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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zarcmp | Structured version Visualization version GIF version |
Description: The Zariski topology is compact. Proposition 1.1.10(ii) of [EGA], p. 82. (Contributed by Thierry Arnoux, 2-Jul-2024.) |
Ref | Expression |
---|---|
zartop.1 | β’ π = (Specβπ ) |
zartop.2 | β’ π½ = (TopOpenβπ) |
Ref | Expression |
---|---|
zarcmp | β’ (π β CRing β π½ β Comp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zartop.1 | . 2 β’ π = (Specβπ ) | |
2 | zartop.2 | . 2 β’ π½ = (TopOpenβπ) | |
3 | sseq1 4005 | . . . . 5 β’ (π = π β (π β π β π β π)) | |
4 | 3 | rabbidv 3437 | . . . 4 β’ (π = π β {π β (PrmIdealβπ ) β£ π β π} = {π β (PrmIdealβπ ) β£ π β π}) |
5 | sseq2 4006 | . . . . 5 β’ (π = π β (π β π β π β π)) | |
6 | 5 | cbvrabv 3439 | . . . 4 β’ {π β (PrmIdealβπ ) β£ π β π} = {π β (PrmIdealβπ ) β£ π β π} |
7 | 4, 6 | eqtrdi 2784 | . . 3 β’ (π = π β {π β (PrmIdealβπ ) β£ π β π} = {π β (PrmIdealβπ ) β£ π β π}) |
8 | 7 | cbvmptv 5261 | . 2 β’ (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) = (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) |
9 | 1, 2, 8 | zarcmplem 33482 | 1 β’ (π β CRing β π½ β Comp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {crab 3429 β wss 3947 β¦ cmpt 5231 βcfv 6548 TopOpenctopn 17403 CRingccrg 20174 LIdealclidl 21102 Compccmp 23303 PrmIdealcprmidl 33164 Speccrspec 33463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-reg 9616 ax-inf2 9665 ax-ac2 10487 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218 ax-mulf 11219 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-rpss 7728 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-oi 9534 df-r1 9788 df-rank 9789 df-dju 9925 df-card 9963 df-ac 10140 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-rhm 20411 df-nzr 20452 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lmhm 20907 df-lbs 20960 df-sra 21058 df-rgmod 21059 df-lidl 21104 df-rsp 21105 df-lpidl 21212 df-cnfld 21280 df-zring 21373 df-zrh 21429 df-dsmm 21666 df-frlm 21681 df-uvc 21717 df-top 22809 df-topon 22826 df-cld 22936 df-cmp 23304 df-prmidl 33165 df-mxidl 33186 df-idlsrg 33225 df-rspec 33464 |
This theorem is referenced by: (None) |
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