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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 2736 | . . 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 22180 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = ((𝑁 maDet 𝑃)‘((𝑋 · 1 ) − (𝑇‘𝑀)))) |
13 | eqid 2736 | . . . 4 ⊢ (𝑁 Mat 𝑃) = (𝑁 Mat 𝑃) | |
14 | 5 | fveq2i 6845 | . . . . 5 ⊢ (-g‘𝑌) = (-g‘(𝑁 Mat 𝑃)) |
15 | 7, 14 | eqtri 2764 | . . . 4 ⊢ − = (-g‘(𝑁 Mat 𝑃)) |
16 | 5 | fveq2i 6845 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
17 | 9, 16 | eqtri 2764 | . . . 4 ⊢ · = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
18 | 5 | fveq2i 6845 | . . . . 5 ⊢ (1r‘𝑌) = (1r‘(𝑁 Mat 𝑃)) |
19 | 11, 18 | eqtri 2764 | . . . 4 ⊢ 1 = (1r‘(𝑁 Mat 𝑃)) |
20 | eqid 2736 | . . . 4 ⊢ ((𝑋 · 1 ) − (𝑇‘𝑀)) = ((𝑋 · 1 ) − (𝑇‘𝑀)) | |
21 | 2, 3, 4, 13, 8, 10, 15, 17, 19, 20 | chmatcl 22177 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 ) − (𝑇‘𝑀)) ∈ (Base‘(𝑁 Mat 𝑃))) |
22 | 5 | eqcomi 2745 | . . . . 5 ⊢ (𝑁 Mat 𝑃) = 𝑌 |
23 | 22 | fveq2i 6845 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘𝑌) |
24 | chpmatval2.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐺) | |
25 | chpmatval2.g | . . . . . 6 ⊢ 𝐺 = (SymGrp‘𝑁) | |
26 | 25 | fveq2i 6845 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘(SymGrp‘𝑁)) |
27 | 24, 26 | eqtri 2764 | . . . 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 21936 | . . 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 2776 | 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 1087 = wceq 1541 ∈ wcel 2106 ↦ cmpt 5188 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 Basecbs 17083 .rcmulr 17134 ·𝑠 cvsca 17137 Σg cgsu 17322 -gcsg 18750 SymGrpcsymg 19148 pmSgncpsgn 19271 mulGrpcmgp 19896 1rcur 19913 Ringcrg 19964 ℤRHomczrh 20900 var1cv1 21547 Poly1cpl1 21548 Mat cmat 21754 maDet cmdat 21933 matToPolyMat cmat2pmat 22053 CharPlyMat cchpmat 22175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-subrg 20220 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-dsmm 21138 df-frlm 21153 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-opsr 21315 df-psr1 21551 df-vr1 21552 df-ply1 21553 df-mamu 21733 df-mat 21755 df-mdet 21934 df-mat2pmat 22056 df-chpmat 22176 |
This theorem is referenced by: (None) |
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