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Theorem zart0 32859
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 = (Specβ€˜π‘…)
zartop.2 𝐽 = (TopOpenβ€˜π‘†)
Assertion
Ref Expression
zart0 (𝑅 ∈ CRing β†’ 𝐽 ∈ Kol2)

Proof of Theorem zart0
Dummy variables 𝑖 𝑗 π‘˜ 𝑑 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zartop.1 . . . 4 𝑆 = (Specβ€˜π‘…)
2 zartop.2 . . . 4 𝐽 = (TopOpenβ€˜π‘†)
31, 2zartop 32856 . . 3 (𝑅 ∈ CRing β†’ 𝐽 ∈ Top)
4 sseq2 4009 . . . . . . . . . . 11 (𝑗 = π‘₯ β†’ (π‘₯ βŠ† 𝑗 ↔ π‘₯ βŠ† π‘₯))
5 simpr 486 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ ∈ (PrmIdealβ€˜π‘…))
6 ssidd 4006 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ βŠ† π‘₯)
74, 5, 6elrabd 3686 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
87ad2antrr 725 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
9 sseq1 4008 . . . . . . . . . . . . . 14 (π‘˜ = 𝑖 β†’ (π‘˜ βŠ† 𝑗 ↔ 𝑖 βŠ† 𝑗))
109rabbidv 3441 . . . . . . . . . . . . 13 (π‘˜ = 𝑖 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
1110cbvmptv 5262 . . . . . . . . . . . 12 (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
12 crngring 20068 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
1312ad2antrr 725 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑅 ∈ Ring)
14 simplr 768 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ ∈ (PrmIdealβ€˜π‘…))
15 prmidlidl 32562 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ ∈ (LIdealβ€˜π‘…))
1613, 14, 15syl2anc 585 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ ∈ (LIdealβ€˜π‘…))
17 fvex 6905 . . . . . . . . . . . . . 14 (PrmIdealβ€˜π‘…) ∈ V
1817rabex 5333 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ∈ V
1918a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ∈ V)
20 sseq1 4008 . . . . . . . . . . . . . . 15 (𝑖 = π‘₯ β†’ (𝑖 βŠ† 𝑗 ↔ π‘₯ βŠ† 𝑗))
2120rabbidv 3441 . . . . . . . . . . . . . 14 (𝑖 = π‘₯ β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
2221eqcomd 2739 . . . . . . . . . . . . 13 (𝑖 = π‘₯ β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
2322adantl 483 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑖 = π‘₯) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
2411, 16, 19, 23elrnmptdv 5962 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}))
25 simpr 486 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}) β†’ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
2625eleq2d 2820 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}) β†’ (π‘₯ ∈ 𝑑 ↔ π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}))
2725eleq2d 2820 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}) β†’ (𝑦 ∈ 𝑑 ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}))
2826, 27bibi12d 346 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}) β†’ ((π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) ↔ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})))
2924, 28rspcdv 3605 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ (βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})))
3029imp 408 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗}))
318, 30mpbid 231 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
32 sseq2 4009 . . . . . . . . . 10 (𝑗 = 𝑦 β†’ (π‘₯ βŠ† 𝑗 ↔ π‘₯ βŠ† 𝑦))
3332elrab 3684 . . . . . . . . 9 (𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} ↔ (𝑦 ∈ (PrmIdealβ€˜π‘…) ∧ π‘₯ βŠ† 𝑦))
3433simprbi 498 . . . . . . . 8 (𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗} β†’ π‘₯ βŠ† 𝑦)
3531, 34syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ π‘₯ βŠ† 𝑦)
36 sseq2 4009 . . . . . . . . . . 11 (𝑗 = 𝑦 β†’ (𝑦 βŠ† 𝑗 ↔ 𝑦 βŠ† 𝑦))
37 simpr 486 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 ∈ (PrmIdealβ€˜π‘…))
38 ssidd 4006 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 βŠ† 𝑦)
3936, 37, 38elrabd 3686 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
4039ad4ant13 750 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
41 simpr 486 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 ∈ (PrmIdealβ€˜π‘…))
42 prmidlidl 32562 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 ∈ (LIdealβ€˜π‘…))
4313, 41, 42syl2anc 585 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 ∈ (LIdealβ€˜π‘…))
4417rabex 5333 . . . . . . . . . . . . 13 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ∈ V
4544a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ∈ V)
46 sseq1 4008 . . . . . . . . . . . . . . 15 (𝑖 = 𝑦 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑦 βŠ† 𝑗))
4746rabbidv 3441 . . . . . . . . . . . . . 14 (𝑖 = 𝑦 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
4847eqcomd 2739 . . . . . . . . . . . . 13 (𝑖 = 𝑦 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
4948adantl 483 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑖 = 𝑦) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
5011, 43, 45, 49elrnmptdv 5962 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}))
51 simpr 486 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}) β†’ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
5251eleq2d 2820 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}) β†’ (π‘₯ ∈ 𝑑 ↔ π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}))
5351eleq2d 2820 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}) β†’ (𝑦 ∈ 𝑑 ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}))
5452, 53bibi12d 346 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑑 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}) β†’ ((π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) ↔ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})))
5550, 54rspcdv 3605 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ (βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})))
5655imp 408 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ↔ 𝑦 ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗}))
5740, 56mpbird 257 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
58 sseq2 4009 . . . . . . . . . 10 (𝑗 = π‘₯ β†’ (𝑦 βŠ† 𝑗 ↔ 𝑦 βŠ† π‘₯))
5958elrab 3684 . . . . . . . . 9 (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} ↔ (π‘₯ ∈ (PrmIdealβ€˜π‘…) ∧ 𝑦 βŠ† π‘₯))
6059simprbi 498 . . . . . . . 8 (π‘₯ ∈ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗} β†’ 𝑦 βŠ† π‘₯)
6157, 60syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ 𝑦 βŠ† π‘₯)
6235, 61eqssd 4000 . . . . . 6 ((((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) ∧ βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑)) β†’ π‘₯ = 𝑦)
6362ex 414 . . . . 5 (((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ (βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦))
6463anasss 468 . . . 4 ((𝑅 ∈ CRing ∧ (π‘₯ ∈ (PrmIdealβ€˜π‘…) ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…))) β†’ (βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦))
6564ralrimivva 3201 . . 3 (𝑅 ∈ CRing β†’ βˆ€π‘₯ ∈ (PrmIdealβ€˜π‘…)βˆ€π‘¦ ∈ (PrmIdealβ€˜π‘…)(βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦))
663, 65jca 513 . 2 (𝑅 ∈ CRing β†’ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ (PrmIdealβ€˜π‘…)βˆ€π‘¦ ∈ (PrmIdealβ€˜π‘…)(βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦)))
67 eqid 2733 . . . . 5 (PrmIdealβ€˜π‘…) = (PrmIdealβ€˜π‘…)
681, 2, 67zartopon 32857 . . . 4 (𝑅 ∈ CRing β†’ 𝐽 ∈ (TopOnβ€˜(PrmIdealβ€˜π‘…)))
69 toponuni 22416 . . . 4 (𝐽 ∈ (TopOnβ€˜(PrmIdealβ€˜π‘…)) β†’ (PrmIdealβ€˜π‘…) = βˆͺ 𝐽)
7068, 69syl 17 . . 3 (𝑅 ∈ CRing β†’ (PrmIdealβ€˜π‘…) = βˆͺ 𝐽)
711, 2, 67, 11zartopn 32855 . . . 4 (𝑅 ∈ CRing β†’ (𝐽 ∈ (TopOnβ€˜(PrmIdealβ€˜π‘…)) ∧ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) = (Clsdβ€˜π½)))
7271simprd 497 . . 3 (𝑅 ∈ CRing β†’ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) = (Clsdβ€˜π½))
7370, 72ist0cld 32813 . 2 (𝑅 ∈ CRing β†’ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ (PrmIdealβ€˜π‘…)βˆ€π‘¦ ∈ (PrmIdealβ€˜π‘…)(βˆ€π‘‘ ∈ ran (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗})(π‘₯ ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) β†’ π‘₯ = 𝑦))))
7466, 73mpbird 257 1 (𝑅 ∈ CRing β†’ 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆͺ cuni 4909   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  TopOpenctopn 17367  Ringcrg 20056  CRingccrg 20057  LIdealclidl 20783  Topctop 22395  TopOnctopon 22412  Clsdccld 22520  Kol2ct0 22810  PrmIdealcprmidl 32553  Speccrspec 32842
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-ac2 10458  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rpss 7713  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-ac 10111  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-rest 17368  df-topn 17369  df-0g 17387  df-mre 17530  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-cntz 19181  df-lsm 19504  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-sra 20785  df-rgmod 20786  df-lidl 20787  df-rsp 20788  df-lpidl 20881  df-top 22396  df-topon 22413  df-cld 22523  df-t0 22817  df-prmidl 32554  df-mxidl 32576  df-idlsrg 32615  df-rspec 32843
This theorem is referenced by: (None)
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