Theorem chpmatval 22724

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 = (Poly1‘𝑅)
chpmatfval.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chpmatfval.d	⊢ 𝐷 = (𝑁 maDet 𝑃)
chpmatfval.s	⊢ − = (-g‘𝑌)
chpmatfval.x	⊢ 𝑋 = (var1‘𝑅)
chpmatfval.m	⊢ · = ( ·𝑠 ‘𝑌)
chpmatfval.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chpmatfval.i	⊢ 1 = (1r‘𝑌)

Assertion

Ref	Expression
chpmatval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

Proof of Theorem chpmatval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	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 = (Poly1‘𝑅)
5		chpmatfval.y	. . . 4 ⊢ 𝑌 = (𝑁 Mat 𝑃)
6		chpmatfval.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑃)
7		chpmatfval.s	. . . 4 ⊢ − = (-g‘𝑌)
8		chpmatfval.x	. . . 4 ⊢ 𝑋 = (var1‘𝑅)
9		chpmatfval.m	. . . 4 ⊢ · = ( ·𝑠 ‘𝑌)
10		chpmatfval.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
11		chpmatfval.i	. . . 4 ⊢ 1 = (1r‘𝑌)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	chpmatfval 22723	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
13	12	3adant3 1132	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
14		fveq2 6860	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀))
15	14	oveq2d 7405	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀)))
16	15	fveq2d 6864	. . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
17	16	adantl 481	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
18		simp3 1138	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
19		fvexd 6875	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V)
20	13, 17, 18, 19	fvmptd 6977	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ↦ cmpt 5190  ‘cfv 6513  (class class class)co 7389  Fincfn 8920  Basecbs 17185   ·_𝑠 cvsca 17230  -_gcsg 18873  1_rcur 20096  var₁cv1 22066  Poly₁cpl1 22067   Mat cmat 22300   maDet cmdat 22477   matToPolyMat cmat2pmat 22597   CharPlyMat cchpmat 22719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-chpmat 22720

This theorem is referenced by:  chpmatply1  22725  chpmatval2  22726  chpmat0d  22727  chpmat1d  22729  chpdmat  22734  cpmadurid  22760  cpmidgsum2  22772