![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cpmidgsum | Structured version Visualization version GIF version |
Description: Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.) |
Ref | Expression |
---|---|
cpmidgsum.a | β’ π΄ = (π Mat π ) |
cpmidgsum.b | β’ π΅ = (Baseβπ΄) |
cpmidgsum.p | β’ π = (Poly1βπ ) |
cpmidgsum.y | β’ π = (π Mat π) |
cpmidgsum.x | β’ π = (var1βπ ) |
cpmidgsum.e | β’ β = (.gβ(mulGrpβπ)) |
cpmidgsum.m | β’ Β· = ( Β·π βπ) |
cpmidgsum.1 | β’ 1 = (1rβπ) |
cpmidgsum.u | β’ π = (algScβπ) |
cpmidgsum.c | β’ πΆ = (π CharPlyMat π ) |
cpmidgsum.k | β’ πΎ = (πΆβπ) |
cpmidgsum.h | β’ π» = (πΎ Β· 1 ) |
Ref | Expression |
---|---|
cpmidgsum | β’ ((π β Fin β§ π β CRing β§ π β π΅) β π» = (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmidgsum.k | . . 3 β’ πΎ = (πΆβπ) | |
2 | cpmidgsum.c | . . . 4 β’ πΆ = (π CharPlyMat π ) | |
3 | cpmidgsum.a | . . . 4 β’ π΄ = (π Mat π ) | |
4 | cpmidgsum.b | . . . 4 β’ π΅ = (Baseβπ΄) | |
5 | cpmidgsum.p | . . . 4 β’ π = (Poly1βπ ) | |
6 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
7 | 2, 3, 4, 5, 6 | chpmatply1 22334 | . . 3 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΆβπ) β (Baseβπ)) |
8 | 1, 7 | eqeltrid 2838 | . 2 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΎ β (Baseβπ)) |
9 | cpmidgsum.y | . . 3 β’ π = (π Mat π) | |
10 | eqid 2733 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
11 | cpmidgsum.m | . . 3 β’ Β· = ( Β·π βπ) | |
12 | cpmidgsum.e | . . 3 β’ β = (.gβ(mulGrpβπ)) | |
13 | cpmidgsum.x | . . 3 β’ π = (var1βπ ) | |
14 | eqid 2733 | . . 3 β’ (π matToPolyMat π ) = (π matToPolyMat π ) | |
15 | cpmidgsum.u | . . 3 β’ π = (algScβπ) | |
16 | eqid 2733 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
17 | cpmidgsum.1 | . . 3 β’ 1 = (1rβπ) | |
18 | cpmidgsum.h | . . 3 β’ π» = (πΎ Β· 1 ) | |
19 | 5, 9, 10, 11, 12, 13, 14, 3, 4, 15, 16, 6, 15, 17, 18 | pmatcollpwscmat 22293 | . 2 β’ ((π β Fin β§ π β CRing β§ πΎ β (Baseβπ)) β π» = (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 ))))) |
20 | 8, 19 | syld3an3 1410 | 1 β’ ((π β Fin β§ π β CRing β§ π β π΅) β π» = (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β¦ cmpt 5232 βcfv 6544 (class class class)co 7409 Fincfn 8939 β0cn0 12472 Basecbs 17144 Β·π cvsca 17201 Ξ£g cgsu 17386 .gcmg 18950 mulGrpcmgp 19987 1rcur 20004 CRingccrg 20057 algSccascl 21407 var1cv1 21700 Poly1cpl1 21701 coe1cco1 21702 Mat cmat 21907 matToPolyMat cmat2pmat 22206 CharPlyMat cchpmat 22328 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-splice 14700 df-reverse 14709 df-s2 14799 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-efmnd 18750 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-gim 19133 df-cntz 19181 df-oppg 19210 df-symg 19235 df-pmtr 19310 df-psgn 19359 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-srg 20010 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-lmod 20473 df-lss 20543 df-sra 20785 df-rgmod 20786 df-cnfld 20945 df-zring 21018 df-zrh 21053 df-dsmm 21287 df-frlm 21302 df-assa 21408 df-ascl 21410 df-psr 21462 df-mvr 21463 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-vr1 21705 df-ply1 21706 df-coe1 21707 df-mamu 21886 df-mat 21908 df-mdet 22087 df-mat2pmat 22209 df-decpmat 22265 df-chpmat 22329 |
This theorem is referenced by: cpmidgsumm2pm 22371 cpmidg2sum 22382 |
Copyright terms: Public domain | W3C validator |