Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zartopn | Structured version Visualization version GIF version |
Description: The Zariski topology is a topology, and its closed sets are images by 𝑉 of the ideals of 𝑅. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
zartop.1 | ⊢ 𝑆 = (Spec‘𝑅) |
zartop.2 | ⊢ 𝐽 = (TopOpen‘𝑆) |
zarcls.1 | ⊢ 𝑃 = (PrmIdeal‘𝑅) |
zarcls.2 | ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) |
Ref | Expression |
---|---|
zartopn | ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3979 | . . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃 | |
2 | zarcls.1 | . . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅) | |
3 | 2 | fvexi 6709 | . . . . . . . . 9 ⊢ 𝑃 ∈ V |
4 | 3 | elpw2 5223 | . . . . . . . 8 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃) |
5 | 1, 4 | mpbir 234 | . . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 |
6 | 5 | rgenw 3063 | . . . . . 6 ⊢ ∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 |
7 | zarcls.2 | . . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) | |
8 | 7 | rnmptss 6917 | . . . . . 6 ⊢ (∀𝑖 ∈ (LIdeal‘𝑅){𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 → ran 𝑉 ⊆ 𝒫 𝑃) |
9 | 6, 8 | ax-mp 5 | . . . . 5 ⊢ ran 𝑉 ⊆ 𝒫 𝑃 |
10 | 9 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ CRing → ran 𝑉 ⊆ 𝒫 𝑃) |
11 | crngring 19528 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 2 | rabeqi 3382 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} |
13 | 12 | mpteq2i 5132 | . . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
14 | 7, 13 | eqtri 2759 | . . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
15 | eqid 2736 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
16 | 14, 2, 15 | zarcls0 31486 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃) |
17 | 7 | funmpt2 6397 | . . . . . . 7 ⊢ Fun 𝑉 |
18 | eqid 2736 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
19 | 18, 15 | lidl0 20211 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
20 | 3 | rabex 5210 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V |
21 | 20, 7 | dmmpti 6500 | . . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅) |
22 | 19, 21 | eleqtrrdi 2842 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ dom 𝑉) |
23 | fvelrn 6875 | . . . . . . 7 ⊢ ((Fun 𝑉 ∧ {(0g‘𝑅)} ∈ dom 𝑉) → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉) | |
24 | 17, 22, 23 | sylancr 590 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉) |
25 | 16, 24 | eqeltrrd 2832 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉) |
26 | 11, 25 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉) |
27 | 14 | zarclsint 31490 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉) |
28 | 10, 26, 27 | ismred 17059 | . . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃)) |
29 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
30 | 21, 29 | lidl1 20212 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉) |
31 | 11, 30 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉) |
32 | 31, 21 | eleqtrdi 2841 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅)) |
33 | 14, 29 | zarcls1 31487 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅))) |
34 | 29, 33 | mpbiri 261 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → (𝑉‘(Base‘𝑅)) = ∅) |
35 | 32, 34 | mpdan 687 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅) |
36 | 17 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ CRing → Fun 𝑉) |
37 | fvelrn 6875 | . . . . 5 ⊢ ((Fun 𝑉 ∧ (Base‘𝑅) ∈ dom 𝑉) → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉) | |
38 | 36, 31, 37 | syl2anc 587 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉) |
39 | 35, 38 | eqeltrrd 2832 | . . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉) |
40 | 14 | zarclsun 31488 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉) |
41 | eqid 2736 | . . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} | |
42 | 28, 39, 40, 41 | mretopd 21943 | . 2 ⊢ (𝑅 ∈ CRing → ({𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))) |
43 | zartop.1 | . . . . . 6 ⊢ 𝑆 = (Spec‘𝑅) | |
44 | zartop.2 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑆) | |
45 | 43, 44, 2, 7 | zarcls 31492 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}) |
46 | 11, 45 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝐽 = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}) |
47 | 46 | eleq1d 2815 | . . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃))) |
48 | 46 | fveq2d 6699 | . . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})) |
49 | 48 | eqeq2d 2747 | . . 3 ⊢ (𝑅 ∈ CRing → (ran 𝑉 = (Clsd‘𝐽) ↔ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))) |
50 | 47, 49 | anbi12d 634 | . 2 ⊢ (𝑅 ∈ CRing → ((𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽)) ↔ ({𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})))) |
51 | 42, 50 | mpbird 260 | 1 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 {crab 3055 ∖ cdif 3850 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 {csn 4527 ↦ cmpt 5120 dom cdm 5536 ran crn 5537 Fun wfun 6352 ‘cfv 6358 Basecbs 16666 TopOpenctopn 16880 0gc0g 16898 Ringcrg 19516 CRingccrg 19517 LIdealclidl 20161 TopOnctopon 21761 Clsdccld 21867 PrmIdealcprmidl 31278 Speccrspec 31480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-ac2 10042 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-rpss 7489 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-ac 9695 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-rest 16881 df-topn 16882 df-0g 16900 df-mre 17043 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-cntz 18665 df-lsm 18979 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-sra 20163 df-rgmod 20164 df-lidl 20165 df-rsp 20166 df-lpidl 20235 df-top 21745 df-topon 21762 df-cld 21870 df-prmidl 31279 df-mxidl 31300 df-idlsrg 31314 df-rspec 31481 |
This theorem is referenced by: zartop 31494 zartopon 31495 zart0 31497 zarmxt1 31498 zarcmplem 31499 rhmpreimacn 31503 |
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