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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 4090 | . . . . . . . 8 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ⊆ 𝑃 | |
2 | zarcls.1 | . . . . . . . . . 10 ⊢ 𝑃 = (PrmIdeal‘𝑅) | |
3 | 2 | fvexi 6921 | . . . . . . . . 9 ⊢ 𝑃 ∈ V |
4 | 3 | elpw2 5340 | . . . . . . . 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 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 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 20263 | . . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
12 | 2 | rabeqi 3447 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} |
13 | 12 | mpteq2i 5253 | . . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
14 | 7, 13 | eqtri 2763 | . . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
15 | eqid 2735 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
16 | 14, 2, 15 | zarcls0 33829 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑉‘{(0g‘𝑅)}) = 𝑃) |
17 | 7 | funmpt2 6607 | . . . . . . 7 ⊢ Fun 𝑉 |
18 | eqid 2735 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
19 | 18, 15 | lidl0 21258 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → {(0g‘𝑅)} ∈ (LIdeal‘𝑅)) |
20 | 3 | rabex 5345 | . . . . . . . . 9 ⊢ {𝑗 ∈ 𝑃 ∣ 𝑖 ⊆ 𝑗} ∈ V |
21 | 20, 7 | dmmpti 6713 | . . . . . . . 8 ⊢ dom 𝑉 = (LIdeal‘𝑅) |
22 | 19, 21 | eleqtrrdi 2850 | . . . . . . 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 2840 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ran 𝑉) |
26 | 11, 25 | syl 17 | . . . 4 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ ran 𝑉) |
27 | 14 | zarclsint 33833 | . . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑧 ⊆ ran 𝑉 ∧ 𝑧 ≠ ∅) → ∩ 𝑧 ∈ ran 𝑉) |
28 | 10, 26, 27 | ismred 17647 | . . 3 ⊢ (𝑅 ∈ CRing → ran 𝑉 ∈ (Moore‘𝑃)) |
29 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
30 | 21, 29 | lidl1 21261 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ dom 𝑉) |
31 | 11, 30 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ dom 𝑉) |
32 | 31, 21 | eleqtrdi 2849 | . . . . 5 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅)) |
33 | 14, 29 | zarcls1 33830 | . . . . . 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 2840 | . . 3 ⊢ (𝑅 ∈ CRing → ∅ ∈ ran 𝑉) |
40 | 14 | zarclsun 33831 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑥 ∈ ran 𝑉 ∧ 𝑦 ∈ ran 𝑉) → (𝑥 ∪ 𝑦) ∈ ran 𝑉) |
41 | eqid 2735 | . . 3 ⊢ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} = {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} | |
42 | 28, 39, 40, 41 | mretopd 23116 | . 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 33835 | . . . . 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 2824 | . . 3 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝑃) ↔ {𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉} ∈ (TopOn‘𝑃))) |
48 | 46 | fveq2d 6911 | . . . 4 ⊢ (𝑅 ∈ CRing → (Clsd‘𝐽) = (Clsd‘{𝑠 ∈ 𝒫 𝑃 ∣ (𝑃 ∖ 𝑠) ∈ ran 𝑉})) |
49 | 48 | eqeq2d 2746 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 Fun wfun 6557 ‘cfv 6563 Basecbs 17245 TopOpenctopn 17468 0gc0g 17486 Ringcrg 20251 CRingccrg 20252 LIdealclidl 21234 TopOnctopon 22932 Clsdccld 23040 PrmIdealcprmidl 33443 Speccrspec 33823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-rest 17469 df-topn 17470 df-0g 17488 df-mre 17631 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-lpidl 21350 df-top 22916 df-topon 22933 df-cld 23043 df-prmidl 33444 df-mxidl 33468 df-idlsrg 33509 df-rspec 33824 |
This theorem is referenced by: zartop 33837 zartopon 33838 zart0 33840 zarmxt1 33841 zarcmplem 33842 rhmpreimacn 33846 |
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