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Theorem zart0 32500
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 32497 . . 3 (𝑅 ∈ CRing β†’ 𝐽 ∈ Top)
4 sseq2 3975 . . . . . . . . . . 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 3972 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ π‘₯ ∈ (PrmIdealβ€˜π‘…)) β†’ π‘₯ βŠ† π‘₯)
74, 5, 6elrabd 3652 . . . . . . . . . 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 3974 . . . . . . . . . . . . . 14 (π‘˜ = 𝑖 β†’ (π‘˜ βŠ† 𝑗 ↔ 𝑖 βŠ† 𝑗))
109rabbidv 3418 . . . . . . . . . . . . 13 (π‘˜ = 𝑖 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
1110cbvmptv 5223 . . . . . . . . . . . 12 (π‘˜ ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘˜ βŠ† 𝑗}) = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
12 crngring 19983 . . . . . . . . . . . . . 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 32256 . . . . . . . . . . . . 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 6860 . . . . . . . . . . . . . 14 (PrmIdealβ€˜π‘…) ∈ V
1817rabex 5294 . . . . . . . . . . . . 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 3974 . . . . . . . . . . . . . . 15 (𝑖 = π‘₯ β†’ (𝑖 βŠ† 𝑗 ↔ π‘₯ βŠ† 𝑗))
2120rabbidv 3418 . . . . . . . . . . . . . 14 (𝑖 = π‘₯ β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ π‘₯ βŠ† 𝑗})
2221eqcomd 2743 . . . . . . . . . . . . 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 5922 . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 3576 . . . . . . . . . 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 3975 . . . . . . . . . 10 (𝑗 = 𝑦 β†’ (π‘₯ βŠ† 𝑗 ↔ π‘₯ βŠ† 𝑦))
3332elrab 3650 . . . . . . . . 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 3975 . . . . . . . . . . 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 3972 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑦 βŠ† 𝑦)
3936, 37, 38elrabd 3652 . . . . . . . . . 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 32256 . . . . . . . . . . . . 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 5294 . . . . . . . . . . . . 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 3974 . . . . . . . . . . . . . . 15 (𝑖 = 𝑦 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑦 βŠ† 𝑗))
4746rabbidv 3418 . . . . . . . . . . . . . 14 (𝑖 = 𝑦 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑦 βŠ† 𝑗})
4847eqcomd 2743 . . . . . . . . . . . . 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 5922 . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 2824 . . . . . . . . . . . 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 3576 . . . . . . . . . 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 3975 . . . . . . . . . 10 (𝑗 = π‘₯ β†’ (𝑦 βŠ† 𝑗 ↔ 𝑦 βŠ† π‘₯))
5958elrab 3650 . . . . . . . . 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 3966 . . . . . 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 3198 . . 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 2737 . . . . 5 (PrmIdealβ€˜π‘…) = (PrmIdealβ€˜π‘…)
681, 2, 67zartopon 32498 . . . 4 (𝑅 ∈ CRing β†’ 𝐽 ∈ (TopOnβ€˜(PrmIdealβ€˜π‘…)))
69 toponuni 22279 . . . 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 32496 . . . 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 32454 . 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 3065  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆͺ cuni 4870   ↦ cmpt 5193  ran crn 5639  β€˜cfv 6501  TopOpenctopn 17310  Ringcrg 19971  CRingccrg 19972  LIdealclidl 20647  Topctop 22258  TopOnctopon 22275  Clsdccld 22383  Kol2ct0 22673  PrmIdealcprmidl 32247  Speccrspec 32483
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-rpss 7665  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-rest 17311  df-topn 17312  df-0g 17330  df-mre 17473  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-sra 20649  df-rgmod 20650  df-lidl 20651  df-rsp 20652  df-lpidl 20729  df-top 22259  df-topon 22276  df-cld 22386  df-t0 22680  df-prmidl 32248  df-mxidl 32269  df-idlsrg 32283  df-rspec 32484
This theorem is referenced by: (None)
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