝑥𝑗𝑥𝑦))
3332elrab 3631 . . . . . . . . 9 (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥𝑦))
3433simprbi 500 . . . . . . . 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 3944 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑦𝑗𝑦𝑦))
37 simpr 488 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38 ssidd 3941 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦𝑦)
3936, 37, 38elrabd 3633 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4039ad4ant13 750 . . . . . . . . 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 488 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42 prmidlidl 31027 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
4313, 41, 42syl2anc 587 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (LIdeal‘𝑅))
4417rabex 5202 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V
4544a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V)
46 sseq1 3943 . . . . . . . . . . . . . . 15 (𝑖 = 𝑦 → (𝑖𝑗𝑦𝑗))
4746rabbidv 3430 . . . . . . . . . . . . . 14 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4847eqcomd 2807 . . . . . . . . . . . . 13 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
4948adantl 485 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑦) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
5011, 43, 45, 49elrnmptdv 5802 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
51 simpr 488 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
5251eleq2d 2878 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5351eleq2d 2878 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5452, 53bibi12d 349 . . . . . . . . . . 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 3566 . . . . . . . . . 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 410 . . . . . . . . 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 260 . . . . . . . 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 3944 . . . . . . . . . 10 (𝑗 = 𝑥 → (𝑦𝑗𝑦𝑥))
5958elrab 3631 . . . . . . . . 9 (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦𝑥))
6059simprbi 500 . . . . . . . 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 3935 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥 = 𝑦)
6362ex 416 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6463anasss 470 . . . 4 ((𝑅 ∈ CRing ∧ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ∈ (PrmIdeal‘𝑅))) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6564ralrimivva 3159 . . 3 (𝑅 ∈ CRing → ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
663, 65jca 515 . 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 2801 . . . . 5 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
681, 2, 67zartopon 31230 . . . 4 (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69 toponuni 21522 . . . 4 (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = 𝐽)
7068, 69syl 17 . . 3 (𝑅 ∈ CRing → (PrmIdeal‘𝑅) = 𝐽)
711, 2, 67, 11zartopn 31228 . . . 4 (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽)))
7271simprd 499 . . 3 (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽))
7370, 72ist0cld 31186 . 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 260 1 (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  Vcvv 3444  wss 3884   cuni 4803  cmpt 5113  ran crn 5524  cfv 6328  TopOpenctopn 16690  Ringcrg 19293  CRingccrg 19294  LIdealclidl 19938  Topctop 21501  TopOnctopon 21518  Clsdccld 21624  Kol2ct0 21914  PrmIdealcprmidl 31018  Speccrspec 31215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-ac2 9878  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rpss 7433  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-ac 9531  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-rest 16691  df-topn 16692  df-0g 16710  df-mre 16852  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18271  df-cntz 18442  df-lsm 18756  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-rsp 19943  df-lpidl 20012  df-top 21502  df-topon 21519  df-cld 21627  df-t0 21921  df-prmidl 31019  df-mxidl 31040  df-idlsrg 31054  df-rspec 31216
This theorem is referenced by: (None)
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