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 | ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
zarcls0.1 | ⊢ 𝑃 = (PrmIdeal‘𝑅) |
zarcls0.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
zarcls0 | ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zarclsx.1 | . . 3 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) |
3 | zarcls0.1 | . . 3 ⊢ 𝑃 = (PrmIdeal‘𝑅) | |
4 | simplr 766 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑖 = { 0 }) | |
5 | simpll 764 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑅 ∈ Ring) | |
6 | prmidlidl 31615 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) | |
7 | 5, 6 | sylancom 588 | . . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) |
8 | eqid 2740 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
9 | zarcls0.2 | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
10 | 8, 9 | lidl0cl 20481 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → 0 ∈ 𝑗) |
11 | 5, 7, 10 | syl2anc 584 | . . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 0 ∈ 𝑗) |
12 | 11 | snssd 4748 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → { 0 } ⊆ 𝑗) |
13 | 4, 12 | eqsstrd 3964 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑖 ⊆ 𝑗) |
14 | 13 | ralrimiva 3110 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) → ∀𝑗 ∈ (PrmIdeal‘𝑅)𝑖 ⊆ 𝑗) |
15 | rabid2 3313 | . . . 4 ⊢ ((PrmIdeal‘𝑅) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ↔ ∀𝑗 ∈ (PrmIdeal‘𝑅)𝑖 ⊆ 𝑗) | |
16 | 14, 15 | sylibr 233 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) → (PrmIdeal‘𝑅) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
17 | 3, 16 | eqtr2id 2793 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑖 = { 0 }) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = 𝑃) |
18 | 8, 9 | lidl0 20488 | . 2 ⊢ (𝑅 ∈ Ring → { 0 } ∈ (LIdeal‘𝑅)) |
19 | 3 | fvexi 6785 | . . 3 ⊢ 𝑃 ∈ V |
20 | 19 | a1i 11 | . 2 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ V) |
21 | 2, 17, 18, 20 | fvmptd 6879 | 1 ⊢ (𝑅 ∈ Ring → (𝑉‘{ 0 }) = 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 {crab 3070 Vcvv 3431 ⊆ wss 3892 {csn 4567 ↦ cmpt 5162 ‘cfv 6432 0gc0g 17148 Ringcrg 19781 LIdealclidl 20430 PrmIdealcprmidl 31606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-mgp 19719 df-ur 19736 df-ring 19783 df-subrg 20020 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-lidl 20434 df-prmidl 31607 |
This theorem is referenced by: zartopn 31821 |
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