Theorem chcoeffeq 22796

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 2731	. . 3 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌)
12		eqid 2731	. . 3 ⊢ (1r‘𝑌) = (1r‘𝑌)
13		eqid 2731	. . 3 ⊢ (var1‘𝑅) = (var1‘𝑅)
14		eqid 2731	. . 3 ⊢ (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀)) = (((var1‘𝑅)( ·𝑠 ‘𝑌)(1r‘𝑌)) − (𝑇‘𝑀))
15		eqid 2731	. . 3 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16		chcoeffeq.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
17		eqid 2731	. . 3 ⊢ (Poly1‘𝐴) = (Poly1‘𝐴)
18		eqid 2731	. . 3 ⊢ (var1‘𝐴) = (var1‘𝐴)
19		eqid 2731	. . 3 ⊢ ( ·𝑠 ‘(Poly1‘𝐴)) = ( ·𝑠 ‘(Poly1‘𝐴))
20		eqid 2731	. . 3 ⊢ (.g‘(mulGrp‘(Poly1‘𝐴))) = (.g‘(mulGrp‘(Poly1‘𝐴)))
21		chcoeffeq.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22		eqid 2731	. . 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 22793	. 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 2731	. . . . . . 7 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25		eqid 2731	. . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26		chcoeffeq.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
27		chcoeffeq.k	. . . . . . 7 ⊢ 𝐾 = (𝐶‘𝑀)
28		eqid 2731	. . . . . . 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 22783	. . . . . 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 2731	. . . . . . . 8 ⊢ (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
33	1, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6	cpmadurid 22777	. . . . . . 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 6818	. . . . . . . . . . 11 ⊢ (𝐶‘𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
35	27, 34	eqtri 2754	. . . . . . . . . 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 2737	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
38	37	oveq1d 7356	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠 ‘𝑌)(1r‘𝑌)) = (𝐾( ·𝑠 ‘𝑌)(1r‘𝑌)))
39	33, 38	eqtrd 2766	. . . . . 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 6817	. . . . . . . . 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 2747	. . . . . . . . . . . 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 22795	. . . . . . . . . . . . . 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 3145	. . 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 3145	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ifcif 4470   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745  Fincfn 8864  0cc0 11001  1c1 11002   + caddc 11004   < clt 11141   − cmin 11339  ℕcn 12120  ℕ₀cn0 12376  ...cfz 13402  Basecbs 17115  ._rcmulr 17157   ·_𝑠 cvsca 17160  0_gc0g 17338   Σ_g cgsu 17339  -_gcsg 18843  ._gcmg 18975  mulGrpcmgp 20053  1_rcur 20094  CRingccrg 20147  algSccascl 21784  var₁cv1 22083  Poly₁cpl1 22084  coe₁cco1 22085   Mat cmat 22317   maAdju cmadu 22542   ConstPolyMat ccpmat 22613   matToPolyMat cmat2pmat 22614   cPolyMatToMat ccpmat2mat 22615   pMatToMatPoly cpm2mp 22702   CharPlyMat cchpmat 22736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-addf 11080  ax-mulf 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1513  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-ot 4580  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-cur 8192  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-oi 9391  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-xnn0 12450  df-z 12464  df-dec 12584  df-uz 12728  df-rp 12886  df-fz 13403  df-fzo 13550  df-seq 13904  df-exp 13964  df-hash 14233  df-word 14416  df-lsw 14465  df-concat 14473  df-s1 14499  df-substr 14544  df-pfx 14574  df-splice 14652  df-reverse 14661  df-s2 14750  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-starv 17171  df-sca 17172  df-vsca 17173  df-ip 17174  df-tset 17175  df-ple 17176  df-ds 17178  df-unif 17179  df-hom 17180  df-cco 17181  df-0g 17340  df-gsum 17341  df-prds 17346  df-pws 17348  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-efmnd 18772  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-ghm 19120  df-gim 19166  df-cntz 19224  df-oppg 19253  df-symg 19277  df-pmtr 19349  df-psgn 19398  df-evpm 19399  df-cmn 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-srg 20100  df-ring 20148  df-cring 20149  df-oppr 20250  df-dvdsr 20270  df-unit 20271  df-invr 20301  df-dvr 20314  df-rhm 20385  df-subrng 20456  df-subrg 20480  df-drng 20641  df-lmod 20790  df-lss 20860  df-sra 21102  df-rgmod 21103  df-cnfld 21287  df-zring 21379  df-zrh 21435  df-dsmm 21664  df-frlm 21679  df-assa 21785  df-ascl 21787  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-psr1 22087  df-vr1 22088  df-ply1 22089  df-coe1 22090  df-mamu 22301  df-mat 22318  df-mdet 22495  df-madu 22544  df-cpmat 22616  df-mat2pmat 22617  df-cpmat2mat 22618  df-decpmat 22673  df-pm2mp 22703  df-chpmat 22737

This theorem is referenced by:  cayhamlem3  22797