Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  zart0 Structured version   Visualization version   GIF version

Theorem zart0 33878
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 33875 . . 3 (𝑅 ∈ CRing → 𝐽 ∈ Top)
4 sseq2 4010 . . . . . . . . . . 11 (𝑗 = 𝑥 → (𝑥𝑗𝑥𝑥))
5 simpr 484 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
6 ssidd 4007 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥𝑥)
74, 5, 6elrabd 3694 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
87ad2antrr 726 . . . . . . . . 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 4009 . . . . . . . . . . . . . 14 (𝑘 = 𝑖 → (𝑘𝑗𝑖𝑗))
109rabbidv 3444 . . . . . . . . . . . . 13 (𝑘 = 𝑖 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
1110cbvmptv 5255 . . . . . . . . . . . 12 (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
12 crngring 20242 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1312ad2antrr 726 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring)
14 simplr 769 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑥 ∈ (PrmIdeal‘𝑅))
15 prmidlidl 33472 . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . 14 (PrmIdeal‘𝑅) ∈ V
1817rabex 5339 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V
1918a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ V)
20 sseq1 4009 . . . . . . . . . . . . . . 15 (𝑖 = 𝑥 → (𝑖𝑗𝑥𝑗))
2120rabbidv 3444 . . . . . . . . . . . . . 14 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2221eqcomd 2743 . . . . . . . . . . . . 13 (𝑖 = 𝑥 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2322adantl 481 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑥) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2411, 16, 19, 23elrnmptdv 5976 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
25 simpr 484 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗})
2625eleq2d 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2725eleq2d 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗}))
2826, 27bibi12d 345 . . . . . . . . . . 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 3614 . . . . . . . . . 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 406 . . . . . . . . 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 232 . . . . . . . 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 4010 . . . . . . . . . 10 (𝑗 = 𝑦 → (𝑥𝑗𝑥𝑦))
3332elrab 3692 . . . . . . . . 9 (𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑥𝑗} ↔ (𝑦 ∈ (PrmIdeal‘𝑅) ∧ 𝑥𝑦))
3433simprbi 496 . . . . . . . 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 4010 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑦𝑗𝑦𝑦))
37 simpr 484 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
38 ssidd 4007 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦𝑦)
3936, 37, 38elrabd 3694 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4039ad4ant13 751 . . . . . . . . 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 484 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → 𝑦 ∈ (PrmIdeal‘𝑅))
42 prmidlidl 33472 . . . . . . . . . . . . 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 5339 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V
4544a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ V)
46 sseq1 4009 . . . . . . . . . . . . . . 15 (𝑖 = 𝑦 → (𝑖𝑗𝑦𝑗))
4746rabbidv 3444 . . . . . . . . . . . . . 14 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
4847eqcomd 2743 . . . . . . . . . . . . 13 (𝑖 = 𝑦 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
4948adantl 481 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑖 = 𝑦) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
5011, 43, 45, 49elrnmptdv 5976 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}))
51 simpr 484 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗})
5251eleq2d 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑥𝑑𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5351eleq2d 2827 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ 𝑑 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}) → (𝑦𝑑𝑦 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗}))
5452, 53bibi12d 345 . . . . . . . . . . 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 3614 . . . . . . . . . 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 406 . . . . . . . . 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 257 . . . . . . . 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 4010 . . . . . . . . . 10 (𝑗 = 𝑥 → (𝑦𝑗𝑦𝑥))
5958elrab 3692 . . . . . . . . 9 (𝑥 ∈ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑦𝑗} ↔ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦𝑥))
6059simprbi 496 . . . . . . . 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 4001 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) ∧ ∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑)) → 𝑥 = 𝑦)
6362ex 412 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑥 ∈ (PrmIdeal‘𝑅)) ∧ 𝑦 ∈ (PrmIdeal‘𝑅)) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6463anasss 466 . . . 4 ((𝑅 ∈ CRing ∧ (𝑥 ∈ (PrmIdeal‘𝑅) ∧ 𝑦 ∈ (PrmIdeal‘𝑅))) → (∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
6564ralrimivva 3202 . . 3 (𝑅 ∈ CRing → ∀𝑥 ∈ (PrmIdeal‘𝑅)∀𝑦 ∈ (PrmIdeal‘𝑅)(∀𝑑 ∈ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗})(𝑥𝑑𝑦𝑑) → 𝑥 = 𝑦))
663, 65jca 511 . 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 2737 . . . . 5 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
681, 2, 67zartopon 33876 . . . 4 (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
69 toponuni 22920 . . . 4 (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = 𝐽)
7068, 69syl 17 . . 3 (𝑅 ∈ CRing → (PrmIdeal‘𝑅) = 𝐽)
711, 2, 67, 11zartopn 33874 . . . 4 (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽)))
7271simprd 495 . . 3 (𝑅 ∈ CRing → ran (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑘𝑗}) = (Clsd‘𝐽))
7370, 72ist0cld 33832 . 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 257 1 (𝑅 ∈ CRing → 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   cuni 4907  cmpt 5225  ran crn 5686  cfv 6561  TopOpenctopn 17466  Ringcrg 20230  CRingccrg 20231  LIdealclidl 21216  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  Kol2ct0 23314  PrmIdealcprmidl 33463  Speccrspec 33861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-rest 17467  df-topn 17468  df-0g 17486  df-mre 17629  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-lpidl 21332  df-top 22900  df-topon 22917  df-cld 23027  df-t0 23321  df-prmidl 33464  df-mxidl 33488  df-idlsrg 33529  df-rspec 33862
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator