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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zartopon | Structured version Visualization version GIF version |
Description: The points of the Zariski topology are the prime ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
zartop.1 | ⊢ 𝑆 = (Spec‘𝑅) |
zartop.2 | ⊢ 𝐽 = (TopOpen‘𝑆) |
zartop.3 | ⊢ 𝑃 = (PrmIdeal‘𝑅) |
Ref | Expression |
---|---|
zartopon | ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zartop.1 | . . 3 ⊢ 𝑆 = (Spec‘𝑅) | |
2 | zartop.2 | . . 3 ⊢ 𝐽 = (TopOpen‘𝑆) | |
3 | zartop.3 | . . 3 ⊢ 𝑃 = (PrmIdeal‘𝑅) | |
4 | sseq1 4005 | . . . . 5 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗)) | |
5 | 4 | rabbidv 3441 | . . . 4 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝑃 ∣ 𝑘 ⊆ 𝑗}) |
6 | 5 | cbvmptv 5257 | . . 3 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑘 ⊆ 𝑗}) |
7 | 1, 2, 3, 6 | zartopn 32786 | . 2 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽))) |
8 | 7 | simpld 496 | 1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 ⊆ wss 3946 ↦ cmpt 5227 ran crn 5673 ‘cfv 6535 TopOpenctopn 17354 CRingccrg 20039 LIdealclidl 20760 TopOnctopon 22381 Clsdccld 22489 PrmIdealcprmidl 32504 Speccrspec 32773 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-ac2 10445 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
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-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-rpss 7700 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-dju 9883 df-card 9921 df-ac 10098 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-rest 17355 df-topn 17356 df-0g 17374 df-mre 17517 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-cntz 19166 df-lsm 19488 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-cring 20041 df-subrg 20338 df-lmod 20450 df-lss 20520 df-lsp 20560 df-sra 20762 df-rgmod 20763 df-lidl 20764 df-rsp 20765 df-lpidl 20857 df-top 22365 df-topon 22382 df-cld 22492 df-prmidl 32505 df-mxidl 32527 df-idlsrg 32560 df-rspec 32774 |
This theorem is referenced by: zart0 32790 rspectps 32794 rhmpreimacn 32796 |
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