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Theorem chcoeffeqlem 23003

Description: Lemma for chcoeffeq 23004. (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 2765	. . . . 5 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
2		eqid 2765	. . . . 5 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
3		eqid 2765	. . . . 5 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
4		crngring 20318	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5		chcoeffeq.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matring 22561	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
7	4, 6	sylan2 604	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
8	7	3adant3 1148	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ Ring)
9	8	adantr 485	. . . . 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 2765	. . . . 5 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
12		eqid 2765	. . . . 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 2765	. . . . . . . 8 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
21		eqid 2765	. . . . . . . 8 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
22		eqid 2765	. . . . . . . 8 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
23		eqid 2765	. . . . . . . 8 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
24		eqid 2765	. . . . . . . 8 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
25		eqid 2765	. . . . . . . 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 22999	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀))
29	28	anasss 471	. . . . . 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 22973	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
31	4, 30	syl3anl2 1436	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅))
32	31	adantr 485	. . . . . . . . 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 6971	. . . . . . . . . 10 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → ((𝑈 ∘ 𝐺)‘𝑙) = (𝑈‘(𝐺‘𝑙)))
34	33	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅) ∧ 𝑙 ∈ ℕ₀) → (𝑈‘(𝐺‘𝑙)) = ((𝑈 ∘ 𝐺)‘𝑙))
35	32, 34	sylan 591	. . . . . . . 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 8834	. . . . . . . . . 10 ⊢ ((𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀) → (𝑈 ∘ 𝐺):ℕ₀⟶𝐵)
37	36	adantl 486	. . . . . . . . 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 7069	. . . . . . . 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 2865	. . . . . . 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 3157	. . . . . 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 699	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (𝑈‘(𝐺‘𝑙)) ∈ 𝐵)
42	4	anim2i 628	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
43	42	3adant3 1148	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
44	43	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
45	5, 10, 20, 27	cpm2mf 22870	. . . . . . . 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 7119	. . . . . . 7 ⊢ ((𝑈:(𝑁 ConstPolyMat 𝑅)⟶𝐵 ∧ 𝐺:ℕ₀⟶(𝑁 ConstPolyMat 𝑅)) → (𝑈 ∘ 𝐺) = (𝑙 ∈ ℕ₀ ↦ (𝑈‘(𝐺‘𝑙))))
48	46, 31, 47	syl2anc 595	. . . . . 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 23000	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
50	49	anasss 471	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈 ∘ 𝐺) finSupp (0_g‘𝐴))
51	48, 50	eqbrtrrd 5129	. . . . 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 1229	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → 𝑁 ∈ Fin)
53	4	3ad2ant2 1150	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
54	53	ad2antrr 738	. . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
58	56, 5, 10, 13, 57	chpmatply1 22950	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ (Base‘𝑃))
59	55, 58	eqeltrid 2869	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘𝑃))
60		eqid 2765	. . . . . . . . . 10 ⊢ (coe₁‘𝐾) = (coe₁‘𝐾)
61		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
62	60, 57, 13, 61	coe1fvalcl 22332	. . . . . . . . 9 ⊢ ((𝐾 ∈ (Base‘𝑃) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
63	59, 62	sylan 591	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅))
64	63	adantlr 727	. . . . . . 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 20339	. . . . . . . . 9 ⊢ (𝐴 ∈ Ring → 1 ∈ 𝐵)
67	8, 66	syl 18	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 1 ∈ 𝐵)
68	67	ad2antrr 738	. . . . . . 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 22549	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (((coe₁‘𝐾)‘𝑙) ∈ (Base‘𝑅) ∧ 1 ∈ 𝐵)) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
71	52, 54, 64, 68, 70	syl22anc 851	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
72	71	ralrimiva 3157	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ∀𝑙 ∈ ℕ₀ (((coe₁‘𝐾)‘𝑙) ∗ 1 ) ∈ 𝐵)
73		nn0ex 12501	. . . . . . 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 22547	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
76	4, 75	sylan2 604	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ LMod)
77	76	3adant3 1148	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ LMod)
78	77	adantr 485	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐴 ∈ LMod)
79		eqidd 2766	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (Scalar‘𝐴) = (Scalar‘𝐴))
80		fvexd 6886	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ 𝑙 ∈ ℕ₀) → ((coe₁‘𝐾)‘𝑙) ∈ V)
81		eqid 2765	. . . . . 6 ⊢ (0_g‘(Scalar‘𝐴)) = (0_g‘(Scalar‘𝐴))
82	5	matsca2 22538	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
83	82	3adant3 1148	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
84	83, 53	eqeltrrd 2866	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) ∈ Ring)
85	83	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝐴) = 𝑅)
86	85	fveq2d 6875	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘𝑅))
87	86, 13	eqtr4di 2818	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Poly₁‘(Scalar‘𝐴)) = 𝑃)
88	87	fveq2d 6875	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘𝑃))
89	59, 88	eleqtrrd 2868	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴))))
90		eqid 2765	. . . . . . . . 9 ⊢ (Poly₁‘(Scalar‘𝐴)) = (Poly₁‘(Scalar‘𝐴))
91		eqid 2765	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘(Scalar‘𝐴))) = (Base‘(Poly₁‘(Scalar‘𝐴)))
92	90, 91, 81	mptcoe1fsupp 22335	. . . . . . . 8 ⊢ (((Scalar‘𝐴) ∈ Ring ∧ 𝐾 ∈ (Base‘(Poly₁‘(Scalar‘𝐴)))) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
93	84, 89, 92	syl2anc 595	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑙 ∈ ℕ₀ ↦ ((coe₁‘𝐾)‘𝑙)) finSupp (0_g‘(Scalar‘𝐴)))
94	93	adantr 485	. . . . . 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 21017	. . . . 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 6876	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑈‘(𝐺‘𝑛)) = (𝑈‘(𝐺‘𝑙)))
97		oveq1 7407	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)) = (𝑙(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))
98	96, 97	oveq12d 7418	. . . . . . . 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 5209	. . . . . . 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 7411	. . . . . 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 6871	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((coe₁‘𝐾)‘𝑛) = ((coe₁‘𝐾)‘𝑙))
103	102	oveq1d 7415	. . . . . . . . 9 ⊢ (𝑛 = 𝑙 → (((coe₁‘𝐾)‘𝑛) ∗ 1 ) = (((coe₁‘𝐾)‘𝑙) ∗ 1 ))
104	103, 97	oveq12d 7418	. . . . . . . 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 5209	. . . . . . 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 7411	. . . . . 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 22430	. . . 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 481	. . 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 2781	. . . 4 ⊢ (𝑛 = 𝑙 → ((𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ) ↔ (𝑈‘(𝐺‘𝑙)) = (((coe₁‘𝐾)‘𝑙) ∗ 1 )))
111	110	cbvralvw 3243	. . 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 417	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 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 Vcvv 3457 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 ∘ ccom 5656 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 ↑_m cmap 8812 Fincfn 8931 finSupp cfsupp 9309 0cc0 11088 1c1 11089 + caddc 11091 < clt 11231 − cmin 11429 ℕcn 12224 ℕ₀cn0 12495 ...cfz 13526 Basecbs 17259 ._rcmulr 17301 Scalarcsca 17303 ·_𝑠 cvsca 17304 0_gc0g 17482 Σ_g cgsu 17483 -_gcsg 18992 ._gcmg 19124 mulGrpcmgp 20207 1_rcur 20254 Ringcrg 20306 CRingccrg 20307 LModclmod 20950 var₁cv1 22296 Poly₁cpl1 22297 coe₁cco1 22298 Mat cmat 22525 maAdju cmadu 22750 ConstPolyMat ccpmat 22821 matToPolyMat cmat2pmat 22822 cPolyMatToMat ccpmat2mat 22823 CharPlyMat cchpmat 22944

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-addf 11167 ax-mulf 11168

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-xor 1535 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-ot 4594 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-ofr 7665 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-sup 9390 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-xnn0 12569 df-z 12583 df-dec 12703 df-uz 12854 df-rp 13008 df-fz 13527 df-fzo 13674 df-seq 14029 df-exp 14089 df-hash 14358 df-word 14541 df-lsw 14590 df-concat 14598 df-s1 14624 df-substr 14669 df-pfx 14699 df-splice 14777 df-reverse 14786 df-s2 14875 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-0g 17484 df-gsum 17485 df-prds 17490 df-pws 17492 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-submnd 18832 df-efmnd 18918 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-ghm 19275 df-gim 19320 df-cntz 19378 df-oppg 19407 df-symg 19431 df-pmtr 19503 df-psgn 19552 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-srg 20260 df-ring 20308 df-cring 20309 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-rhm 20545 df-subrng 20622 df-subrg 20646 df-drng 20806 df-lmod 20952 df-lss 21022 df-sra 21263 df-rgmod 21264 df-cnfld 21483 df-zring 21557 df-zrh 21613 df-dsmm 21842 df-frlm 21857 df-ascl 21965 df-psr 22019 df-mvr 22020 df-mpl 22021 df-opsr 22023 df-psr1 22300 df-vr1 22301 df-ply1 22302 df-coe1 22303 df-mamu 22509 df-mat 22526 df-mdet 22703 df-cpmat 22824 df-mat2pmat 22825 df-cpmat2mat 22826 df-chpmat 22945

This theorem is referenced by: chcoeffeq 23004

W3C validator