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Mirrors > Home > MPE Home > Th. List > chpmatval2 | Structured version Visualization version GIF version |
Description: The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.) |
Ref | Expression |
---|---|
chpmatply1.c | ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) |
chpmatply1.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
chpmatply1.b | ⊢ 𝐵 = (Base‘𝐴) |
chpmatply1.p | ⊢ 𝑃 = (Poly1‘𝑅) |
chpmatval2.y | ⊢ 𝑌 = (𝑁 Mat 𝑃) |
chpmatval2.m1 | ⊢ − = (-g‘𝑌) |
chpmatval2.x | ⊢ 𝑋 = (var1‘𝑅) |
chpmatval2.t1 | ⊢ · = ( ·𝑠 ‘𝑌) |
chpmatval2.t | ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) |
chpmatval2.i | ⊢ 1 = (1r‘𝑌) |
chpmatval2.g | ⊢ 𝐺 = (SymGrp‘𝑁) |
chpmatval2.h | ⊢ 𝐻 = (Base‘𝐺) |
chpmatval2.z | ⊢ 𝑍 = (ℤRHom‘𝑃) |
chpmatval2.s | ⊢ 𝑆 = (pmSgn‘𝑁) |
chpmatval2.u | ⊢ 𝑈 = (mulGrp‘𝑃) |
chpmatval2.rm | ⊢ × = (.r‘𝑃) |
Ref | Expression |
---|---|
chpmatval2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpmatply1.c | . . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) | |
2 | chpmatply1.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | chpmatply1.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
4 | chpmatply1.p | . . 3 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | chpmatval2.y | . . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃) | |
6 | eqid 2610 | . . 3 ⊢ (𝑁 maDet 𝑃) = (𝑁 maDet 𝑃) | |
7 | chpmatval2.m1 | . . 3 ⊢ − = (-g‘𝑌) | |
8 | chpmatval2.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
9 | chpmatval2.t1 | . . 3 ⊢ · = ( ·𝑠 ‘𝑌) | |
10 | chpmatval2.t | . . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) | |
11 | chpmatval2.i | . . 3 ⊢ 1 = (1r‘𝑌) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | chpmatval 20455 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
13 | eqid 2610 | . . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
14 | 5 | fveq2i 6106 | . . . . 5 ⊢ (-g‘𝑌) = (-g‘(𝑁 Mat 𝑃)) |
15 | 7, 14 | eqtri 2632 | . . . 4 ⊢ − = (-g‘(𝑁 Mat 𝑃)) |
16 | 5 | fveq2i 6106 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
17 | 9, 16 | eqtri 2632 | . . . 4 ⊢ · = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
18 | 5 | fveq2i 6106 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘(𝑁 Mat 𝑃)) |
19 | 11, 18 | eqtri 2632 | . . . 4 ⊢ 1 = (1r‘(𝑁 Mat 𝑃)) |
20 | eqid 2610 | . . . 4 ⊢ ((𝑋 · 1 ) − (𝑇‘𝑀)) = ((𝑋 · 1 ) − (𝑇‘𝑀)) | |
21 | 2, 3, 4, 13, 8, 10, 15, 17, 19, 20 | chmatcl 20452 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
22 | 5 | eqcomi 2619 | . . . . 5 ⊢ (𝑁 Mat 𝑃) = 𝑌 |
23 | 22 | fveq2i 6106 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘𝑌) |
24 | chpmatval2.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐺) | |
25 | chpmatval2.g | . . . . . 6 ⊢ 𝐺 = (SymGrp‘𝑁) | |
26 | 25 | fveq2i 6106 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘(SymGrp‘𝑁)) |
27 | 24, 26 | eqtri 2632 | . . . 4 ⊢ 𝐻 = (Base‘(SymGrp‘𝑁)) |
28 | chpmatval2.z | . . . 4 ⊢ 𝑍 = (ℤRHom‘𝑃) | |
29 | chpmatval2.s | . . . 4 ⊢ 𝑆 = (pmSgn‘𝑁) | |
30 | chpmatval2.rm | . . . 4 ⊢ × = (.r‘𝑃) | |
31 | chpmatval2.u | . . . 4 ⊢ 𝑈 = (mulGrp‘𝑃) | |
32 | 6, 5, 23, 27, 28, 29, 30, 31 | mdetleib 20212 | . . 3 ⊢ (((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃)) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥))))))) |
33 | 21, 32 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀))) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥))))))) |
34 | 12, 33 | eqtrd 2644 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥))))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 .rcmulr 15769 ·𝑠 cvsca 15772 Σg cgsu 15924 -gcsg 17247 SymGrpcsymg 17620 pmSgncpsgn 17732 mulGrpcmgp 18312 1rcur 18324 Ringcrg 18370 var1cv1 19367 Poly1cpl1 19368 ℤRHomczrh 19667 Mat cmat 20032 maDet cmdat 20209 matToPolyMat cmat2pmat 20328 CharPlyMat cchpmat 20450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-subrg 18601 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-ascl 19135 df-psr 19177 df-mvr 19178 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-vr1 19372 df-ply1 19373 df-dsmm 19895 df-frlm 19910 df-mamu 20009 df-mat 20033 df-mdet 20210 df-mat2pmat 20331 df-chpmat 20451 |
This theorem is referenced by: (None) |
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