MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chpmatval Structured version   Visualization version   GIF version

Theorem chpmatval 22180
Description: The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
Hypotheses
Ref Expression
chpmatfval.c 𝐶 = (𝑁 CharPlyMat 𝑅)
chpmatfval.a 𝐴 = (𝑁 Mat 𝑅)
chpmatfval.b 𝐵 = (Base‘𝐴)
chpmatfval.p 𝑃 = (Poly1𝑅)
chpmatfval.y 𝑌 = (𝑁 Mat 𝑃)
chpmatfval.d 𝐷 = (𝑁 maDet 𝑃)
chpmatfval.s = (-g𝑌)
chpmatfval.x 𝑋 = (var1𝑅)
chpmatfval.m · = ( ·𝑠𝑌)
chpmatfval.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chpmatfval.i 1 = (1r𝑌)
Assertion
Ref Expression
chpmatval ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐶𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))

Proof of Theorem chpmatval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 chpmatfval.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
2 chpmatfval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
3 chpmatfval.b . . . 4 𝐵 = (Base‘𝐴)
4 chpmatfval.p . . . 4 𝑃 = (Poly1𝑅)
5 chpmatfval.y . . . 4 𝑌 = (𝑁 Mat 𝑃)
6 chpmatfval.d . . . 4 𝐷 = (𝑁 maDet 𝑃)
7 chpmatfval.s . . . 4 = (-g𝑌)
8 chpmatfval.x . . . 4 𝑋 = (var1𝑅)
9 chpmatfval.m . . . 4 · = ( ·𝑠𝑌)
10 chpmatfval.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
11 chpmatfval.i . . . 4 1 = (1r𝑌)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11chpmatfval 22179 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
13123adant3 1132 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
14 fveq2 6842 . . . . 5 (𝑚 = 𝑀 → (𝑇𝑚) = (𝑇𝑀))
1514oveq2d 7373 . . . 4 (𝑚 = 𝑀 → ((𝑋 · 1 ) (𝑇𝑚)) = ((𝑋 · 1 ) (𝑇𝑀)))
1615fveq2d 6846 . . 3 (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) (𝑇𝑚))) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
1716adantl 482 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) (𝑇𝑚))) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
18 simp3 1138 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
19 fvexd 6857 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐷‘((𝑋 · 1 ) (𝑇𝑀))) ∈ V)
2013, 17, 18, 19fvmptd 6955 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐶𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cmpt 5188  cfv 6496  (class class class)co 7357  Fincfn 8883  Basecbs 17083   ·𝑠 cvsca 17137  -gcsg 18750  1rcur 19913  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-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3065  df-rex 3074  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-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  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-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-chpmat 22176
This theorem is referenced by:  chpmatply1  22181  chpmatval2  22182  chpmat0d  22183  chpmat1d  22185  chpdmat  22190  cpmadurid  22216  cpmidgsum2  22228
  Copyright terms: Public domain W3C validator