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Theorem chcoeffeqlem 22850

Description: Lemma for chcoeffeq 22851. (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 2736	. . . . 5 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
2		eqid 2736	. . . . 5 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
3		eqid 2736	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
4		crngring 20226	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5		chcoeffeq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 22408	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	4, 6	sylan2 594	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
8	7	3adant3 1133	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
9	8	adantr 480	. . . . 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 2736	. . . . 5 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
12		eqid 2736	. . . . 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 2736	. . . . . . . 8 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21		eqid 2736	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
22		eqid 2736	. . . . . . . 8 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
23		eqid 2736	. . . . . . . 8 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24		eqid 2736	. . . . . . . 8 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
25		eqid 2736	. . . . . . . 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 22846	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀))
29	28	anasss 466	. . . . . 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 22820	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
31	4, 30	syl3anl2 1416	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
32	31	adantr 480	. . . . . . . . 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 6939	. . . . . . . . . 10 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → ((𝑈 ∘ 𝐺)‘𝑙) = (𝑈‘(𝐺‘𝑙)))
34	33	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → (𝑈‘(𝐺‘𝑙)) = ((𝑈 ∘ 𝐺)‘𝑙))
35	32, 34	sylan 581	. . . . . . . 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 8796	. . . . . . . . . 10 ⊢ ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀) → (𝑈 ∘ 𝐺):ℕ₀⟶𝐵)
37	36	adantl 481	. . . . . . . . 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 7036	. . . . . . . 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 2836	. . . . . . 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 3129	. . . . . 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 688	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
42	4	anim2i 618	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43	42	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
44	43	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
45	5, 10, 20, 27	cpm2mf 22717	. . . . . . . 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 7086	. . . . . . 7 ⊢ ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵 ∧ 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈 ∘ 𝐺) = (𝑙 ∈ ℕ₀ ↦ (𝑈‘(𝐺‘𝑙))))
48	46, 31, 47	syl2anc 585	. . . . . 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 22847	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
50	49	anasss 466	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
51	48, 50	eqbrtrrd 5109	. . . . 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 1214	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → 𝑁 ∈ Fin)
53	4	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
54	53	ad2antrr 727	. . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
58	56, 5, 10, 13, 57	chpmatply1 22797	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
59	55, 58	eqeltrid 2840	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
60		eqid 2736	. . . . . . . . . 10 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
61		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
62	60, 57, 13, 61	coe1fvalcl 22176	. . . . . . . . 9 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
63	59, 62	sylan 581	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
64	63	adantlr 716	. . . . . . 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 20246	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵)
67	8, 66	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐵)
68	67	ad2antrr 727	. . . . . . 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 22396	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
71	52, 54, 64, 68, 70	syl22anc 839	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
72	71	ralrimiva 3129	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
73		nn0ex 12443	. . . . . . 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 22394	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
76	4, 75	sylan2 594	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77	76	3adant3 1133	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ LMod)
78	77	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐴 ∈ LMod)
79		eqidd 2737	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80		fvexd 6855	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ V)
81		eqid 2736	. . . . . 6 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
82	5	matsca2 22385	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83	82	3adant3 1133	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
84	83, 53	eqeltrrd 2837	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) ∈ Ring)
85	83	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) = 𝑅)
86	85	fveq2d 6844	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘𝑅))
87	86, 13	eqtr4di 2789	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = 𝑃)
88	87	fveq2d 6844	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘𝑃))
89	59, 88	eleqtrrd 2839	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴))))
90		eqid 2736	. . . . . . . . 9 ⊢ (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘(Scalar‘𝐴))
91		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘(Poly₁‘(Scalar‘𝐴)))
92	90, 91, 81	mptcoe1fsupp 22179	. . . . . . . 8 ⊢ (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
93	84, 89, 92	syl2anc 585	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
94	93	adantr 480	. . . . . 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 20922	. . . . 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 6845	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑙)))
97		oveq1 7374	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)) = (𝑙(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))
98	96, 97	oveq12d 7385	. . . . . . . 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 5189	. . . . . . 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 7378	. . . . . 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 6840	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((coe₁‘𝐾)‘𝑛) = ((coe₁‘𝐾)‘𝑙))
103	102	oveq1d 7382	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (((coe₁‘𝐾)‘𝑛) ∗ 1 ) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
104	103, 97	oveq12d 7385	. . . . . . . 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 5189	. . . . . . 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 7378	. . . . . 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 22274	. . . 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 476	. . 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 2752	. . . 4 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 )))
111	110	cbvralvw 3215	. . 3 ⊢ (∀𝑛 ∈ ℕ₀ (𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ ∀𝑙 ∈ ℕ₀ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
112	109, 111	sylibr 234	. 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 412	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 Vcvv 3429 ifcif 4466 class class class wbr 5085 ↦ cmpt 5166 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ↑_m cmap 8773 Fincfn 8893 finSupp cfsupp 9274 0cc0 11038 1c1 11039 + caddc 11041 < clt 11179 − cmin 11377 ℕcn 12174 ℕ₀cn0 12437 ...cfz 13461 Basecbs 17179 ._rcmulr 17221 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 Σ_g cgsu 17403 -_gcsg 18911 ._gcmg 19043 mulGrpcmgp 20121 1_rcur 20162 Ringcrg 20214 CRingccrg 20215 LModclmod 20855 var₁cv1 22139 Poly₁cpl1 22140 coe₁cco1 22141 Mat cmat 22372 maAdju cmadu 22597 ConstPolyMat ccpmat 22668 matToPolyMat cmat2pmat 22669 cPolyMatToMat ccpmat2mat 22670 CharPlyMat cchpmat 22791

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-word 14476 df-lsw 14525 df-concat 14533 df-s1 14559 df-substr 14604 df-pfx 14634 df-splice 14712 df-reverse 14721 df-s2 14810 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-efmnd 18837 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-gim 19234 df-cntz 19292 df-oppg 19321 df-symg 19345 df-pmtr 19417 df-psgn 19466 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-srg 20168 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-sra 21168 df-rgmod 21169 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-dsmm 21712 df-frlm 21727 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-mamu 22356 df-mat 22373 df-mdet 22550 df-cpmat 22671 df-mat2pmat 22672 df-cpmat2mat 22673 df-chpmat 22792

This theorem is referenced by: chcoeffeq 22851

W3C validator