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| Mirrors > Home > MPE Home > Th. List > cpmidgsumm2pm | Structured version Visualization version GIF version | ||
| Description: Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-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 ) |
| cpmidgsumm2pm.o | β’ π = (1rβπ΄) |
| cpmidgsumm2pm.m | β’ β = ( Β·π βπ΄) |
| cpmidgsumm2pm.t | β’ π = (π matToPolyMat π ) |
| Ref | Expression |
|---|---|
| cpmidgsumm2pm | 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 π)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cpmidgsum.a | . . 3 β’ π΄ = (π Mat π ) | |
| 2 | cpmidgsum.b | . . 3 β’ π΅ = (Baseβπ΄) | |
| 3 | cpmidgsum.p | . . 3 β’ π = (Poly1βπ ) | |
| 4 | cpmidgsum.y | . . 3 β’ π = (π Mat π) | |
| 5 | cpmidgsum.x | . . 3 β’ π = (var1βπ ) | |
| 6 | cpmidgsum.e | . . 3 β’ β = (.gβ(mulGrpβπ)) | |
| 7 | cpmidgsum.m | . . 3 β’ Β· = ( Β·π βπ) | |
| 8 | cpmidgsum.1 | . . 3 β’ 1 = (1rβπ) | |
| 9 | cpmidgsum.u | . . 3 β’ π = (algScβπ) | |
| 10 | cpmidgsum.c | . . 3 β’ πΆ = (π CharPlyMat π ) | |
| 11 | cpmidgsum.k | . . 3 β’ πΎ = (πΆβπ) | |
| 12 | cpmidgsum.h | . . 3 β’ π» = (πΎ Β· 1 ) | |
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cpmidgsum 22786 | . 2 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 ))))) |
| 14 | 3simpa 1145 | . . . . . . . 8 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (π β Fin β§ π β CRing)) | |
| 15 | 14 | adantr 479 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β (π β Fin β§ π β CRing)) |
| 16 | eqid 2725 | . . . . . . . . . 10 β’ (Baseβπ) = (Baseβπ) | |
| 17 | 10, 1, 2, 3, 16 | chpmatply1 22750 | . . . . . . . . 9 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΆβπ) β (Baseβπ)) |
| 18 | 11, 17 | eqeltrid 2829 | . . . . . . . 8 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΎ β (Baseβπ)) |
| 19 | eqid 2725 | . . . . . . . . 9 β’ (coe1βπΎ) = (coe1βπΎ) | |
| 20 | eqid 2725 | . . . . . . . . 9 β’ (Baseβπ ) = (Baseβπ ) | |
| 21 | 19, 16, 3, 20 | coe1fvalcl 22138 | . . . . . . . 8 β’ ((πΎ β (Baseβπ) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
| 22 | 18, 21 | sylan 578 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
| 23 | crngring 20187 | . . . . . . . . . . 11 β’ (π β CRing β π β Ring) | |
| 24 | 23 | anim2i 615 | . . . . . . . . . 10 β’ ((π β Fin β§ π β CRing) β (π β Fin β§ π β Ring)) |
| 25 | 1 | matring 22361 | . . . . . . . . . 10 β’ ((π β Fin β§ π β Ring) β π΄ β Ring) |
| 26 | cpmidgsumm2pm.o | . . . . . . . . . . 11 β’ π = (1rβπ΄) | |
| 27 | 2, 26 | ringidcl 20204 | . . . . . . . . . 10 β’ (π΄ β Ring β π β π΅) |
| 28 | 24, 25, 27 | 3syl 18 | . . . . . . . . 9 β’ ((π β Fin β§ π β CRing) β π β π΅) |
| 29 | 28 | 3adant3 1129 | . . . . . . . 8 β’ ((π β Fin β§ π β CRing β§ π β π΅) β π β π΅) |
| 30 | 29 | adantr 479 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β π β π΅) |
| 31 | cpmidgsumm2pm.t | . . . . . . . 8 β’ π = (π matToPolyMat π ) | |
| 32 | eqid 2725 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
| 33 | cpmidgsumm2pm.m | . . . . . . . 8 β’ β = ( Β·π βπ΄) | |
| 34 | 31, 1, 2, 3, 4, 32, 20, 9, 33, 7 | mat2pmatlin 22653 | . . . . . . 7 β’ (((π β Fin β§ π β CRing) β§ (((coe1βπΎ)βπ) β (Baseβπ ) β§ π β π΅)) β (πβ(((coe1βπΎ)βπ) β π)) = ((πβ((coe1βπΎ)βπ)) Β· (πβπ))) |
| 35 | 15, 22, 30, 34 | syl12anc 835 | . . . . . 6 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β (πβ(((coe1βπΎ)βπ) β π)) = ((πβ((coe1βπΎ)βπ)) Β· (πβπ))) |
| 36 | 31, 1, 2, 3, 4, 32 | mat2pmatrhm 22652 | . . . . . . . . 9 β’ ((π β Fin β§ π β CRing) β π β (π΄ RingHom π)) |
| 37 | 26, 8 | rhm1 20430 | . . . . . . . . 9 β’ (π β (π΄ RingHom π) β (πβπ) = 1 ) |
| 38 | 14, 36, 37 | 3syl 18 | . . . . . . . 8 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πβπ) = 1 ) |
| 39 | 38 | adantr 479 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β (πβπ) = 1 ) |
| 40 | 39 | oveq2d 7431 | . . . . . 6 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((πβ((coe1βπΎ)βπ)) Β· (πβπ)) = ((πβ((coe1βπΎ)βπ)) Β· 1 )) |
| 41 | 35, 40 | eqtr2d 2766 | . . . . 5 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((πβ((coe1βπΎ)βπ)) Β· 1 ) = (πβ(((coe1βπΎ)βπ) β π))) |
| 42 | 41 | oveq2d 7431 | . . . 4 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 )) = ((π β π) Β· (πβ(((coe1βπΎ)βπ) β π)))) |
| 43 | 42 | mpteq2dva 5243 | . . 3 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (π β β0 β¦ ((π β π) Β· ((πβ((coe1βπΎ)βπ)) Β· 1 ))) = (π β β0 β¦ ((π β π) Β· (πβ(((coe1βπΎ)βπ) β π))))) |
| 44 | 43 | oveq2d 7431 | . 2 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 )))) = (π Ξ£g (π β β0 β¦ ((π β π) Β· (πβ(((coe1βπΎ)βπ) β π)))))) |
| 45 | 13, 44 | eqtrd 2765 | 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 π)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β¦ cmpt 5226 βcfv 6542 (class class class)co 7415 Fincfn 8960 β0cn0 12500 Basecbs 17177 Β·π cvsca 17234 Ξ£g cgsu 17419 .gcmg 19025 mulGrpcmgp 20076 1rcur 20123 Ringcrg 20175 CRingccrg 20176 RingHom crh 20410 algSccascl 21788 var1cv1 22101 Poly1cpl1 22102 coe1cco1 22103 Mat cmat 22323 matToPolyMat cmat2pmat 22622 CharPlyMat cchpmat 22744 |
| 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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-addf 11215 ax-mulf 11216 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-ofr 7682 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-word 14495 df-lsw 14543 df-concat 14551 df-s1 14576 df-substr 14621 df-pfx 14651 df-splice 14730 df-reverse 14739 df-s2 14829 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-efmnd 18823 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-gim 19215 df-cntz 19270 df-oppg 19299 df-symg 19324 df-pmtr 19399 df-psgn 19448 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-srg 20129 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-drng 20628 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-dsmm 21668 df-frlm 21683 df-assa 21789 df-ascl 21791 df-psr 21844 df-mvr 21845 df-mpl 21846 df-opsr 21848 df-psr1 22105 df-vr1 22106 df-ply1 22107 df-coe1 22108 df-mamu 22307 df-mat 22324 df-mdet 22503 df-mat2pmat 22625 df-decpmat 22681 df-chpmat 22745 |
| This theorem is referenced by: cpmidpmat 22791 |
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