Theorem chpmatfval 23153

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 23150	. . . 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 7429	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
5		chpmatfval.a	. . . . . . . 8 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	4, 5	eqtr4di 2813	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = 𝐴)
7	6	fveq2d 6889	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘𝐴))
8		chpmatfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
9	7, 8	eqtr4di 2813	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
10		simpl 488	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑛 = 𝑁)
11		simpr 490	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
12	11	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = (Poly1‘𝑅))
13		chpmatfval.p	. . . . . . . . 9 ⊢ 𝑃 = (Poly1‘𝑅)
14	12, 13	eqtr4di 2813	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = 𝑃)
15	10, 14	oveq12d 7438	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 maDet (Poly1‘𝑟)) = (𝑁 maDet 𝑃))
16		chpmatfval.d	. . . . . . 7 ⊢ 𝐷 = (𝑁 maDet 𝑃)
17	15, 16	eqtr4di 2813	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 maDet (Poly1‘𝑟)) = 𝐷)
18		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
19	18	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = (Poly1‘𝑅))
20	19, 13	eqtr4di 2813	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Poly1‘𝑟) = 𝑃)
21	10, 20	oveq12d 7438	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat (Poly1‘𝑟)) = (𝑁 Mat 𝑃))
22		chpmatfval.y	. . . . . . . . . 10 ⊢ 𝑌 = (𝑁 Mat 𝑃)
23	21, 22	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat (Poly1‘𝑟)) = 𝑌)
24	23	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1‘𝑟))) = (-g‘𝑌))
25		chpmatfval.s	. . . . . . . 8 ⊢ − = (-g‘𝑌)
26	24, 25	eqtr4di 2813	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (-g‘(𝑛 Mat (Poly1‘𝑟))) = − )
27	23	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟))) = ( ·𝑠 ‘𝑌))
28		chpmatfval.m	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑌)
29	27, 28	eqtr4di 2813	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟))) = · )
30		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (var1‘𝑟) = (var1‘𝑅))
31		chpmatfval.x	. . . . . . . . . 10 ⊢ 𝑋 = (var1‘𝑅)
32	30, 31	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (var1‘𝑟) = 𝑋)
33	32	adantl 487	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (var1‘𝑟) = 𝑋)
34	23	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1‘𝑟))) = (1r‘𝑌))
35		chpmatfval.i	. . . . . . . . 9 ⊢ 1 = (1r‘𝑌)
36	34, 35	eqtr4di 2813	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat (Poly1‘𝑟))) = 1 )
37	29, 33, 36	oveq123d 7441	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟)))) = (𝑋 · 1 ))
38		oveq12 7429	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = (𝑁 matToPolyMat 𝑅))
39		chpmatfval.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
40	38, 39	eqtr4di 2813	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 matToPolyMat 𝑟) = 𝑇)
41	40	fveq1d 6887	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑛 matToPolyMat 𝑟)‘𝑚) = (𝑇‘𝑚))
42	26, 37, 41	oveq123d 7441	. . . . . 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 6892	. . . . 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 5191	. . . 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 487	. . 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 488	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
47		elex 3471	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
48	47	adantl 487	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
49	8	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
50		mptexg 7227	. . . 4 ⊢ (𝐵 ∈ V → (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))) ∈ V)
51	49, 50	mp1i 14	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))) ∈ V)
52	3, 45, 46, 48, 51	ovmpod 7572	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 CharPlyMat 𝑅) = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))
53	1, 52	eqtrid 2807	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ↦ cmpt 5185  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Fincfn 8976  Basecbs 17392   ·_𝑠 cvsca 17437  -_gcsg 19151  1_rcur 20412  var₁cv1 22499  Poly₁cpl1 22500   Mat cmat 22727   maDet cmdat 22904   matToPolyMat cmat2pmat 23027   CharPlyMat cchpmat 23149

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 23150

This theorem is used by:  chpmatval  23154