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Theorem zarcls 33411
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 2727 . . . 4 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
4 zarcls.1 . . . 4 𝑃 = (PrmIdealβ€˜π‘…)
5 eqid 2727 . . . 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 33404 . . 3 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (TopOpenβ€˜π‘†))
71, 6eqtr4id 2786 . 2 (𝑅 ∈ Ring β†’ 𝐽 = ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}))
8 nfv 1910 . . 3 Ⅎ𝑠 𝑅 ∈ Ring
9 nfcv 2898 . . 3 Ⅎ𝑠ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
10 nfrab1 3446 . . 3 Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉}
11 notrab 4307 . . . . . . . . . 10 (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}
1211eqeq2i 2740 . . . . . . . . 9 (𝑠 = (𝑃 βˆ– {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗}) ↔ 𝑠 = {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
13 ssrab2 4073 . . . . . . . . . . . 12 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃
1413a1i 11 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃)
15 elpwi 4605 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ 𝑠 βŠ† 𝑃)
16 ssdifsym 4259 . . . . . . . . . . 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 2734 . . . . . . . . . 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 3171 . . . . . 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 6905 . . . . . . . 8 𝑃 ∈ V
2524rabex 5328 . . . . . . 7 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} ∈ V
2623, 25elrnmpti 5956 . . . . . 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 4073 . . . . . . . . . 10 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† 𝑃
3024elpw2 5341 . . . . . . . . . 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 3060 . . . . . . . 8 βˆ€π‘– ∈ (LIdealβ€˜π‘…){𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} ∈ 𝒫 𝑃
33 eqid 2727 . . . . . . . . 9 (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗})
3433rnmptss 7127 . . . . . . . 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 3974 . . . . . 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 3473 . . . . . . 7 𝑠 ∈ V
3933elrnmpt 5952 . . . . . . 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 3447 . . . 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 3997 . 2 (𝑅 ∈ Ring β†’ ran (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗}) = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉})
467, 45eqtrd 2767 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 1534   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065  {crab 3427  Vcvv 3469   βˆ– cdif 3941   βŠ† wss 3944  π’« cpw 4598   ↦ cmpt 5225  ran crn 5673  β€˜cfv 6542  TopOpenctopn 17394  Ringcrg 20164  LIdealclidl 21091  PrmIdealcprmidl 33086  Speccrspec 33399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-tset 17243  df-ple 17244  df-rest 17395  df-topn 17396  df-prmidl 33087  df-idlsrg 33148  df-rspec 33400
This theorem is referenced by:  zartopn  33412
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