Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > krull | Structured version Visualization version GIF version |
Description: Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.) |
Ref | Expression |
---|---|
krull | β’ (π β NzRing β βπ π β (MaxIdealβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nzrring 20654 | . . 3 β’ (π β NzRing β π β Ring) | |
2 | eqid 2738 | . . . . 5 β’ (LIdealβπ ) = (LIdealβπ ) | |
3 | eqid 2738 | . . . . 5 β’ (0gβπ ) = (0gβπ ) | |
4 | 2, 3 | lidl0 20612 | . . . 4 β’ (π β Ring β {(0gβπ )} β (LIdealβπ )) |
5 | 1, 4 | syl 17 | . . 3 β’ (π β NzRing β {(0gβπ )} β (LIdealβπ )) |
6 | fvex 6851 | . . . . . . 7 β’ (0gβπ ) β V | |
7 | hashsng 14197 | . . . . . . 7 β’ ((0gβπ ) β V β (β―β{(0gβπ )}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 β’ (β―β{(0gβπ )}) = 1 |
9 | simpr 486 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β {(0gβπ )} = (Baseβπ )) | |
10 | 9 | fveq2d 6842 | . . . . . 6 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β (β―β{(0gβπ )}) = (β―β(Baseβπ ))) |
11 | 8, 10 | eqtr3id 2792 | . . . . 5 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 = (β―β(Baseβπ ))) |
12 | 1red 11090 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 β β) | |
13 | eqid 2738 | . . . . . . . . . 10 β’ (Baseβπ ) = (Baseβπ ) | |
14 | 13 | isnzr2hash 20657 | . . . . . . . . 9 β’ (π β NzRing β (π β Ring β§ 1 < (β―β(Baseβπ )))) |
15 | 14 | simprbi 498 | . . . . . . . 8 β’ (π β NzRing β 1 < (β―β(Baseβπ ))) |
16 | 15 | adantr 482 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 < (β―β(Baseβπ ))) |
17 | 12, 16 | ltned 11225 | . . . . . 6 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 β (β―β(Baseβπ ))) |
18 | 17 | neneqd 2947 | . . . . 5 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β Β¬ 1 = (β―β(Baseβπ ))) |
19 | 11, 18 | pm2.65da 816 | . . . 4 β’ (π β NzRing β Β¬ {(0gβπ )} = (Baseβπ )) |
20 | 19 | neqned 2949 | . . 3 β’ (π β NzRing β {(0gβπ )} β (Baseβπ )) |
21 | 13 | ssmxidl 32016 | . . 3 β’ ((π β Ring β§ {(0gβπ )} β (LIdealβπ ) β§ {(0gβπ )} β (Baseβπ )) β βπ β (MaxIdealβπ ){(0gβπ )} β π) |
22 | 1, 5, 20, 21 | syl3anc 1372 | . 2 β’ (π β NzRing β βπ β (MaxIdealβπ ){(0gβπ )} β π) |
23 | df-rex 3073 | . . 3 β’ (βπ β (MaxIdealβπ ){(0gβπ )} β π β βπ(π β (MaxIdealβπ ) β§ {(0gβπ )} β π)) | |
24 | exsimpl 1872 | . . 3 β’ (βπ(π β (MaxIdealβπ ) β§ {(0gβπ )} β π) β βπ π β (MaxIdealβπ )) | |
25 | 23, 24 | sylbi 216 | . 2 β’ (βπ β (MaxIdealβπ ){(0gβπ )} β π β βπ π β (MaxIdealβπ )) |
26 | 22, 25 | syl 17 | 1 β’ (π β NzRing β βπ π β (MaxIdealβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 β wne 2942 βwrex 3072 Vcvv 3444 β wss 3909 {csn 4585 class class class wbr 5104 βcfv 6492 1c1 10986 < clt 11123 β―chash 14158 Basecbs 17018 0gc0g 17256 Ringcrg 19888 LIdealclidl 20554 NzRingcnzr 20650 MaxIdealcmxidl 32005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-ac2 10333 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-rpss 7651 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-oadd 8384 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-dju 9771 df-card 9809 df-ac 9986 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-n0 12348 df-xnn0 12420 df-z 12434 df-uz 12697 df-fz 13354 df-hash 14159 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-sca 17084 df-vsca 17085 df-ip 17086 df-0g 17258 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-grp 18686 df-minusg 18687 df-sbg 18688 df-subg 18858 df-mgp 19826 df-ur 19843 df-ring 19890 df-subrg 20143 df-lmod 20247 df-lss 20316 df-sra 20556 df-rgmod 20557 df-lidl 20558 df-nzr 20651 df-mxidl 32006 |
This theorem is referenced by: mxidlnzrb 32018 |
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