Theorem chcoeffeq 23043

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 2763	. . 3 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
12		eqid 2763	. . 3 ⊢ (1r‘𝑌) = (1r‘𝑌)
13		eqid 2763	. . 3 ⊢ (var1‘𝑅) = (var1‘𝑅)
14		eqid 2763	. . 3 ⊢ (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))
15		eqid 2763	. . 3 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16		chcoeffeq.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
17		eqid 2763	. . 3 ⊢ (Poly1‘𝐴) = (Poly1‘𝐴)
18		eqid 2763	. . 3 ⊢ (var1‘𝐴) = (var1‘𝐴)
19		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘(Poly1‘𝐴)) = ( ·𝑠 ‘(Poly1‘𝐴))
20		eqid 2763	. . 3 ⊢ (.g‘(mulGrp‘(Poly1‘𝐴))) = (.g‘(mulGrp‘(Poly1‘𝐴)))
21		chcoeffeq.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22		eqid 2763	. . 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 23040	. 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 2763	. . . . . . 7 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25		eqid 2763	. . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26		chcoeffeq.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
27		chcoeffeq.k	. . . . . . 7 ⊢ 𝐾 = (𝐶‘𝑀)
28		eqid 2763	. . . . . . 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 23030	. . . . . 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 2763	. . . . . . . 8 ⊢ (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
33	1, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6	cpmadurid 23024	. . . . . . 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 6882	. . . . . . . . . . 11 ⊢ (𝐶‘𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
35	27, 34	eqtri 2786	. . . . . . . . . 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 2769	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
38	37	oveq1d 7425	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠 ‘𝑌)(1r‘𝑌)) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)))
39	33, 38	eqtrd 2798	. . . . . 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 6881	. . . . . . . . 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 489	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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 23042	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 243	. . . . . . . . . . 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 424	. . . . . . . . . 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 96	. . . . . . . . 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 35	. . . . . . . 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 417	. . . . . . 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 96	. . . . . 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 50	. . . . 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 460	. . . 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 3178	. . 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 3178	. 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 16	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 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  Fincfn 8939  0cc0 11095  1c1 11096   + caddc 11098   < clt 11238   − cmin 11436  ℕcn 12228  ℕ₀cn0 12499  ...cfz 13530  Basecbs 17264  ._rcmulr 17306   ·_𝑠 cvsca 17309  0_gc0g 17487   Σ_g cgsu 17488  -_gcsg 18997  ._gcmg 19128  mulGrpcmgp 20211  1_rcur 20258  CRingccrg 20311  algSccascl 22002  var₁cv1 22336  Poly₁cpl1 22337  coe₁cco1 22338   Mat cmat 22564   maAdju cmadu 22789   ConstPolyMat ccpmat 22860   matToPolyMat cmat2pmat 22861   cPolyMatToMat ccpmat2mat 22862   pMatToMatPoly cpm2mp 22949   CharPlyMat cchpmat 22983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-addf 11174  ax-mulf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-ofr 7675  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-cur 8259  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-word 14547  df-lsw 14596  df-concat 14604  df-s1 14630  df-substr 14675  df-pfx 14705  df-splice 14783  df-reverse 14792  df-s2 14881  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-efmnd 18923  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-gim 19324  df-cntz 19382  df-oppg 19411  df-symg 19435  df-pmtr 19507  df-psgn 19556  df-evpm 19557  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-srg 20264  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-invr 20466  df-dvr 20479  df-rhm 20550  df-subrng 20645  df-subrg 20669  df-drng 20829  df-lmod 20983  df-lss 21053  df-sra 21294  df-rgmod 21295  df-cnfld 21523  df-zring 21597  df-zrh 21653  df-dsmm 21882  df-frlm 21897  df-assa 22003  df-ascl 22005  df-psr 22059  df-mvr 22060  df-mpl 22061  df-opsr 22063  df-psr1 22340  df-vr1 22341  df-ply1 22342  df-coe1 22343  df-mamu 22548  df-mat 22565  df-mdet 22742  df-madu 22791  df-cpmat 22863  df-mat2pmat 22864  df-cpmat2mat 22865  df-decpmat 22920  df-pm2mp 22950  df-chpmat 22984

This theorem is referenced by:  cayhamlem3  23044