Theorem chpmatfval 22768

Description: Value of the characteristic polynomial function. (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
chpmatfval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))

Distinct variable groups:   𝐵,𝑚   𝐷,𝑚   1 ,𝑚   𝑚,𝑁   𝑅,𝑚   𝑚,𝑋   𝑇,𝑚   · ,𝑚   − ,𝑚

Allowed substitution hints:   𝐴(𝑚)   𝐶(𝑚)   𝑃(𝑚)   𝑉(𝑚)   𝑌(𝑚)

Proof of Theorem chpmatfval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		chpmatfval.c	. 2 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
2		df-chpmat 22765	. . . 4 ⊢ CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))))))
4		oveq12 7414	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5		chpmatfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	4, 5	eqtr4di 2788	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
7	6	fveq2d 6880	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8		chpmatfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
9	7, 8	eqtr4di 2788	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10		simpl 482	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
11		simpr 484	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
12	11	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = (Poly1‘𝑅))
13		chpmatfval.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
14	12, 13	eqtr4di 2788	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = 𝑃)
15	10, 14	oveq12d 7423	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 maDet (Poly1‘𝑟)) = (𝑁 maDet 𝑃))
16		chpmatfval.d	. . . . . . 7 ⊢ 𝐷 = (𝑁 maDet 𝑃)
17	15, 16	eqtr4di 2788	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 maDet (Poly1‘𝑟)) = 𝐷)
18		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = (Poly1‘𝑅))
20	19, 13	eqtr4di 2788	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = 𝑃)
21	10, 20	oveq12d 7423	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat (Poly1‘𝑟)) = (𝑁 Mat 𝑃))
22		chpmatfval.y	. . . . . . . . . 10 ⊢ 𝑌 = (𝑁 Mat 𝑃)
23	21, 22	eqtr4di 2788	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat (Poly1‘𝑟)) = 𝑌)
24	23	fveq2d 6880	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1‘𝑟))) = (-g‘𝑌))
25		chpmatfval.s	. . . . . . . 8 ⊢ − = (-g‘𝑌)
26	24, 25	eqtr4di 2788	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1‘𝑟))) = − )
27	23	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟))) = ( ·𝑠 ‘𝑌))
28		chpmatfval.m	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑌)
29	27, 28	eqtr4di 2788	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟))) = · )
30		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (var1‘𝑟) = (var1‘𝑅))
31		chpmatfval.x	. . . . . . . . . 10 ⊢ 𝑋 = (var1‘𝑅)
32	30, 31	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (var1‘𝑟) = 𝑋)
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (var1‘𝑟) = 𝑋)
34	23	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1‘𝑟))) = (1r‘𝑌))
35		chpmatfval.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑌)
36	34, 35	eqtr4di 2788	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1‘𝑟))) = 1 )
37	29, 33, 36	oveq123d 7426	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟)))) = (𝑋 · 1 ))
38		oveq12 7414	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39		chpmatfval.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
40	38, 39	eqtr4di 2788	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
41	40	fveq1d 6878	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇‘𝑚))
42	26, 37, 41	oveq123d 7426	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)) = ((𝑋 · 1 ) − (𝑇‘𝑚)))
43	17, 42	fveq12d 6883	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))
44	9, 43	mpteq12dv 5207	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))) = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
45	44	adantl 481	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))) = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
46		simpl 482	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
47		elex 3480	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
48	47	adantl 481	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
49	8	fvexi 6890	. . . 4 ⊢ 𝐵 ∈ V
50		mptexg 7213	. . . 4 ⊢ (𝐵 ∈ V → (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))) ∈ V)
51	49, 50	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))) ∈ V)
52	3, 45, 46, 48, 51	ovmpod 7559	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
53	1, 52	eqtrid 2782	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ↦ cmpt 5201  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8959  Basecbs 17228   ·_𝑠 cvsca 17275  -_gcsg 18918  1_rcur 20141  var₁cv1 22111  Poly₁cpl1 22112   Mat cmat 22345   maDet cmdat 22522   matToPolyMat cmat2pmat 22642   CharPlyMat cchpmat 22764

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-chpmat 22765

This theorem is referenced by:  chpmatval  22769