Theorem chcoeffeqlem 23116

Description: Lemma for chcoeffeq 23117. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
chcoeffeq.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
chcoeffeq.b	⊢ 𝐵 = (Base‘𝐴)
chcoeffeq.p	⊢ 𝑃 = (Poly1‘𝑅)
chcoeffeq.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
chcoeffeq.r	⊢ × = (.r‘𝑌)
chcoeffeq.s	⊢ − = (-g‘𝑌)
chcoeffeq.0	⊢ 0 = (0g‘𝑌)
chcoeffeq.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
chcoeffeq.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
chcoeffeq.k	⊢ 𝐾 = (𝐶‘𝑀)
chcoeffeq.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
chcoeffeq.w	⊢ 𝑊 = (Base‘𝑌)
chcoeffeq.1	⊢ 1 = (1r‘𝐴)
chcoeffeq.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
chcoeffeq.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)

Assertion

Ref	Expression
chcoeffeqlem	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝐺   𝑛,𝐾   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑈,𝑛   𝑛,𝑌   1 ,𝑛   ∗ ,𝑛   𝑛,𝑏   𝑛,𝑠

Allowed substitution hints:   𝐴(𝑠, 𝑏)   𝐵(𝑠, 𝑏)   𝐶(𝑛, 𝑠, 𝑏)   𝑃(𝑛, 𝑠, 𝑏)   𝑅(𝑠, 𝑏)   𝑇(𝑛, 𝑠, 𝑏)   × (𝑛, 𝑠, 𝑏)   𝑈(𝑠, 𝑏)   1 (𝑠, 𝑏)   𝐺(𝑠, 𝑏)   ∗ (𝑠, 𝑏)   𝐾(𝑠, 𝑏)   𝑀(𝑠, 𝑏)   − (𝑛, 𝑠, 𝑏)   𝑁(𝑠, 𝑏)   𝑊(𝑛, 𝑠, 𝑏)   𝑌(𝑠, 𝑏)   0 (𝑛, 𝑠, 𝑏)

Proof of Theorem chcoeffeqlem

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2762	. . . . 5 ⊢ (Poly1‘𝐴) = (Poly1‘𝐴)
2		eqid 2762	. . . . 5 ⊢ (var1‘𝐴) = (var1‘𝐴)
3		eqid 2762	. . . . 5 ⊢ (.g‘(mulGrp‘(Poly1‘𝐴))) = (.g‘(mulGrp‘(Poly1‘𝐴)))
4		crngring 20390	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5		chcoeffeq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 22671	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	4, 6	sylan2 605	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
8	7	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
9	8	adantr 486	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐴 ∈ Ring)
10		chcoeffeq.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
11		eqid 2762	. . . . 5 ⊢ ( ·𝑠 ‘(Poly1‘𝐴)) = ( ·𝑠 ‘(Poly1‘𝐴))
12		eqid 2762	. . . . 5 ⊢ (0g‘𝐴) = (0g‘𝐴)
13		chcoeffeq.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
14		chcoeffeq.y	. . . . . . . 8 ⊢ 𝑌 = (𝑁 Mat 𝑃)
15		chcoeffeq.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
16		chcoeffeq.r	. . . . . . . 8 ⊢ × = (.r‘𝑌)
17		chcoeffeq.s	. . . . . . . 8 ⊢ − = (-g‘𝑌)
18		chcoeffeq.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑌)
19		chcoeffeq.g	. . . . . . . 8 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
20		eqid 2762	. . . . . . . 8 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21		eqid 2762	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
22		eqid 2762	. . . . . . . 8 ⊢ (1r‘𝑌) = (1r‘𝑌)
23		eqid 2762	. . . . . . . 8 ⊢ (var1‘𝑅) = (var1‘𝑅)
24		eqid 2762	. . . . . . . 8 ⊢ (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))
25		eqid 2762	. . . . . . . 8 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
26		chcoeffeq.w	. . . . . . . 8 ⊢ 𝑊 = (Base‘𝑌)
27		chcoeffeq.u	. . . . . . . 8 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
28	5, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27	cpmadumatpolylem1 23112	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0))
29	28	anasss 472	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0))
30	5, 10, 13, 14, 16, 17, 18, 15, 19, 20	chfacfisfcpmat 23086	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
31	4, 30	syl3anl2 1440	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
32	31	adantr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) → 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅))
33		fvco3 6982	. . . . . . . . . 10 ⊢ ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → ((𝑈 ∘ 𝐺)‘𝑙) = (𝑈‘(𝐺‘𝑙)))
34	33	eqcomd 2768	. . . . . . . . 9 ⊢ ((𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺‘𝑙)) = ((𝑈 ∘ 𝐺)‘𝑙))
35	32, 34	sylan 592	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺‘𝑙)) = ((𝑈 ∘ 𝐺)‘𝑙))
36		elmapi 8852	. . . . . . . . . 10 ⊢ ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵)
37	36	adantl 487	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) → (𝑈 ∘ 𝐺):ℕ0⟶𝐵)
38	37	ffvelcdmda 7081	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) ∧ 𝑙 ∈ ℕ0) → ((𝑈 ∘ 𝐺)‘𝑙) ∈ 𝐵)
39	35, 38	eqeltrd 2862	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) ∧ 𝑙 ∈ ℕ0) → (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
40	39	ralrimiva 3156	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑈 ∘ 𝐺) ∈ (𝐵 ↑m ℕ0)) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
41	29, 40	mpdan 700	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
42	4	anim2i 629	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43	42	3adant3 1150	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
44	43	adantr 486	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
45	5, 10, 20, 27	cpm2mf 22983	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
46	44, 45	syl 18	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
47		fcompt 7131	. . . . . . 7 ⊢ ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵 ∧ 𝐺:ℕ0⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈 ∘ 𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺‘𝑙))))
48	46, 31, 47	syl2anc 596	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑈 ∘ 𝐺) = (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺‘𝑙))))
49	5, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1, 2, 11, 3, 27	cpmadumatpolylem2 23113	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0g‘𝐴))
50	49	anasss 472	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑈 ∘ 𝐺) finSupp (0g‘𝐴))
51	48, 50	eqbrtrrd 5133	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (𝑈‘(𝐺‘𝑙))) finSupp (0g‘𝐴))
52		simpll1 1231	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑁 ∈ Fin)
53	4	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
54	53	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 𝑅 ∈ Ring)
55		chcoeffeq.k	. . . . . . . . . 10 ⊢ 𝐾 = (𝐶‘𝑀)
56		chcoeffeq.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
57		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
58	56, 5, 10, 13, 57	chpmatply1 23063	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
59	55, 58	eqeltrid 2866	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
60		eqid 2762	. . . . . . . . . 10 ⊢ (coe1‘𝐾) = (coe1‘𝐾)
61		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
62	60, 57, 13, 61	coe1fvalcl 22443	. . . . . . . . 9 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ0) → ((coe1‘𝐾)‘𝑙) ∈ (Base‘𝑅))
63	59, 62	sylan 592	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑙 ∈ ℕ0) → ((coe1‘𝐾)‘𝑙) ∈ (Base‘𝑅))
64	63	adantlr 728	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1‘𝐾)‘𝑙) ∈ (Base‘𝑅))
65		chcoeffeq.1	. . . . . . . . . 10 ⊢ 1 = (1r‘𝐴)
66	10, 65	ringidcl 20412	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵)
67	8, 66	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐵)
68	67	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → 1 ∈ 𝐵)
69		chcoeffeq.m	. . . . . . . 8 ⊢ ∗ = ( ·𝑠 ‘𝐴)
70	61, 5, 10, 69	matvscl 22659	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe1‘𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵)) → (((coe1‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
71	52, 54, 64, 68, 70	syl22anc 852	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → (((coe1‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
72	71	ralrimiva 3156	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ∀𝑙 ∈ ℕ0 (((coe1‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
73		nn0ex 12538	. . . . . . 7 ⊢ ℕ0 ∈ V
74	73	a1i 11	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ℕ0 ∈ V)
75	5	matlmod 22657	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
76	4, 75	sylan2 605	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77	76	3adant3 1150	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ LMod)
78	77	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐴 ∈ LMod)
79		eqidd 2763	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80		fvexd 6897	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑙 ∈ ℕ0) → ((coe1‘𝐾)‘𝑙) ∈ V)
81		eqid 2762	. . . . . 6 ⊢ (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
82	5	matsca2 22648	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83	82	3adant3 1150	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
84	83, 53	eqeltrrd 2863	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) ∈ Ring)
85	83	eqcomd 2768	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) = 𝑅)
86	85	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly1‘(Scalar‘𝐴)) = (Poly1‘𝑅))
87	86, 13	eqtr4di 2815	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly1‘(Scalar‘𝐴)) = 𝑃)
88	87	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘𝑃))
89	59, 88	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴))))
90		eqid 2762	. . . . . . . . 9 ⊢ (Poly1‘(Scalar‘𝐴)) = (Poly1‘(Scalar‘𝐴))
91		eqid 2762	. . . . . . . . 9 ⊢ (Base‘(Poly1‘(Scalar‘𝐴))) = (Base‘(Poly1‘(Scalar‘𝐴)))
92	90, 91, 81	mptcoe1fsupp 22446	. . . . . . . 8 ⊢ (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly1‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ0 ↦ ((coe1‘𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
93	84, 89, 92	syl2anc 596	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑙 ∈ ℕ0 ↦ ((coe1‘𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
94	93	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ ((coe1‘𝐾)‘𝑙)) finSupp (0g‘(Scalar‘𝐴)))
95	74, 78, 79, 10, 80, 68, 12, 81, 69, 94	mptscmfsupp0 21117	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑙 ∈ ℕ0 ↦ (((coe1‘𝐾)‘𝑙) ∗ 1 )) finSupp (0g‘𝐴))
96		2fveq3 6887	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑙)))
97		oveq1 7424	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)) = (𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))
98	96, 97	oveq12d 7435	. . . . . . . 8 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))) = ((𝑈‘(𝐺‘𝑙))( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))
99	98	cbvmptv 5213	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑙))( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))
100	99	oveq2i 7428	. . . . . 6 ⊢ ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑙))( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))
101	100	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑙))( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
102		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((coe1‘𝐾)‘𝑛) = ((coe1‘𝐾)‘𝑙))
103	102	oveq1d 7432	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (((coe1‘𝐾)‘𝑛) ∗ 1 ) = (((coe1‘𝐾)‘𝑙) ∗ 1 ))
104	103, 97	oveq12d 7435	. . . . . . . 8 ⊢ (𝑛 = 𝑙 → ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))) = ((((coe1‘𝐾)‘𝑙) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))
105	104	cbvmptv 5213	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))) = (𝑙 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑙) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))
106	105	oveq2i 7428	. . . . . 6 ⊢ ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑙) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))
107	106	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑙 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑙) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑙(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
108	1, 2, 3, 9, 10, 11, 12, 41, 51, 72, 95, 101, 107	gsumply1eq 22540	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺‘𝑙)) = (((coe1‘𝐾)‘𝑙) ∗ 1 )))
109	108	biimpa 482	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺‘𝑙)) = (((coe1‘𝐾)‘𝑙) ∗ 1 ))
110	96, 103	eqeq12d 2778	. . . 4 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ) ↔ (𝑈‘(𝐺‘𝑙)) = (((coe1‘𝐾)‘𝑙) ∗ 1 )))
111	110	cbvralvw 3242	. . 3 ⊢ (∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ) ↔ ∀𝑙 ∈ ℕ0 (𝑈‘(𝐺‘𝑙)) = (((coe1‘𝐾)‘𝑙) ∗ 1 ))
112	109, 111	sylibr 237	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))
113	112	ex 418	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453  ifcif 4485   class class class wbr 5107   ↦ cmpt 5190   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ↑_m cmap 8830  Fincfn 8956   finSupp cfsupp 9335  0cc0 11128  1c1 11129   + caddc 11131   < clt 11271   − cmin 11469  ℕcn 12261  ℕ₀cn0 12532  ...cfz 13565  Basecbs 17307  ._rcmulr 17349  Scalarcsca 17351   ·_𝑠 cvsca 17352  0_gc0g 17530   Σ_g cgsu 17531  -_gcsg 19065  ._gcmg 19196  mulGrpcmgp 20279  1_rcur 20326  Ringcrg 20378  CRingccrg 20379  LModclmod 21050  var₁cv1 22407  Poly₁cpl1 22408  coe₁cco1 22409   Mat cmat 22635   maAdju cmadu 22860   ConstPolyMat ccpmat 22934   matToPolyMat cmat2pmat 22935   cPolyMatToMat ccpmat2mat 22936   CharPlyMat cchpmat 23057

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-addf 11207  ax-mulf 11208

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-ofr 7683  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8163  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fsupp 9336  df-sup 9416  df-oi 9486  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-xnn0 12606  df-z 12620  df-dec 12741  df-uz 12892  df-rp 13047  df-fz 13566  df-fzo 13714  df-seq 14070  df-exp 14130  df-hash 14399  df-word 14583  df-lsw 14632  df-concat 14640  df-s1 14667  df-substr 14713  df-pfx 14745  df-splice 14823  df-reverse 14832  df-s2 14923  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-starv 17363  df-sca 17364  df-vsca 17365  df-ip 17366  df-tset 17367  df-ple 17368  df-ds 17370  df-unif 17371  df-hom 17372  df-cco 17373  df-0g 17532  df-gsum 17533  df-prds 17538  df-pws 17540  df-mre 17676  df-mrc 17677  df-acs 17679  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-mhm 18897  df-submnd 18898  df-efmnd 18984  df-grp 19066  df-minusg 19067  df-sbg 19068  df-mulg 19197  df-subg 19252  df-ghm 19347  df-gim 19392  df-cntz 19450  df-oppg 19479  df-symg 19503  df-pmtr 19575  df-psgn 19624  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-srg 20332  df-ring 20380  df-cring 20381  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-dvr 20548  df-rhm 20619  df-subrng 20714  df-subrg 20738  df-drng 20898  df-lmod 21052  df-lss 21122  df-sra 21363  df-rgmod 21364  df-cnfld 21592  df-zring 21666  df-zrh 21722  df-dsmm 21951  df-frlm 21966  df-ascl 22076  df-psr 22130  df-mvr 22131  df-mpl 22132  df-opsr 22134  df-psr1 22411  df-vr1 22412  df-ply1 22413  df-coe1 22414  df-mamu 22619  df-mat 22636  df-mdet 22813  df-cpmat 22937  df-mat2pmat 22938  df-cpmat2mat 22939  df-chpmat 23058

This theorem is used by:  chcoeffeq  23117