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Theorem zarcls 32854
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 32847 . . 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 3452 . . 3 Ⅎ𝑠{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 βˆ– 𝑠) ∈ ran 𝑉}
11 notrab 4312 . . . . . . . . . 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 4078 . . . . . . . . . . . 12 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃
1413a1i 11 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} βŠ† 𝑃)
15 elpwi 4610 . . . . . . . . . . 11 (𝑠 ∈ 𝒫 𝑃 β†’ 𝑠 βŠ† 𝑃)
16 ssdifsym 4264 . . . . . . . . . . 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 3177 . . . . . 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 6906 . . . . . . . 8 𝑃 ∈ V
2524rabex 5333 . . . . . . 7 {𝑗 ∈ 𝑃 ∣ 𝑖 βŠ† 𝑗} ∈ V
2623, 25elrnmpti 5960 . . . . . 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 4078 . . . . . . . . . 10 {𝑗 ∈ 𝑃 ∣ Β¬ 𝑖 βŠ† 𝑗} βŠ† 𝑃
3024elpw2 5346 . . . . . . . . . 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 3066 . . . . . . . 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 7122 . . . . . . . 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 3979 . . . . . 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 3479 . . . . . . 7 𝑠 ∈ V
3933elrnmpt 5956 . . . . . . 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 3453 . . . 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 4002 . 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 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βˆ– cdif 3946   βŠ† wss 3949  π’« cpw 4603   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  TopOpenctopn 17367  Ringcrg 20056  LIdealclidl 20783  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-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-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-iun 5000  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-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-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  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-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  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-tset 17216  df-ple 17217  df-rest 17368  df-topn 17369  df-prmidl 32554  df-idlsrg 32615  df-rspec 32843
This theorem is referenced by:  zartopn  32855
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