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Mirrors > Home > MPE Home > Th. List > cpmidg2sum | Structured version Visualization version GIF version |
Description: Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.) |
Ref | Expression |
---|---|
cpmadugsum.a | β’ π΄ = (π Mat π ) |
cpmadugsum.b | β’ π΅ = (Baseβπ΄) |
cpmadugsum.p | β’ π = (Poly1βπ ) |
cpmadugsum.y | β’ π = (π Mat π) |
cpmadugsum.t | β’ π = (π matToPolyMat π ) |
cpmadugsum.x | β’ π = (var1βπ ) |
cpmadugsum.e | β’ β = (.gβ(mulGrpβπ)) |
cpmadugsum.m | β’ Β· = ( Β·π βπ) |
cpmadugsum.r | β’ Γ = (.rβπ) |
cpmadugsum.1 | β’ 1 = (1rβπ) |
cpmadugsum.g | β’ + = (+gβπ) |
cpmadugsum.s | β’ β = (-gβπ) |
cpmadugsum.i | β’ πΌ = ((π Β· 1 ) β (πβπ)) |
cpmadugsum.j | β’ π½ = (π maAdju π) |
cpmadugsum.0 | β’ 0 = (0gβπ) |
cpmadugsum.g2 | β’ πΊ = (π β β0 β¦ if(π = 0, ( 0 β ((πβπ) Γ (πβ(πβ0)))), if(π = (π + 1), (πβ(πβπ )), if((π + 1) < π, 0 , ((πβ(πβ(π β 1))) β ((πβπ) Γ (πβ(πβπ)))))))) |
cpmidgsum2.c | β’ πΆ = (π CharPlyMat π ) |
cpmidgsum2.k | β’ πΎ = (πΆβπ) |
cpmidg2sum.u | β’ π = (algScβπ) |
Ref | Expression |
---|---|
cpmidg2sum | β’ ((π β Fin β§ π β CRing β§ π β π΅) β βπ β β βπ β (π΅ βm (0...π ))(π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )))) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmadugsum.a | . . . . . 6 β’ π΄ = (π Mat π ) | |
2 | cpmadugsum.b | . . . . . 6 β’ π΅ = (Baseβπ΄) | |
3 | cpmadugsum.p | . . . . . 6 β’ π = (Poly1βπ ) | |
4 | cpmadugsum.y | . . . . . 6 β’ π = (π Mat π) | |
5 | cpmadugsum.x | . . . . . 6 β’ π = (var1βπ ) | |
6 | cpmadugsum.e | . . . . . 6 β’ β = (.gβ(mulGrpβπ)) | |
7 | cpmadugsum.m | . . . . . 6 β’ Β· = ( Β·π βπ) | |
8 | cpmadugsum.1 | . . . . . 6 β’ 1 = (1rβπ) | |
9 | cpmidg2sum.u | . . . . . 6 β’ π = (algScβπ) | |
10 | cpmidgsum2.c | . . . . . 6 β’ πΆ = (π CharPlyMat π ) | |
11 | cpmidgsum2.k | . . . . . 6 β’ πΎ = (πΆβπ) | |
12 | eqid 2726 | . . . . . 6 β’ (πΎ Β· 1 ) = (πΎ Β· 1 ) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cpmidgsum 22720 | . . . . 5 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 ))))) |
14 | 13 | eqcomd 2732 | . . . 4 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )))) = (πΎ Β· 1 )) |
15 | 14 | ad3antrrr 727 | . . 3 β’ (((((π β Fin β§ π β CRing β§ π β π΅) β§ π β β) β§ π β (π΅ βm (0...π ))) β§ (πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) β (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )))) = (πΎ Β· 1 )) |
16 | simpr 484 | . . 3 β’ (((((π β Fin β§ π β CRing β§ π β π΅) β§ π β β) β§ π β (π΅ βm (0...π ))) β§ (πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) β (πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) | |
17 | 15, 16 | eqtrd 2766 | . 2 β’ (((((π β Fin β§ π β CRing β§ π β π΅) β§ π β β) β§ π β (π΅ βm (0...π ))) β§ (πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) β (π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )))) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) |
18 | cpmadugsum.t | . . 3 β’ π = (π matToPolyMat π ) | |
19 | cpmadugsum.r | . . 3 β’ Γ = (.rβπ) | |
20 | cpmadugsum.g | . . 3 β’ + = (+gβπ) | |
21 | cpmadugsum.s | . . 3 β’ β = (-gβπ) | |
22 | cpmadugsum.i | . . 3 β’ πΌ = ((π Β· 1 ) β (πβπ)) | |
23 | cpmadugsum.j | . . 3 β’ π½ = (π maAdju π) | |
24 | cpmadugsum.0 | . . 3 β’ 0 = (0gβπ) | |
25 | cpmadugsum.g2 | . . 3 β’ πΊ = (π β β0 β¦ if(π = 0, ( 0 β ((πβπ) Γ (πβ(πβ0)))), if(π = (π + 1), (πβ(πβπ )), if((π + 1) < π, 0 , ((πβ(πβ(π β 1))) β ((πβπ) Γ (πβ(πβπ)))))))) | |
26 | 1, 2, 3, 4, 18, 5, 6, 7, 19, 8, 20, 21, 22, 23, 24, 25, 10, 11, 12 | cpmidgsum2 22731 | . 2 β’ ((π β Fin β§ π β CRing β§ π β π΅) β βπ β β βπ β (π΅ βm (0...π ))(πΎ Β· 1 ) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) |
27 | 17, 26 | reximddv2 3206 | 1 β’ ((π β Fin β§ π β CRing β§ π β π΅) β βπ β β βπ β (π΅ βm (0...π ))(π Ξ£g (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )))) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πΊβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwrex 3064 ifcif 4523 class class class wbr 5141 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βm cmap 8819 Fincfn 8938 0cc0 11109 1c1 11110 + caddc 11112 < clt 11249 β cmin 11445 βcn 12213 β0cn0 12473 ...cfz 13487 Basecbs 17150 +gcplusg 17203 .rcmulr 17204 Β·π cvsca 17207 0gc0g 17391 Ξ£g cgsu 17392 -gcsg 18862 .gcmg 18992 mulGrpcmgp 20036 1rcur 20083 CRingccrg 20136 algSccascl 21742 var1cv1 22045 Poly1cpl1 22046 coe1cco1 22047 Mat cmat 22257 maAdju cmadu 22484 matToPolyMat cmat2pmat 22556 CharPlyMat cchpmat 22678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-tpos 8209 df-cur 8250 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14030 df-hash 14293 df-word 14468 df-lsw 14516 df-concat 14524 df-s1 14549 df-substr 14594 df-pfx 14624 df-splice 14703 df-reverse 14712 df-s2 14802 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-submnd 18711 df-efmnd 18791 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-ghm 19136 df-gim 19181 df-cntz 19230 df-oppg 19259 df-symg 19284 df-pmtr 19359 df-psgn 19408 df-evpm 19409 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-srg 20089 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-drng 20586 df-lmod 20705 df-lss 20776 df-sra 21018 df-rgmod 21019 df-cnfld 21236 df-zring 21329 df-zrh 21385 df-dsmm 21622 df-frlm 21637 df-assa 21743 df-ascl 21745 df-psr 21798 df-mvr 21799 df-mpl 21800 df-opsr 21802 df-psr1 22049 df-vr1 22050 df-ply1 22051 df-coe1 22052 df-mamu 22236 df-mat 22258 df-mdet 22437 df-madu 22486 df-mat2pmat 22559 df-decpmat 22615 df-chpmat 22679 |
This theorem is referenced by: (None) |
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