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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 22725 | . 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 480 | . . . . . . 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 2726 | . . . . . . . . . 10 β’ (Baseβπ) = (Baseβπ) | |
| 17 | 10, 1, 2, 3, 16 | chpmatply1 22689 | . . . . . . . . 9 β’ ((π β Fin β§ π β CRing β§ π β π΅) β (πΆβπ) β (Baseβπ)) |
| 18 | 11, 17 | eqeltrid 2831 | . . . . . . . 8 β’ ((π β Fin β§ π β CRing β§ π β π΅) β πΎ β (Baseβπ)) |
| 19 | eqid 2726 | . . . . . . . . 9 β’ (coe1βπΎ) = (coe1βπΎ) | |
| 20 | eqid 2726 | . . . . . . . . 9 β’ (Baseβπ ) = (Baseβπ ) | |
| 21 | 19, 16, 3, 20 | coe1fvalcl 22086 | . . . . . . . 8 β’ ((πΎ β (Baseβπ) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
| 22 | 18, 21 | sylan 579 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β ((coe1βπΎ)βπ) β (Baseβπ )) |
| 23 | crngring 20150 | . . . . . . . . . . 11 β’ (π β CRing β π β Ring) | |
| 24 | 23 | anim2i 616 | . . . . . . . . . 10 β’ ((π β Fin β§ π β CRing) β (π β Fin β§ π β Ring)) |
| 25 | 1 | matring 22300 | . . . . . . . . . 10 β’ ((π β Fin β§ π β Ring) β π΄ β Ring) |
| 26 | cpmidgsumm2pm.o | . . . . . . . . . . 11 β’ π = (1rβπ΄) | |
| 27 | 2, 26 | ringidcl 20165 | . . . . . . . . . 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 480 | . . . . . . 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 2726 | . . . . . . . 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 22592 | . . . . . . 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 834 | . . . . . 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 22591 | . . . . . . . . 9 β’ ((π β Fin β§ π β CRing) β π β (π΄ RingHom π)) |
| 37 | 26, 8 | rhm1 20391 | . . . . . . . . 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 480 | . . . . . . 7 β’ (((π β Fin β§ π β CRing β§ π β π΅) β§ π β β0) β (πβπ) = 1 ) |
| 40 | 39 | oveq2d 7421 | . . . . . 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 2767 | . . . . 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 7421 | . . . 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 5241 | . . 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 7421 | . 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 2766 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 Fincfn 8941 β0cn0 12476 Basecbs 17153 Β·π cvsca 17210 Ξ£g cgsu 17395 .gcmg 18995 mulGrpcmgp 20039 1rcur 20086 Ringcrg 20138 CRingccrg 20139 RingHom crh 20371 algSccascl 21747 var1cv1 22050 Poly1cpl1 22051 coe1cco1 22052 Mat cmat 22262 matToPolyMat cmat2pmat 22561 CharPlyMat cchpmat 22683 |
| 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 7722 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 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-splice 14706 df-reverse 14715 df-s2 14805 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-efmnd 18794 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-gim 19184 df-cntz 19233 df-oppg 19262 df-symg 19287 df-pmtr 19362 df-psgn 19411 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-dsmm 21627 df-frlm 21642 df-assa 21748 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-mamu 22241 df-mat 22263 df-mdet 22442 df-mat2pmat 22564 df-decpmat 22620 df-chpmat 22684 |
| This theorem is referenced by: cpmidpmat 22730 |
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