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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zarcls0 | Structured version Visualization version GIF version |
Description: The closure of the identity ideal in the Zariski topology. Proposition 1.1.2(i) of [EGA] p. 80. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
zarclsx.1 | β’ π = (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) |
zarcls0.1 | β’ π = (PrmIdealβπ ) |
zarcls0.2 | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
zarcls0 | β’ (π β Ring β (πβ{ 0 }) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zarclsx.1 | . . 3 β’ π = (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π}) | |
2 | 1 | a1i 11 | . 2 β’ (π β Ring β π = (π β (LIdealβπ ) β¦ {π β (PrmIdealβπ ) β£ π β π})) |
3 | zarcls0.1 | . . 3 β’ π = (PrmIdealβπ ) | |
4 | simplr 767 | . . . . . 6 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β π = { 0 }) | |
5 | simpll 765 | . . . . . . . 8 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β π β Ring) | |
6 | prmidlidl 33178 | . . . . . . . . 9 β’ ((π β Ring β§ π β (PrmIdealβπ )) β π β (LIdealβπ )) | |
7 | 5, 6 | sylancom 586 | . . . . . . . 8 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β π β (LIdealβπ )) |
8 | eqid 2727 | . . . . . . . . 9 β’ (LIdealβπ ) = (LIdealβπ ) | |
9 | zarcls0.2 | . . . . . . . . 9 β’ 0 = (0gβπ ) | |
10 | 8, 9 | lidl0cl 21121 | . . . . . . . 8 β’ ((π β Ring β§ π β (LIdealβπ )) β 0 β π) |
11 | 5, 7, 10 | syl2anc 582 | . . . . . . 7 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β 0 β π) |
12 | 11 | snssd 4815 | . . . . . 6 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β { 0 } β π) |
13 | 4, 12 | eqsstrd 4018 | . . . . 5 β’ (((π β Ring β§ π = { 0 }) β§ π β (PrmIdealβπ )) β π β π) |
14 | 13 | ralrimiva 3142 | . . . 4 β’ ((π β Ring β§ π = { 0 }) β βπ β (PrmIdealβπ )π β π) |
15 | rabid2 3461 | . . . 4 β’ ((PrmIdealβπ ) = {π β (PrmIdealβπ ) β£ π β π} β βπ β (PrmIdealβπ )π β π) | |
16 | 14, 15 | sylibr 233 | . . 3 β’ ((π β Ring β§ π = { 0 }) β (PrmIdealβπ ) = {π β (PrmIdealβπ ) β£ π β π}) |
17 | 3, 16 | eqtr2id 2780 | . 2 β’ ((π β Ring β§ π = { 0 }) β {π β (PrmIdealβπ ) β£ π β π} = π) |
18 | 8, 9 | lidl0 21131 | . 2 β’ (π β Ring β { 0 } β (LIdealβπ )) |
19 | 3 | fvexi 6914 | . . 3 β’ π β V |
20 | 19 | a1i 11 | . 2 β’ (π β Ring β π β V) |
21 | 2, 17, 18, 20 | fvmptd 7015 | 1 β’ (π β Ring β (πβ{ 0 }) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3057 {crab 3428 Vcvv 3471 β wss 3947 {csn 4630 β¦ cmpt 5233 βcfv 6551 0gc0g 17426 Ringcrg 20178 LIdealclidl 21107 PrmIdealcprmidl 33169 |
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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 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 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-sca 17254 df-vsca 17255 df-ip 17256 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-minusg 18899 df-sbg 18900 df-subg 19083 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-subrg 20513 df-lmod 20750 df-lss 20821 df-sra 21063 df-rgmod 21064 df-lidl 21109 df-prmidl 33170 |
This theorem is referenced by: zartopn 33481 |
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