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Theorem chcoeffeqlem 22378

Description: Lemma for chcoeffeq 22379. (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 2732	. . . . 5 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
2		eqid 2732	. . . . 5 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
3		eqid 2732	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
4		crngring 20061	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5		chcoeffeq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 21936	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	4, 6	sylan2 593	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
8	7	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
9	8	adantr 481	. . . . 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 2732	. . . . 5 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
12		eqid 2732	. . . . 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 2732	. . . . . . . 8 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21		eqid 2732	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
22		eqid 2732	. . . . . . . 8 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
23		eqid 2732	. . . . . . . 8 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24		eqid 2732	. . . . . . . 8 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
25		eqid 2732	. . . . . . . 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 22374	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀))
29	28	anasss 467	. . . . . 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 22348	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
31	4, 30	syl3anl2 1413	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
32	31	adantr 481	. . . . . . . . 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 6987	. . . . . . . . . 10 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → ((𝑈 ∘ 𝐺)‘𝑙) = (𝑈‘(𝐺‘𝑙)))
34	33	eqcomd 2738	. . . . . . . . 9 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → (𝑈‘(𝐺‘𝑙)) = ((𝑈 ∘ 𝐺)‘𝑙))
35	32, 34	sylan 580	. . . . . . . 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 8839	. . . . . . . . . 10 ⊢ ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀) → (𝑈 ∘ 𝐺):ℕ₀⟶𝐵)
37	36	adantl 482	. . . . . . . . 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 7083	. . . . . . . 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 2833	. . . . . . 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 3146	. . . . . 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 685	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
42	4	anim2i 617	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43	42	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
44	43	adantr 481	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
45	5, 10, 20, 27	cpm2mf 22245	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
46	44, 45	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵)
47		fcompt 7127	. . . . . . 7 ⊢ ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵 ∧ 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈 ∘ 𝐺) = (𝑙 ∈ ℕ₀ ↦ (𝑈‘(𝐺‘𝑙))))
48	46, 31, 47	syl2anc 584	. . . . . 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 22375	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
50	49	anasss 467	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
51	48, 50	eqbrtrrd 5171	. . . . 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 1212	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → 𝑁 ∈ Fin)
53	4	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
54	53	ad2antrr 724	. . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
58	56, 5, 10, 13, 57	chpmatply1 22325	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
59	55, 58	eqeltrid 2837	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
60		eqid 2732	. . . . . . . . . 10 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
61		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
62	60, 57, 13, 61	coe1fvalcl 21727	. . . . . . . . 9 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
63	59, 62	sylan 580	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
64	63	adantlr 713	. . . . . . 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 20076	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵)
67	8, 66	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐵)
68	67	ad2antrr 724	. . . . . . 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 21924	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
71	52, 54, 64, 68, 70	syl22anc 837	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
72	71	ralrimiva 3146	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
73		nn0ex 12474	. . . . . . 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 21922	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
76	4, 75	sylan2 593	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77	76	3adant3 1132	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ LMod)
78	77	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐴 ∈ LMod)
79		eqidd 2733	. . . . . 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 2732	. . . . . 6 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
82	5	matsca2 21913	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83	82	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
84	83, 53	eqeltrrd 2834	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) ∈ Ring)
85	83	eqcomd 2738	. . . . . . . . . . . 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 2790	. . . . . . . . . 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 2836	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴))))
90		eqid 2732	. . . . . . . . 9 ⊢ (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘(Scalar‘𝐴))
91		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘(Poly₁‘(Scalar‘𝐴)))
92	90, 91, 81	mptcoe1fsupp 21730	. . . . . . . 8 ⊢ (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
93	84, 89, 92	syl2anc 584	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
94	93	adantr 481	. . . . . 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 20529	. . . . 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 7412	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)) = (𝑙(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))
98	96, 97	oveq12d 7423	. . . . . . . 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 5260	. . . . . . 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 7416	. . . . . 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 7420	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (((coe₁‘𝐾)‘𝑛) ∗ 1 ) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
104	103, 97	oveq12d 7423	. . . . . . . 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 5260	. . . . . . 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 7416	. . . . . 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 21820	. . . 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 477	. . 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 2748	. . . 4 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 )))
111	110	cbvralvw 3234	. . 3 ⊢ (∀𝑛 ∈ ℕ₀ (𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ ∀𝑙 ∈ ℕ₀ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
112	109, 111	sylibr 233	. 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 413	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

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 − cmin 11440 ℕcn 12208 ℕ₀cn0 12468 ...cfz 13480 Basecbs 17140 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Σ_g cgsu 17382 -_gcsg 18817 ._gcmg 18944 mulGrpcmgp 19981 1_rcur 19998 Ringcrg 20049 CRingccrg 20050 LModclmod 20463 var₁cv1 21691 Poly₁cpl1 21692 coe₁cco1 21693 Mat cmat 21898 maAdju cmadu 22125 ConstPolyMat ccpmat 22196 matToPolyMat cmat2pmat 22197 cPolyMatToMat ccpmat2mat 22198 CharPlyMat cchpmat 22319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-splice 14696 df-reverse 14705 df-s2 14795 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-efmnd 18746 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-gim 19127 df-cntz 19175 df-oppg 19204 df-symg 19229 df-pmtr 19304 df-psgn 19353 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-zring 21010 df-zrh 21044 df-dsmm 21278 df-frlm 21293 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mamu 21877 df-mat 21899 df-mdet 22078 df-cpmat 22199 df-mat2pmat 22200 df-cpmat2mat 22201 df-chpmat 22320

This theorem is referenced by: chcoeffeq 22379

W3C validator