Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22078 | . 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 6822 | . . . . 5 ⊢ (-g‘𝑌) = (-g‘(𝑁 Mat 𝑃)) |
15 | 7, 14 | eqtri 2764 | . . . 4 ⊢ − = (-g‘(𝑁 Mat 𝑃)) |
16 | 5 | fveq2i 6822 | . . . . 5 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
17 | 9, 16 | eqtri 2764 | . . . 4 ⊢ · = ( ·𝑠 ‘(𝑁 Mat 𝑃)) |
18 | 5 | fveq2i 6822 | . . . . 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 22075 | . . 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 6822 | . . . 4 ⊢ (Base‘(𝑁 Mat 𝑃)) = (Base‘𝑌) |
24 | chpmatval2.h | . . . . 5 ⊢ 𝐻 = (Base‘𝐺) | |
25 | chpmatval2.g | . . . . . 6 ⊢ 𝐺 = (SymGrp‘𝑁) | |
26 | 25 | fveq2i 6822 | . . . . 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 21834 | . . 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 1086 = wceq 1540 ∈ wcel 2105 ↦ cmpt 5172 ∘ ccom 5618 ‘cfv 6473 (class class class)co 7329 Fincfn 8796 Basecbs 17001 .rcmulr 17052 ·𝑠 cvsca 17055 Σg cgsu 17240 -gcsg 18667 SymGrpcsymg 19062 pmSgncpsgn 19185 mulGrpcmgp 19807 1rcur 19824 Ringcrg 19870 ℤRHomczrh 20799 var1cv1 21445 Poly1cpl1 21446 Mat cmat 21652 maDet cmdat 21831 matToPolyMat cmat2pmat 21951 CharPlyMat cchpmat 22073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-ot 4581 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-ofr 7588 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-pm 8681 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-fz 13333 df-fzo 13476 df-seq 13815 df-hash 14138 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-hom 17075 df-cco 17076 df-0g 17241 df-gsum 17242 df-prds 17247 df-pws 17249 df-mre 17384 df-mrc 17385 df-acs 17387 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-mulg 18789 df-subg 18840 df-ghm 18920 df-cntz 19011 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-subrg 20119 df-lmod 20223 df-lss 20292 df-sra 20532 df-rgmod 20533 df-dsmm 21037 df-frlm 21052 df-ascl 21160 df-psr 21210 df-mvr 21211 df-mpl 21212 df-opsr 21214 df-psr1 21449 df-vr1 21450 df-ply1 21451 df-mamu 21631 df-mat 21653 df-mdet 21832 df-mat2pmat 21954 df-chpmat 22074 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |