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Mirrors > Home > MPE Home > Th. List > chpscmat0 | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.) |
Ref | Expression |
---|---|
chp0mat.c | β’ πΆ = (π CharPlyMat π ) |
chp0mat.p | β’ π = (Poly1βπ ) |
chp0mat.a | β’ π΄ = (π Mat π ) |
chp0mat.x | β’ π = (var1βπ ) |
chp0mat.g | β’ πΊ = (mulGrpβπ) |
chp0mat.m | β’ β = (.gβπΊ) |
chpscmat.d | β’ π· = {π β (Baseβπ΄) β£ βπ β (Baseβπ )βπ β π βπ β π (πππ) = if(π = π, π, (0gβπ ))} |
chpscmat.s | β’ π = (algScβπ) |
chpscmat.m | β’ β = (-gβπ) |
Ref | Expression |
---|---|
chpscmat0 | β’ (((π β Fin β§ π β CRing) β§ (π β π· β§ πΌ β π β§ βπ β π (πππ) = (πΌππΌ))) β (πΆβπ) = ((β―βπ) β (π β (πβ(πΌππΌ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chp0mat.c | . 2 β’ πΆ = (π CharPlyMat π ) | |
2 | chp0mat.p | . 2 β’ π = (Poly1βπ ) | |
3 | chp0mat.a | . 2 β’ π΄ = (π Mat π ) | |
4 | chp0mat.x | . 2 β’ π = (var1βπ ) | |
5 | chp0mat.g | . 2 β’ πΊ = (mulGrpβπ) | |
6 | chp0mat.m | . 2 β’ β = (.gβπΊ) | |
7 | chpscmat.d | . 2 β’ π· = {π β (Baseβπ΄) β£ βπ β (Baseβπ )βπ β π βπ β π (πππ) = if(π = π, π, (0gβπ ))} | |
8 | chpscmat.s | . 2 β’ π = (algScβπ) | |
9 | chpscmat.m | . 2 β’ β = (-gβπ) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | chpscmat 22351 | 1 β’ (((π β Fin β§ π β CRing) β§ (π β π· β§ πΌ β π β§ βπ β π (πππ) = (πΌππΌ))) β (πΆβπ) = ((β―βπ) β (π β (πβ(πΌππΌ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 βwrex 3070 {crab 3432 ifcif 4528 βcfv 6543 (class class class)co 7411 Fincfn 8941 β―chash 14292 Basecbs 17146 0gc0g 17387 -gcsg 18823 .gcmg 18952 mulGrpcmgp 19989 CRingccrg 20059 algSccascl 21413 var1cv1 21706 Poly1cpl1 21707 Mat cmat 21914 CharPlyMat cchpmat 22335 |
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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-word 14467 df-lsw 14515 df-concat 14523 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14702 df-reverse 14711 df-s2 14801 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-0g 17389 df-gsum 17390 df-prds 17395 df-pws 17397 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-efmnd 18752 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-ghm 19092 df-gim 19135 df-cntz 19183 df-oppg 19212 df-symg 19237 df-pmtr 19312 df-psgn 19361 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-rnghom 20255 df-subrg 20321 df-drng 20363 df-lmod 20477 df-lss 20548 df-sra 20791 df-rgmod 20792 df-cnfld 20951 df-zring 21024 df-zrh 21059 df-dsmm 21293 df-frlm 21308 df-ascl 21416 df-psr 21468 df-mvr 21469 df-mpl 21470 df-opsr 21472 df-psr1 21710 df-vr1 21711 df-ply1 21712 df-mamu 21893 df-mat 21915 df-mdet 22094 df-mat2pmat 22216 df-chpmat 22336 |
This theorem is referenced by: chpscmatgsumbin 22353 |
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