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Theorem chcoeffeq 22743

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 = (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
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 = (0_g‘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 2726	. . 3 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11		eqid 2726	. . 3 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
12		eqid 2726	. . 3 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
13		eqid 2726	. . 3 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
14		eqid 2726	. . 3 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
15		eqid 2726	. . 3 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16		chcoeffeq.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
17		eqid 2726	. . 3 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
18		eqid 2726	. . 3 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
19		eqid 2726	. . 3 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
20		eqid 2726	. . 3 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
21		chcoeffeq.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22		eqid 2726	. . 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 22740	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))))
24		eqid 2726	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
25		eqid 2726	. . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26		chcoeffeq.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
27		chcoeffeq.k	. . . . . . 7 ⊢ 𝐾 = (𝐶‘𝑀)
28		eqid 2726	. . . . . . 7 ⊢ (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))
29		chcoeffeq.1	. . . . . . 7 ⊢ 1 = (1_r‘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 22730	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))))
32		eqid 2726	. . . . . . . 8 ⊢ (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
33	1, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6	cpmadurid 22724	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·_𝑠 ‘𝑌)(1_r‘𝑌)))
34	26	fveq1i 6886	. . . . . . . . . . 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 2732	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
38	37	oveq1d 7420	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)))
39	33, 38	eqtrd 2766	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)))
40		fveq2 6885	. . . . . . . . 9 ⊢ (((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)) → ((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))))
44	42, 43	eqeq12d 2742	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) ↔ ((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)))))))
45	1, 2, 3, 4, 6, 7, 8, 5, 26, 27, 9, 16, 29, 30, 21	chcoeffeqlem 22742	. . . . . . . . . . . . . 14 ⊢ (((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 )))
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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((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 ↦ ((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 )))
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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → (((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 )))
48	44, 47	sylbid 239	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → (((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)))) = (𝐾( ·_𝑠 ‘𝑌)(1_r‘𝑌)) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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 3162	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) → (∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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 3162	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) × ((𝑁 maAdju 𝑃)‘(((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))))) = ((Poly₁‘𝐴) Σ_g (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 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 ifcif 4523 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ↑_m cmap 8822 Fincfn 8941 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 − cmin 11448 ℕcn 12216 ℕ₀cn0 12476 ...cfz 13490 Basecbs 17153 ._rcmulr 17207 ·_𝑠 cvsca 17210 0_gc0g 17394 Σ_g cgsu 17395 -_gcsg 18865 ._gcmg 18995 mulGrpcmgp 20039 1_rcur 20086 CRingccrg 20139 algSccascl 21747 var₁cv1 22050 Poly₁cpl1 22051 coe₁cco1 22052 Mat cmat 22262 maAdju cmadu 22489 ConstPolyMat ccpmat 22560 matToPolyMat cmat2pmat 22561 cPolyMatToMat ccpmat2mat 22562 pMatToMatPoly cpm2mp 22649 CharPlyMat cchpmat 22683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-xor 1505 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-cur 8253 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-lsw 14519 df-concat 14527 df-s1 14552 df-substr 14597 df-pfx 14627 df-splice 14706 df-reverse 14715 df-s2 14805 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-efmnd 18794 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-gim 19184 df-cntz 19233 df-oppg 19262 df-symg 19287 df-pmtr 19362 df-psgn 19411 df-evpm 19412 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-dsmm 21627 df-frlm 21642 df-assa 21748 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-mamu 22241 df-mat 22263 df-mdet 22442 df-madu 22491 df-cpmat 22563 df-mat2pmat 22564 df-cpmat2mat 22565 df-decpmat 22620 df-pm2mp 22650 df-chpmat 22684

This theorem is referenced by: cayhamlem3 22744

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