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Theorem zarcls 33343
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 2724 . . . 4 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
4 zarcls.1 . . . 4 𝑃 = (PrmIdealβ€˜π‘…)
5 eqid 2724 . . . 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 33336 . . 3 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopOpenβ€˜π‘†))
71, 6eqtr4id 2783 . 2 (𝑅 ∈ Ring β†’ 𝐽 = ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
8 nfv 1909 . . 3 Ⅎ𝑠 𝑅 ∈ Ring
9 nfcv 2895 . . 3 Ⅎ𝑠ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
10 nfrab1 3443 . . 3 Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉}
11 notrab 4303 . . . . . . . . . 10 (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}
1211eqeq2i 2737 . . . . . . . . 9 (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
13 ssrab2 4069 . . . . . . . . . . . 12 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃
1413a1i 11 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃)
15 elpwi 4601 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ 𝑠 βŠ† 𝑃)
16 ssdifsym 4255 . . . . . . . . . . 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 583 . . . . . . . . . 10 (𝑠 ∈ 𝒫 𝑃 β†’ (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} = (𝑃 βˆ– 𝑠)))
18 eqcom 2731 . . . . . . . . . 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 724 . . . . . . 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 3168 . . . . . 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 6895 . . . . . . . 8 𝑃 ∈ V
2524rabex 5322 . . . . . . 7 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} ∈ V
2623, 25elrnmpti 5949 . . . . . 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 578 . . . 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 4069 . . . . . . . . . 10 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† 𝑃
3024elpw2 5335 . . . . . . . . . 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 3057 . . . . . . . 8 βˆ€π‘– ∈ (LIdealβ€˜π‘…){𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃
33 eqid 2724 . . . . . . . . 9 (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
3433rnmptss 7114 . . . . . . . 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 3970 . . . . . 6 (𝑠 ∈ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) β†’ 𝑠 ∈ 𝒫 𝑃)
3736pm4.71ri 560 . . . . 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 3470 . . . . . . 7 𝑠 ∈ V
3933elrnmpt 5945 . . . . . . 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 622 . . . . 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 3444 . . . 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 3993 . 2 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉})
467, 45eqtrd 2764 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  {crab 3424  Vcvv 3466   βˆ– cdif 3937   βŠ† wss 3940  π’« cpw 4594   ↦ cmpt 5221  ran crn 5667  β€˜cfv 6533  TopOpenctopn 17366  Ringcrg 20128  LIdealclidl 21055  PrmIdealcprmidl 33022  Speccrspec 33331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-tset 17215  df-ple 17216  df-rest 17367  df-topn 17368  df-prmidl 33023  df-idlsrg 33084  df-rspec 33332
This theorem is referenced by:  zartopn  33344
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