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 2737 | . . . . 5 β’ (LIdealβπ ) = (LIdealβπ ) | |
3 | eqid 2737 | . . . . 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 6850 | . . . . . . 7 β’ (0gβπ ) β V | |
7 | hashsng 14196 | . . . . . . 7 β’ ((0gβπ ) β V β (β―β{(0gβπ )}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 β’ (β―β{(0gβπ )}) = 1 |
9 | simpr 485 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β {(0gβπ )} = (Baseβπ )) | |
10 | 9 | fveq2d 6841 | . . . . . 6 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β (β―β{(0gβπ )}) = (β―β(Baseβπ ))) |
11 | 8, 10 | eqtr3id 2791 | . . . . 5 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 = (β―β(Baseβπ ))) |
12 | 1red 11089 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 β β) | |
13 | eqid 2737 | . . . . . . . . . 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 497 | . . . . . . . 8 β’ (π β NzRing β 1 < (β―β(Baseβπ ))) |
16 | 15 | adantr 481 | . . . . . . 7 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 < (β―β(Baseβπ ))) |
17 | 12, 16 | ltned 11224 | . . . . . 6 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β 1 β (β―β(Baseβπ ))) |
18 | 17 | neneqd 2946 | . . . . 5 β’ ((π β NzRing β§ {(0gβπ )} = (Baseβπ )) β Β¬ 1 = (β―β(Baseβπ ))) |
19 | 11, 18 | pm2.65da 815 | . . . 4 β’ (π β NzRing β Β¬ {(0gβπ )} = (Baseβπ )) |
20 | 19 | neqned 2948 | . . 3 β’ (π β NzRing β {(0gβπ )} β (Baseβπ )) |
21 | 13 | ssmxidl 32028 | . . 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 1371 | . 2 β’ (π β NzRing β βπ β (MaxIdealβπ ){(0gβπ )} β π) |
23 | df-rex 3072 | . . 3 β’ (βπ β (MaxIdealβπ ){(0gβπ )} β π β βπ(π β (MaxIdealβπ ) β§ {(0gβπ )} β π)) | |
24 | exsimpl 1871 | . . 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 396 = wceq 1541 βwex 1781 β wcel 2106 β wne 2941 βwrex 3071 Vcvv 3443 β wss 3908 {csn 4584 class class class wbr 5103 βcfv 6491 1c1 10985 < clt 11122 β―chash 14157 Basecbs 17017 0gc0g 17255 Ringcrg 19888 LIdealclidl 20554 NzRingcnzr 20650 MaxIdealcmxidl 32017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-ac2 10332 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-se 5586 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-rpss 7650 df-om 7793 df-1st 7911 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-1o 8379 df-oadd 8383 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-fin 8820 df-dju 9770 df-card 9808 df-ac 9985 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-nn 12087 df-2 12149 df-3 12150 df-4 12151 df-5 12152 df-6 12153 df-7 12154 df-8 12155 df-n0 12347 df-xnn0 12419 df-z 12433 df-uz 12696 df-fz 13353 df-hash 14158 df-sets 16970 df-slot 16988 df-ndx 17000 df-base 17018 df-ress 17047 df-plusg 17080 df-mulr 17081 df-sca 17083 df-vsca 17084 df-ip 17085 df-0g 17257 df-mgm 18431 df-sgrp 18480 df-mnd 18491 df-grp 18685 df-minusg 18686 df-sbg 18687 df-subg 18857 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 32018 |
This theorem is referenced by: mxidlnzrb 32030 |
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