Theorem chpmatval 21434

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 21433	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
13	12	3adant3 1127	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
14		fveq2 6663	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀))
15	14	oveq2d 7165	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀)))
16	15	fveq2d 6667	. . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
17	16	adantl 484	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
18		simp3 1133	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
19		fvexd 6678	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V)
20	13, 17, 18, 19	fvmptd 6768	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

Syntax hints:   → wi 4   ∧ w3a 1082   = wceq 1536   ∈ wcel 2113  Vcvv 3491   ↦ cmpt 5139  ‘cfv 6348  (class class class)co 7149  Fincfn 8502  Basecbs 16478   ·_𝑠 cvsca 16564  -_gcsg 18100  1_rcur 19246  var₁cv1 20339  Poly₁cpl1 20340   Mat cmat 21011   maDet cmdat 21188   matToPolyMat cmat2pmat 21307   CharPlyMat cchpmat 21429

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-chpmat 21430

This theorem is referenced by:  chpmatply1  21435  chpmatval2  21436  chpmat0d  21437  chpmat1d  21439  chpdmat  21444  cpmadurid  21470  cpmidgsum2  21482