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Mirrors > Home > MPE Home > Th. List > chp0mat | Structured version Visualization version Unicode version |
Description: The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.) |
Ref | Expression |
---|---|
chp0mat.c | CharPlyMat |
chp0mat.p | Poly1 |
chp0mat.a | Mat |
chp0mat.x | var1 |
chp0mat.g | mulGrp |
chp0mat.m | .g |
chp0mat.0 |
Ref | Expression |
---|---|
chp0mat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 | |
2 | simpr 477 | . . 3 | |
3 | crngring 18558 | . . . . 5 | |
4 | chp0mat.a | . . . . . 6 Mat | |
5 | 4 | matring 20249 | . . . . 5 |
6 | 3, 5 | sylan2 491 | . . . 4 |
7 | ringgrp 18552 | . . . 4 | |
8 | eqid 2622 | . . . . 5 | |
9 | chp0mat.0 | . . . . 5 | |
10 | 8, 9 | grpidcl 17450 | . . . 4 |
11 | 6, 7, 10 | 3syl 18 | . . 3 |
12 | eqid 2622 | . . . . . . . . . 10 | |
13 | 4, 12 | mat0op 20225 | . . . . . . . . 9 |
14 | 9, 13 | syl5eq 2668 | . . . . . . . 8 |
15 | 3, 14 | sylan2 491 | . . . . . . 7 |
16 | 15 | adantr 481 | . . . . . 6 |
17 | eqidd 2623 | . . . . . 6 | |
18 | simpl 473 | . . . . . . 7 | |
19 | 18 | adantl 482 | . . . . . 6 |
20 | simpr 477 | . . . . . . 7 | |
21 | 20 | adantl 482 | . . . . . 6 |
22 | fvexd 6203 | . . . . . 6 | |
23 | 16, 17, 19, 21, 22 | ovmpt2d 6788 | . . . . 5 |
24 | 23 | a1d 25 | . . . 4 |
25 | 24 | ralrimivva 2971 | . . 3 |
26 | chp0mat.c | . . . 4 CharPlyMat | |
27 | chp0mat.p | . . . 4 Poly1 | |
28 | eqid 2622 | . . . 4 algSc algSc | |
29 | chp0mat.x | . . . 4 var1 | |
30 | chp0mat.g | . . . 4 mulGrp | |
31 | eqid 2622 | . . . 4 | |
32 | 26, 27, 4, 28, 8, 29, 12, 30, 31 | chpdmat 20646 | . . 3 g algSc |
33 | 1, 2, 11, 25, 32 | syl31anc 1329 | . 2 g algSc |
34 | 15 | adantr 481 | . . . . . . . . 9 |
35 | eqidd 2623 | . . . . . . . . 9 | |
36 | simpr 477 | . . . . . . . . 9 | |
37 | fvexd 6203 | . . . . . . . . 9 | |
38 | 34, 35, 36, 36, 37 | ovmpt2d 6788 | . . . . . . . 8 |
39 | 38 | fveq2d 6195 | . . . . . . 7 algSc algSc |
40 | 3 | adantl 482 | . . . . . . . . 9 |
41 | eqid 2622 | . . . . . . . . . 10 | |
42 | 27, 28, 12, 41 | ply1scl0 19660 | . . . . . . . . 9 algSc |
43 | 40, 42 | syl 17 | . . . . . . . 8 algSc |
44 | 43 | adantr 481 | . . . . . . 7 algSc |
45 | 39, 44 | eqtrd 2656 | . . . . . 6 algSc |
46 | 45 | oveq2d 6666 | . . . . 5 algSc |
47 | 27 | ply1ring 19618 | . . . . . . . . . 10 |
48 | ringgrp 18552 | . . . . . . . . . 10 | |
49 | 3, 47, 48 | 3syl 18 | . . . . . . . . 9 |
50 | 49 | adantl 482 | . . . . . . . 8 |
51 | eqid 2622 | . . . . . . . . . 10 | |
52 | 29, 27, 51 | vr1cl 19587 | . . . . . . . . 9 |
53 | 40, 52 | syl 17 | . . . . . . . 8 |
54 | 50, 53 | jca 554 | . . . . . . 7 |
55 | 54 | adantr 481 | . . . . . 6 |
56 | 51, 41, 31 | grpsubid1 17500 | . . . . . 6 |
57 | 55, 56 | syl 17 | . . . . 5 |
58 | 46, 57 | eqtrd 2656 | . . . 4 algSc |
59 | 58 | mpteq2dva 4744 | . . 3 algSc |
60 | 59 | oveq2d 6666 | . 2 g algSc g |
61 | 27 | ply1crng 19568 | . . . . 5 |
62 | 30 | crngmgp 18555 | . . . . 5 CMnd |
63 | cmnmnd 18208 | . . . . 5 CMnd | |
64 | 61, 62, 63 | 3syl 18 | . . . 4 |
65 | 64 | adantl 482 | . . 3 |
66 | 3, 52 | syl 17 | . . . . 5 |
67 | 66 | adantl 482 | . . . 4 |
68 | 30, 51 | mgpbas 18495 | . . . 4 |
69 | 67, 68 | syl6eleq 2711 | . . 3 |
70 | eqid 2622 | . . . 4 | |
71 | chp0mat.m | . . . 4 .g | |
72 | 70, 71 | gsumconst 18334 | . . 3 g |
73 | 65, 1, 69, 72 | syl3anc 1326 | . 2 g |
74 | 33, 60, 73 | 3eqtrd 2660 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 wral 2912 cvv 3200 cmpt 4729 cfv 5888 (class class class)co 6650 cmpt2 6652 cfn 7955 chash 13117 cbs 15857 c0g 16100 g cgsu 16101 cmnd 17294 cgrp 17422 csg 17424 .gcmg 17540 CMndccmn 18193 mulGrpcmgp 18489 crg 18547 ccrg 18548 algSccascl 19311 var1cv1 19546 Poly1cpl1 19547 Mat cmat 20213 CharPlyMat cchpmat 20631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-gim 17701 df-cntz 17750 df-oppg 17776 df-symg 17798 df-pmtr 17862 df-psgn 17911 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-rnghom 18715 df-drng 18749 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-mdet 20391 df-mat2pmat 20512 df-chpmat 20632 |
This theorem is referenced by: (None) |
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