Theorem chcoeffeq 22859

Description: The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-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
chcoeffeq	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀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,𝑠,𝐴   𝐵,𝑏,𝑠   𝑀,𝑏,𝑠   𝑁,𝑏,𝑠   𝑃,𝑏,𝑛,𝑠   𝑅,𝑏,𝑠   𝑇,𝑏,𝑛,𝑠   𝑛,𝑊   𝑌,𝑏,𝑠   0 ,𝑛   × ,𝑛   − ,𝑏,𝑛,𝑠

Allowed substitution hints:   𝐶(𝑛,𝑠,𝑏)   × (𝑠,𝑏)   𝑈(𝑠,𝑏)   1 (𝑠,𝑏)   𝐺(𝑠,𝑏)   ∗ (𝑠,𝑏)   𝐾(𝑠,𝑏)   𝑊(𝑠,𝑏)   0 (𝑠,𝑏)

Proof of Theorem chcoeffeq

Step	Hyp	Ref	Expression
1		chcoeffeq.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		chcoeffeq.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		chcoeffeq.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
4		chcoeffeq.y	. . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		chcoeffeq.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
6		chcoeffeq.r	. . 3 ⊢ × = (.r‘𝑌)
7		chcoeffeq.s	. . 3 ⊢ − = (-g‘𝑌)
8		chcoeffeq.0	. . 3 ⊢ 0 = (0g‘𝑌)
9		chcoeffeq.g	. . 3 ⊢ 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))))))))
10		eqid 2734	. . 3 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11		eqid 2734	. . 3 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
12		eqid 2734	. . 3 ⊢ (1r‘𝑌) = (1r‘𝑌)
13		eqid 2734	. . 3 ⊢ (var1‘𝑅) = (var1‘𝑅)
14		eqid 2734	. . 3 ⊢ (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))
15		eqid 2734	. . 3 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16		chcoeffeq.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
17		eqid 2734	. . 3 ⊢ (Poly1‘𝐴) = (Poly1‘𝐴)
18		eqid 2734	. . 3 ⊢ (var1‘𝐴) = (var1‘𝐴)
19		eqid 2734	. . 3 ⊢ ( ·𝑠 ‘(Poly1‘𝐴)) = ( ·𝑠 ‘(Poly1‘𝐴))
20		eqid 2734	. . 3 ⊢ (.g‘(mulGrp‘(Poly1‘𝐴))) = (.g‘(mulGrp‘(Poly1‘𝐴)))
21		chcoeffeq.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22		eqid 2734	. . 3 ⊢ (𝑁 pMatToMatPoly 𝑅) = (𝑁 pMatToMatPoly 𝑅)
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22	cpmadumatpoly 22856	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
24		eqid 2734	. . . . . . 7 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25		eqid 2734	. . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26		chcoeffeq.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
27		chcoeffeq.k	. . . . . . 7 ⊢ 𝐾 = (𝐶‘𝑀)
28		eqid 2734	. . . . . . 7 ⊢ (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))
29		chcoeffeq.1	. . . . . . 7 ⊢ 1 = (1r‘𝐴)
30		chcoeffeq.m	. . . . . . 7 ⊢ ∗ = ( ·𝑠 ‘𝐴)
31	1, 2, 3, 4, 13, 24, 11, 12, 25, 26, 27, 28, 29, 30, 5, 16, 17, 18, 19, 20, 22	cpmidpmat 22846	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
32		eqid 2734	. . . . . . . 8 ⊢ (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
33	1, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6	cpmadurid 22840	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠 ‘𝑌)(1r‘𝑌)))
34	26	fveq1i 6888	. . . . . . . . . . 11 ⊢ (𝐶‘𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
35	27, 34	eqtri 2757	. . . . . . . . . 10 ⊢ 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀)
36	35	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀))
37	36	eqcomd 2740	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
38	37	oveq1d 7429	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠 ‘𝑌)(1r‘𝑌)) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)))
39	33, 38	eqtrd 2769	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)))
40		fveq2 6887	. . . . . . . . 9 ⊢ (((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))))
41		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
42	41	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
43		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))))
44	42, 43	eqeq12d 2750	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) ↔ ((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)))))))
45	1, 2, 3, 4, 6, 7, 8, 5, 26, 27, 9, 16, 29, 30, 21	chcoeffeqlem 22858	. . . . . . . . . . . . . 14 ⊢ (((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 )))
46	45	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → (((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 )))
47	46	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → (((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 )))
48	44, 47	sylbid 240	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))
49	48	exp31 419	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))))
50	49	com24 95	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))))
51	40, 50	syl5 34	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))))
52	51	ex 412	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))))))
53	52	com24 95	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠 ‘𝑌)(1r‘𝑌))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1‘𝐾)‘𝑛) ∗ 1 )( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → (((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))))))
54	31, 39, 53	mp2d 49	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))))
55	54	impl 455	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))
56	55	reximdva 3155	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (∃𝑏 ∈ (𝐵 ↑m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))
57	56	reximdva 3155	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly1‘𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺‘𝑛))( ·𝑠 ‘(Poly1‘𝐴))(𝑛(.g‘(mulGrp‘(Poly1‘𝐴)))(var1‘𝐴))))) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 )))
58	23, 57	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺‘𝑛)) = (((coe1‘𝐾)‘𝑛) ∗ 1 ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  ifcif 4507   class class class wbr 5125   ↦ cmpt 5207  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8849  Fincfn 8968  0cc0 11138  1c1 11139   + caddc 11141   < clt 11278   − cmin 11475  ℕcn 12249  ℕ₀cn0 12510  ...cfz 13530  Basecbs 17230  ._rcmulr 17278   ·_𝑠 cvsca 17281  0_gc0g 17460   Σ_g cgsu 17461  -_gcsg 18927  ._gcmg 19059  mulGrpcmgp 20110  1_rcur 20151  CRingccrg 20204  algSccascl 21839  var₁cv1 22144  Poly₁cpl1 22145  coe₁cco1 22146   Mat cmat 22378   maAdju cmadu 22605   ConstPolyMat ccpmat 22676   matToPolyMat cmat2pmat 22677   cPolyMatToMat ccpmat2mat 22678   pMatToMatPoly cpm2mp 22765   CharPlyMat cchpmat 22799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-addf 11217  ax-mulf 11218

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1511  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-tp 4613  df-op 4615  df-ot 4617  df-uni 4890  df-int 4929  df-iun 4975  df-iin 4976  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7680  df-ofr 7681  df-om 7871  df-1st 7997  df-2nd 7998  df-supp 8169  df-tpos 8234  df-cur 8275  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-1o 8489  df-2o 8490  df-er 8728  df-map 8851  df-pm 8852  df-ixp 8921  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-fsupp 9385  df-sup 9465  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11477  df-neg 11478  df-div 11904  df-nn 12250  df-2 12312  df-3 12313  df-4 12314  df-5 12315  df-6 12316  df-7 12317  df-8 12318  df-9 12319  df-n0 12511  df-xnn0 12584  df-z 12598  df-dec 12718  df-uz 12862  df-rp 13018  df-fz 13531  df-fzo 13678  df-seq 14026  df-exp 14086  df-hash 14353  df-word 14536  df-lsw 14584  df-concat 14592  df-s1 14617  df-substr 14662  df-pfx 14692  df-splice 14771  df-reverse 14780  df-s2 14870  df-struct 17167  df-sets 17184  df-slot 17202  df-ndx 17214  df-base 17231  df-ress 17257  df-plusg 17290  df-mulr 17291  df-starv 17292  df-sca 17293  df-vsca 17294  df-ip 17295  df-tset 17296  df-ple 17297  df-ds 17299  df-unif 17300  df-hom 17301  df-cco 17302  df-0g 17462  df-gsum 17463  df-prds 17468  df-pws 17470  df-mre 17605  df-mrc 17606  df-acs 17608  df-mgm 18627  df-sgrp 18706  df-mnd 18722  df-mhm 18770  df-submnd 18771  df-efmnd 18856  df-grp 18928  df-minusg 18929  df-sbg 18930  df-mulg 19060  df-subg 19115  df-ghm 19205  df-gim 19251  df-cntz 19309  df-oppg 19338  df-symg 19360  df-pmtr 19433  df-psgn 19482  df-evpm 19483  df-cmn 19773  df-abl 19774  df-mgp 20111  df-rng 20123  df-ur 20152  df-srg 20157  df-ring 20205  df-cring 20206  df-oppr 20307  df-dvdsr 20330  df-unit 20331  df-invr 20361  df-dvr 20374  df-rhm 20445  df-subrng 20519  df-subrg 20543  df-drng 20704  df-lmod 20833  df-lss 20903  df-sra 21145  df-rgmod 21146  df-cnfld 21332  df-zring 21425  df-zrh 21481  df-dsmm 21719  df-frlm 21734  df-assa 21840  df-ascl 21842  df-psr 21896  df-mvr 21897  df-mpl 21898  df-opsr 21900  df-psr1 22148  df-vr1 22149  df-ply1 22150  df-coe1 22151  df-mamu 22362  df-mat 22379  df-mdet 22558  df-madu 22607  df-cpmat 22679  df-mat2pmat 22680  df-cpmat2mat 22681  df-decpmat 22736  df-pm2mp 22766  df-chpmat 22800

This theorem is referenced by:  cayhamlem3  22860