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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 4080 | . . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃 | |
| 2 | zarcls.1 | . . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅) | |
| 3 | 2 | fvexi 6920 | . . . . . . . . 9 ⊢ 𝑃 ∈ V |
| 4 | 3 | elpw2 5334 | . . . . . . . 8 ⊢ ({𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 ↔ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃) |
| 5 | 1, 4 | mpbir 231 | . . . . . . 7 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ 𝒫 𝑃 |
| 6 | 5 | rgenw 3065 | . . . . . 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 7143 | . . . . . 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 20242 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 12 | 2 | rabeqi 3450 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} |
| 13 | 12 | mpteq2i 5247 | . . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
| 14 | 7, 13 | eqtri 2765 | . . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
| 15 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 16 | 14, 2, 15 | zarcls0 33867 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃) |
| 17 | 7 | funmpt2 6605 | . . . . . . 7 ⊢ Fun 𝑉 |
| 18 | eqid 2737 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
| 19 | 18, 15 | lidl0 21240 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
| 20 | 3 | rabex 5339 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V |
| 21 | 20, 7 | dmmpti 6712 | . . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅) |
| 22 | 19, 21 | eleqtrrdi 2852 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ dom 𝑉) |
| 23 | fvelrn 7096 | . . . . . . 7 ⊢ ((Fun 𝑉 ∧ {(0g‘𝑅)} ∈ dom 𝑉) → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉) | |
| 24 | 17, 22, 23 | sylancr 587 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) ∈ ran 𝑉) |
| 25 | 16, 24 | eqeltrrd 2842 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉) |
| 26 | 11, 25 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉) |
| 27 | 14 | zarclsint 33871 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉) |
| 28 | 10, 26, 27 | ismred 17645 | . . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃)) |
| 29 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 30 | 21, 29 | lidl1 21243 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉) |
| 31 | 11, 30 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉) |
| 32 | 31, 21 | eleqtrdi 2851 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅)) |
| 33 | 14, 29 | zarcls1 33868 | . . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅))) |
| 34 | 29, 33 | mpbiri 258 | . . . . 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 7096 | . . . . 5 ⊢ ((Fun 𝑉 ∧ (Base‘𝑅) ∈ dom 𝑉) → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉) | |
| 38 | 36, 31, 37 | syl2anc 584 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) ∈ ran 𝑉) |
| 39 | 35, 38 | eqeltrrd 2842 | . . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉) |
| 40 | 14 | zarclsun 33869 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉) |
| 41 | eqid 2737 | . . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} | |
| 42 | 28, 39, 40, 41 | mretopd 23100 | . 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 33873 | . . . . 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 2826 | . . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃))) |
| 48 | 46 | fveq2d 6910 | . . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})) |
| 49 | 48 | eqeq2d 2748 | . . 3 ⊢ (𝑅 ∈ CRing → (ran 𝑉 = (Clsd‘𝐽) ↔ ran 𝑉 = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉}))) |
| 50 | 47, 49 | anbi12d 632 | . 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 257 | 1 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ∧ ran 𝑉 = (Clsd‘𝐽))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Fun wfun 6555 ‘cfv 6561 Basecbs 17247 TopOpenctopn 17466 0gc0g 17484 Ringcrg 20230 CRingccrg 20231 LIdealclidl 21216 TopOnctopon 22916 Clsdccld 23024 PrmIdealcprmidl 33463 Speccrspec 33861 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-rest 17467 df-topn 17468 df-0g 17486 df-mre 17629 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-lpidl 21332 df-top 22900 df-topon 22917 df-cld 23027 df-prmidl 33464 df-mxidl 33488 df-idlsrg 33529 df-rspec 33862 |
| This theorem is referenced by: zartop 33875 zartopon 33876 zart0 33878 zarmxt1 33879 zarcmplem 33880 rhmpreimacn 33884 |
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