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Theorem zarcls 32519
Description: The open sets of the Zariski topology are the complements of the closed sets. (Contributed by Thierry Arnoux, 16-Jun-2024.)
Hypotheses
Ref Expression
zartop.1 𝑆 = (Specβ€˜π‘…)
zartop.2 𝐽 = (TopOpenβ€˜π‘†)
zarcls.1 𝑃 = (PrmIdealβ€˜π‘…)
zarcls.2 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
zarcls (𝑅 ∈ Ring β†’ 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉})
Distinct variable groups:   𝑃,𝑖,𝑗,𝑠   𝑅,𝑖,𝑗,𝑠   𝑉,𝑠
Allowed substitution hints:   𝑆(𝑖,𝑗,𝑠)   𝐽(𝑖,𝑗,𝑠)   𝑉(𝑖,𝑗)

Proof of Theorem zarcls
StepHypRef Expression
1 zartop.2 . . 3 𝐽 = (TopOpenβ€˜π‘†)
2 zartop.1 . . . 4 𝑆 = (Specβ€˜π‘…)
3 eqid 2733 . . . 4 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
4 zarcls.1 . . . 4 𝑃 = (PrmIdealβ€˜π‘…)
5 eqid 2733 . . . 4 ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
62, 3, 4, 5rspectopn 32512 . . 3 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopOpenβ€˜π‘†))
71, 6eqtr4id 2792 . 2 (𝑅 ∈ Ring β†’ 𝐽 = ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
8 nfv 1918 . . 3 Ⅎ𝑠 𝑅 ∈ Ring
9 nfcv 2904 . . 3 Ⅎ𝑠ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
10 nfrab1 3425 . . 3 Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉}
11 notrab 4275 . . . . . . . . . 10 (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}
1211eqeq2i 2746 . . . . . . . . 9 (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
13 ssrab2 4041 . . . . . . . . . . . 12 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃
1413a1i 11 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃)
15 elpwi 4571 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ 𝑠 βŠ† 𝑃)
16 ssdifsym 4227 . . . . . . . . . . 11 (({𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃 ∧ 𝑠 βŠ† 𝑃) β†’ (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} = (𝑃 βˆ– 𝑠)))
1714, 15, 16syl2anc 585 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝑃 β†’ (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} = (𝑃 βˆ– 𝑠)))
18 eqcom 2740 . . . . . . . . . 10 ({𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} = (𝑃 βˆ– 𝑠) ↔ (𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗})
1917, 18bitrdi 287 . . . . . . . . 9 (𝑠 ∈ 𝒫 𝑃 β†’ (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ (𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}))
2012, 19bitr3id 285 . . . . . . . 8 (𝑠 ∈ 𝒫 𝑃 β†’ (𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ↔ (𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}))
2120ad2antlr 726 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) ∧ 𝑖 ∈ (LIdealβ€˜π‘…)) β†’ (𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ↔ (𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}))
2221rexbidva 3170 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) β†’ (βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)(𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}))
23 zarcls.2 . . . . . . 7 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗})
244fvexi 6860 . . . . . . . 8 𝑃 ∈ V
2524rabex 5293 . . . . . . 7 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} ∈ V
2623, 25elrnmpti 5919 . . . . . 6 ((𝑃 βˆ– 𝑠) ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)(𝑃 βˆ– 𝑠) = {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗})
2722, 26bitr4di 289 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝒫 𝑃) β†’ (βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ↔ (𝑃 βˆ– 𝑠) ∈ ran 𝑉))
2827pm5.32da 580 . . . 4 (𝑅 ∈ Ring β†’ ((𝑠 ∈ 𝒫 𝑃 ∧ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 βˆ– 𝑠) ∈ ran 𝑉)))
29 ssrab2 4041 . . . . . . . . . 10 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† 𝑃
3024elpw2 5306 . . . . . . . . . 10 ({𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† 𝑃)
3129, 30mpbir 230 . . . . . . . . 9 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃
3231rgenw 3065 . . . . . . . 8 βˆ€π‘– ∈ (LIdealβ€˜π‘…){𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃
33 eqid 2733 . . . . . . . . 9 (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
3433rnmptss 7074 . . . . . . . 8 (βˆ€π‘– ∈ (LIdealβ€˜π‘…){𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃 β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) βŠ† 𝒫 𝑃)
3532, 34ax-mp 5 . . . . . . 7 ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) βŠ† 𝒫 𝑃
3635sseli 3944 . . . . . 6 (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†’ 𝑠 ∈ 𝒫 𝑃)
3736pm4.71ri 562 . . . . 5 (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})))
38 vex 3451 . . . . . . 7 𝑠 ∈ V
3933elrnmpt 5915 . . . . . . 7 (𝑠 ∈ V β†’ (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
4038, 39ax-mp 5 . . . . . 6 (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
4140anbi2i 624 . . . . 5 ((𝑠 ∈ 𝒫 𝑃 ∧ 𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
4237, 41bitri 275 . . . 4 (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ (𝑠 ∈ 𝒫 𝑃 ∧ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
43 rabid 3426 . . . 4 (𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉} ↔ (𝑠 ∈ 𝒫 𝑃 ∧ (𝑃 βˆ– 𝑠) ∈ ran 𝑉))
4428, 42, 433bitr4g 314 . . 3 (𝑅 ∈ Ring β†’ (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) ↔ 𝑠 ∈ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉}))
458, 9, 10, 44eqrd 3967 . 2 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉})
467, 45eqtrd 2773 1 (𝑅 ∈ Ring β†’ 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3447   βˆ– cdif 3911   βŠ† wss 3914  π’« cpw 4564   ↦ cmpt 5192  ran crn 5638  β€˜cfv 6500  TopOpenctopn 17311  Ringcrg 19972  LIdealclidl 20676  PrmIdealcprmidl 32262  Speccrspec 32507
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-tset 17160  df-ple 17161  df-rest 17312  df-topn 17313  df-prmidl 32263  df-idlsrg 32298  df-rspec 32508
This theorem is referenced by:  zartopn  32520
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