Theorem chpmatval 22751

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 22750	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
13	12	3adant3 1129	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
14		fveq2 6892	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀))
15	14	oveq2d 7432	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀)))
16	15	fveq2d 6896	. . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
17	16	adantl 480	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
18		simp3 1135	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
19		fvexd 6907	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V)
20	13, 17, 18, 19	fvmptd 7007	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ↦ cmpt 5226  ‘cfv 6543  (class class class)co 7416  Fincfn 8962  Basecbs 17179   ·_𝑠 cvsca 17236  -_gcsg 18896  1_rcur 20125  var₁cv1 22103  Poly₁cpl1 22104   Mat cmat 22325   maDet cmdat 22504   matToPolyMat cmat2pmat 22624   CharPlyMat cchpmat 22746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-chpmat 22747

This theorem is referenced by:  chpmatply1  22752  chpmatval2  22753  chpmat0d  22754  chpmat1d  22756  chpdmat  22761  cpmadurid  22787  cpmidgsum2  22799