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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 | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
cpmidgsum.b | ⊢ 𝐵 = (Base‘𝐴) |
cpmidgsum.p | ⊢ 𝑃 = (Poly1‘𝑅) |
cpmidgsum.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
cpmidgsum.x | ⊢ 𝑋 = (var1‘𝑅) |
cpmidgsum.e | ⊢ ↑ = (.g‘(mulGrp‘𝑃)) |
cpmidgsum.m | ⊢ · = ( ·𝑠 ‘𝑌) |
cpmidgsum.1 | ⊢ 1 = (1r‘𝑌) |
cpmidgsum.u | ⊢ 𝑈 = (algSc‘𝑃) |
cpmidgsum.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
cpmidgsum.k | ⊢ 𝐾 = (𝐶‘𝑀) |
cpmidgsum.h | ⊢ 𝐻 = (𝐾 · 1 ) |
Ref | Expression |
---|---|
cpmidgsum | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑛)) · 1 ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpmidgsum.k | . . 3 ⊢ 𝐾 = (𝐶‘𝑀) | |
2 | cpmidgsum.c | . . . 4 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
3 | cpmidgsum.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
4 | cpmidgsum.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
5 | cpmidgsum.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | eqid 2737 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
7 | 2, 3, 4, 5, 6 | chpmatply1 21729 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃)) |
8 | 1, 7 | eqeltrid 2842 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃)) |
9 | cpmidgsum.y | . . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
10 | eqid 2737 | . . 3 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
11 | cpmidgsum.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑌) | |
12 | cpmidgsum.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑃)) | |
13 | cpmidgsum.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
14 | eqid 2737 | . . 3 ⊢ (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅) | |
15 | cpmidgsum.u | . . 3 ⊢ 𝑈 = (algSc‘𝑃) | |
16 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
17 | cpmidgsum.1 | . . 3 ⊢ 1 = (1r‘𝑌) | |
18 | cpmidgsum.h | . . 3 ⊢ 𝐻 = (𝐾 · 1 ) | |
19 | 5, 9, 10, 11, 12, 13, 14, 3, 4, 15, 16, 6, 15, 17, 18 | pmatcollpwscmat 21688 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐾 ∈ (Base‘𝑃)) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑛)) · 1 ))))) |
20 | 8, 19 | syld3an3 1411 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) · ((𝑈‘((coe1‘𝐾)‘𝑛)) · 1 ))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ↦ cmpt 5135 ‘cfv 6380 (class class class)co 7213 Fincfn 8626 ℕ0cn0 12090 Basecbs 16760 ·𝑠 cvsca 16806 Σg cgsu 16945 .gcmg 18488 mulGrpcmgp 19504 1rcur 19516 CRingccrg 19563 algSccascl 20814 var1cv1 21097 Poly1cpl1 21098 coe1cco1 21099 Mat cmat 21304 matToPolyMat cmat2pmat 21601 CharPlyMat cchpmat 21723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-xor 1508 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-ot 4550 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-word 14070 df-lsw 14118 df-concat 14126 df-s1 14153 df-substr 14206 df-pfx 14236 df-splice 14315 df-reverse 14324 df-s2 14413 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-0g 16946 df-gsum 16947 df-prds 16952 df-pws 16954 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-mhm 18218 df-submnd 18219 df-efmnd 18296 df-grp 18368 df-minusg 18369 df-sbg 18370 df-mulg 18489 df-subg 18540 df-ghm 18620 df-gim 18663 df-cntz 18711 df-oppg 18738 df-symg 18760 df-pmtr 18834 df-psgn 18883 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-srg 19521 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-rnghom 19735 df-drng 19769 df-subrg 19798 df-lmod 19901 df-lss 19969 df-sra 20209 df-rgmod 20210 df-cnfld 20364 df-zring 20436 df-zrh 20470 df-dsmm 20694 df-frlm 20709 df-assa 20815 df-ascl 20817 df-psr 20868 df-mvr 20869 df-mpl 20870 df-opsr 20872 df-psr1 21101 df-vr1 21102 df-ply1 21103 df-coe1 21104 df-mamu 21283 df-mat 21305 df-mdet 21482 df-mat2pmat 21604 df-decpmat 21660 df-chpmat 21724 |
This theorem is referenced by: cpmidgsumm2pm 21766 cpmidg2sum 21777 |
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