Theorem chpmatval 23149

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 23148	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
13	12	3adant3 1150	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
14		fveq2 6885	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑇‘𝑚) = (𝑇‘𝑀))
15	14	oveq2d 7436	. . . 4 ⊢ (𝑚 = 𝑀 → ((𝑋 · 1 ) − (𝑇‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑀)))
16	15	fveq2d 6889	. . 3 ⊢ (𝑚 = 𝑀 → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
17	16	adantl 487	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ 𝑚 = 𝑀) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))
18		simp3 1156	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
19		fvexd 6900	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))) ∈ V)
20	13, 17, 18, 19	fvmptd 7001	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ↦ cmpt 5186  ‘cfv 6538  (class class class)co 7420  Fincfn 8973  Basecbs 17387   ·_𝑠 cvsca 17432  -_gcsg 19146  1_rcur 20407  var₁cv1 22494  Poly₁cpl1 22495   Mat cmat 22722   maDet cmdat 22899   matToPolyMat cmat2pmat 23022   CharPlyMat cchpmat 23144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-chpmat 23145

This theorem is used by:  chpmatply1  23150  chpmatval2  23151  chpmat0d  23152  chpmat1d  23154  chpdmat  23159  cpmadurid  23185  cpmidgsum2  23197