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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zartop | Structured version Visualization version GIF version |
Description: The Zariski topology is a topology. Proposition 1.1.2 of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
zartop.1 | β’ π = (Specβπ ) |
zartop.2 | β’ π½ = (TopOpenβπ) |
Ref | Expression |
---|---|
zartop | β’ (π β CRing β π½ β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zartop.1 | . . . 4 β’ π = (Specβπ ) | |
2 | zartop.2 | . . . 4 β’ π½ = (TopOpenβπ) | |
3 | eqid 2726 | . . . 4 β’ (PrmIdealβπ ) = (PrmIdealβπ ) | |
4 | sseq1 4002 | . . . . . 6 β’ (π = π β (π β π β π β π)) | |
5 | 4 | rabbidv 3434 | . . . . 5 β’ (π = π β {π β (PrmIdealβπ ) β£ π β π} = {π β (PrmIdealβπ ) β£ π β π}) |
6 | 5 | cbvmptv 5254 | . . . 4 β’ (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) = (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) |
7 | 1, 2, 3, 6 | zartopn 33385 | . . 3 β’ (π β CRing β (π½ β (TopOnβ(PrmIdealβπ )) β§ ran (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) = (Clsdβπ½))) |
8 | 7 | simpld 494 | . 2 β’ (π β CRing β π½ β (TopOnβ(PrmIdealβπ ))) |
9 | topontop 22766 | . 2 β’ (π½ β (TopOnβ(PrmIdealβπ )) β π½ β Top) | |
10 | 8, 9 | syl 17 | 1 β’ (π β CRing β π½ β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3426 β wss 3943 β¦ cmpt 5224 ran crn 5670 βcfv 6536 TopOpenctopn 17374 CRingccrg 20137 LIdealclidl 21063 Topctop 22746 TopOnctopon 22763 Clsdccld 22871 PrmIdealcprmidl 33059 Speccrspec 33372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-ac2 10457 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-rpss 7709 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-dju 9895 df-card 9933 df-ac 10110 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-rest 17375 df-topn 17376 df-0g 17394 df-mre 17537 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-sra 21019 df-rgmod 21020 df-lidl 21065 df-rsp 21066 df-lpidl 21173 df-top 22747 df-topon 22764 df-cld 22874 df-prmidl 33060 df-mxidl 33082 df-idlsrg 33121 df-rspec 33373 |
This theorem is referenced by: zart0 33389 zarmxt1 33390 zarcmplem 33391 |
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