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Theorem chpmatval 20397
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 20396 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
13123adant3 1073 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
14 fveq2 6088 . . . . 5 (𝑚 = 𝑀 → (𝑇𝑚) = (𝑇𝑀))
1514oveq2d 6543 . . . 4 (𝑚 = 𝑀 → ((𝑋 · 1 ) (𝑇𝑚)) = ((𝑋 · 1 ) (𝑇𝑀)))
1615fveq2d 6092 . . 3 (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) (𝑇𝑚))) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
1716adantl 480 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) (𝑇𝑚))) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
18 simp3 1055 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → 𝑀𝐵)
19 fvex 6098 . . 3 (𝐷‘((𝑋 · 1 ) (𝑇𝑀))) ∈ V
2019a1i 11 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐷‘((𝑋 · 1 ) (𝑇𝑀))) ∈ V)
2113, 17, 18, 20fvmptd 6182 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐶𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637  cfv 5790  (class class class)co 6527  Fincfn 7818  Basecbs 15641   ·𝑠 cvsca 15718  -gcsg 17193  1rcur 18270  var1cv1 19313  Poly1cpl1 19314   Mat cmat 19974   maDet cmdat 20151   matToPolyMat cmat2pmat 20270   CharPlyMat cchpmat 20392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-chpmat 20393
This theorem is referenced by:  chpmatply1  20398  chpmatval2  20399  chpmat0d  20400  chpmat1d  20402  chpdmat  20407  cpmadurid  20433  cpmidgsum2  20445
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