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Theorem chcoeffeqlem 23079

Description: Lemma for chcoeffeq 23080. (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 = (0_g‘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 = (1_r‘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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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 2766	. . . . 5 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
2		eqid 2766	. . . . 5 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
3		eqid 2766	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
4		crngring 20358	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5		chcoeffeq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 22637	. . . . . . . 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 2766	. . . . 5 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
12		eqid 2766	. . . . 5 ⊢ (0_g‘𝐴) = (0_g‘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 = (0_g‘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 2766	. . . . . . . 8 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21		eqid 2766	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
22		eqid 2766	. . . . . . . 8 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
23		eqid 2766	. . . . . . . 8 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24		eqid 2766	. . . . . . . 8 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
25		eqid 2766	. . . . . . . 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 23075	. . . . . . 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 23049	. . . . . . . . . . 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 6988	. . . . . . . . . 10 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → ((𝑈 ∘ 𝐺)‘𝑙) = (𝑈‘(𝐺‘𝑙)))
34	33	eqcomd 2772	. . . . . . . . 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 8855	. . . . . . . . . 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 7086	. . . . . . . 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 2866	. . . . . . 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 3160	. . . . . 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 22946	. . . . . . . 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 7136	. . . . . . 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 23076	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
50	49	anasss 472	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
51	48, 50	eqbrtrrd 5140	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑙 ∈ ℕ₀ ↦ (𝑈‘(𝐺‘𝑙))) finSupp (0_g‘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 2766	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
58	56, 5, 10, 13, 57	chpmatply1 23026	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
59	55, 58	eqeltrid 2870	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
60		eqid 2766	. . . . . . . . . 10 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
61		eqid 2766	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
62	60, 57, 13, 61	coe1fvalcl 22409	. . . . . . . . 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 = (1_r‘𝐴)
66	10, 65	ringidcl 20380	. . . . . . . . 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 22625	. . . . . . 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 3160	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
73		nn0ex 12528	. . . . . . 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 22623	. . . . . . . . 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 2767	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80		fvexd 6903	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ V)
81		eqid 2766	. . . . . 6 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
82	5	matsca2 22614	. . . . . . . . . 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 2867	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) ∈ Ring)
85	83	eqcomd 2772	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) = 𝑅)
86	85	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘𝑅))
87	86, 13	eqtr4di 2819	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = 𝑃)
88	87	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘𝑃))
89	59, 88	eleqtrrd 2869	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴))))
90		eqid 2766	. . . . . . . . 9 ⊢ (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘(Scalar‘𝐴))
91		eqid 2766	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘(Poly₁‘(Scalar‘𝐴)))
92	90, 91, 81	mptcoe1fsupp 22412	. . . . . . . 8 ⊢ (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
93	84, 89, 92	syl2anc 596	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(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 (0_g‘(Scalar‘𝐴)))
95	74, 78, 79, 10, 80, 68, 12, 81, 69, 94	mptscmfsupp0 21085	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑙 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑙) ∗ 1 )) finSupp (0_g‘𝐴))
96		2fveq3 6893	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑙)))
97		oveq1 7430	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)) = (𝑙(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))
98	96, 97	oveq12d 7441	. . . . . . . 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 5220	. . . . . . 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 7434	. . . . . 6 ⊢ ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑙))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑙(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))))
102		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((coe₁‘𝐾)‘𝑛) = ((coe₁‘𝐾)‘𝑙))
103	102	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (((coe₁‘𝐾)‘𝑛) ∗ 1 ) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
104	103, 97	oveq12d 7441	. . . . . . . 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 5220	. . . . . . 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 7434	. . . . . 6 ⊢ ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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 22506	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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 2782	. . . 4 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 )))
111	110	cbvralvw 3246	. . 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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (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 )))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3082 Vcvv 3458 ifcif 4492 class class class wbr 5114 ↦ cmpt 5197 ∘ ccom 5670 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ↑_m cmap 8833 Fincfn 8952 finSupp cfsupp 9331 0cc0 11118 1c1 11119 + caddc 11121 < clt 11261 − cmin 11459 ℕcn 12251 ℕ₀cn0 12522 ...cfz 13553 Basecbs 17294 ._rcmulr 17336 Scalarcsca 17338 ·_𝑠 cvsca 17339 0_gc0g 17517 Σ_g cgsu 17518 -_gcsg 19033 ._gcmg 19164 mulGrpcmgp 20247 1_rcur 20294 Ringcrg 20346 CRingccrg 20347 LModclmod 21018 var₁cv1 22373 Poly₁cpl1 22374 coe₁cco1 22375 Mat cmat 22601 maAdju cmadu 22826 ConstPolyMat ccpmat 22897 matToPolyMat cmat2pmat 22898 cPolyMatToMat ccpmat2mat 22899 CharPlyMat cchpmat 23020

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-addf 11197 ax-mulf 11198

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-ofr 7688 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-word 14571 df-lsw 14620 df-concat 14628 df-s1 14655 df-substr 14701 df-pfx 14733 df-splice 14811 df-reverse 14820 df-s2 14911 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-0g 17519 df-gsum 17520 df-prds 17525 df-pws 17527 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-efmnd 18959 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-ghm 19315 df-gim 19360 df-cntz 19418 df-oppg 19447 df-symg 19471 df-pmtr 19543 df-psgn 19592 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-srg 20300 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-dvr 20516 df-rhm 20587 df-subrng 20682 df-subrg 20706 df-drng 20866 df-lmod 21020 df-lss 21090 df-sra 21331 df-rgmod 21332 df-cnfld 21560 df-zring 21634 df-zrh 21690 df-dsmm 21919 df-frlm 21934 df-ascl 22042 df-psr 22096 df-mvr 22097 df-mpl 22098 df-opsr 22100 df-psr1 22377 df-vr1 22378 df-ply1 22379 df-coe1 22380 df-mamu 22585 df-mat 22602 df-mdet 22779 df-cpmat 22900 df-mat2pmat 22901 df-cpmat2mat 22902 df-chpmat 23021

This theorem is used by: chcoeffeq 23080

W3C validator