chcoeffeq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chcoeffeq		Structured version Visualization version GIF version

Theorem chcoeffeq 22387

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 2732	. . 3 ⊢ (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11		eqid 2732	. . 3 ⊢ ( ·_𝑠 ‘𝑌) = ( ·_𝑠 ‘𝑌)
12		eqid 2732	. . 3 ⊢ (1_r‘𝑌) = (1_r‘𝑌)
13		eqid 2732	. . 3 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
14		eqid 2732	. . 3 ⊢ (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀)) = (((var₁‘𝑅)( ·_𝑠 ‘𝑌)(1_r‘𝑌)) − (𝑇‘𝑀))
15		eqid 2732	. . 3 ⊢ (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16		chcoeffeq.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
17		eqid 2732	. . 3 ⊢ (Poly₁‘𝐴) = (Poly₁‘𝐴)
18		eqid 2732	. . 3 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
19		eqid 2732	. . 3 ⊢ ( ·_𝑠 ‘(Poly₁‘𝐴)) = ( ·_𝑠 ‘(Poly₁‘𝐴))
20		eqid 2732	. . 3 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐴))) = (._g‘(mulGrp‘(Poly₁‘𝐴)))
21		chcoeffeq.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22		eqid 2732	. . 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 22384	. 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 2732	. . . . . . 7 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
25		eqid 2732	. . . . . . 7 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
26		chcoeffeq.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
27		chcoeffeq.k	. . . . . . 7 ⊢ 𝐾 = (𝐶‘𝑀)
28		eqid 2732	. . . . . . 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 22374	. . . . . 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 2732	. . . . . . . 8 ⊢ (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
33	1, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6	cpmadurid 22368	. . . . . . 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 6892	. . . . . . . . . . 11 ⊢ (𝐶‘𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
35	27, 34	eqtri 2760	. . . . . . . . . 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 2738	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
38	37	oveq1d 7423	. . . . . . 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 2772	. . . . . 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 6891	. . . . . . . . 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 485	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 22386	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 420	. . . . . . . . . 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 413	. . . . . . 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 456	. . . 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 3168	. . 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 3168	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 Fincfn 8938 0cc0 11109 1c1 11110 + caddc 11112 < clt 11247 − cmin 11443 ℕcn 12211 ℕ₀cn0 12471 ...cfz 13483 Basecbs 17143 ._rcmulr 17197 ·_𝑠 cvsca 17200 0_gc0g 17384 Σ_g cgsu 17385 -_gcsg 18820 ._gcmg 18949 mulGrpcmgp 19986 1_rcur 20003 CRingccrg 20056 algSccascl 21406 var₁cv1 21699 Poly₁cpl1 21700 coe₁cco1 21701 Mat cmat 21906 maAdju cmadu 22133 ConstPolyMat ccpmat 22204 matToPolyMat cmat2pmat 22205 cPolyMatToMat ccpmat2mat 22206 pMatToMatPoly cpm2mp 22293 CharPlyMat cchpmat 22327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-cur 8251 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-rp 12974 df-fz 13484 df-fzo 13627 df-seq 13966 df-exp 14027 df-hash 14290 df-word 14464 df-lsw 14512 df-concat 14520 df-s1 14545 df-substr 14590 df-pfx 14620 df-splice 14699 df-reverse 14708 df-s2 14798 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-efmnd 18749 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-gim 19132 df-cntz 19180 df-oppg 19209 df-symg 19234 df-pmtr 19309 df-psgn 19358 df-evpm 19359 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-srg 20009 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-rnghom 20250 df-subrg 20316 df-drng 20358 df-lmod 20472 df-lss 20542 df-sra 20784 df-rgmod 20785 df-cnfld 20944 df-zring 21017 df-zrh 21052 df-dsmm 21286 df-frlm 21301 df-assa 21407 df-ascl 21409 df-psr 21461 df-mvr 21462 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-vr1 21704 df-ply1 21705 df-coe1 21706 df-mamu 21885 df-mat 21907 df-mdet 22086 df-madu 22135 df-cpmat 22207 df-mat2pmat 22208 df-cpmat2mat 22209 df-decpmat 22264 df-pm2mp 22294 df-chpmat 22328

This theorem is referenced by: cayhamlem3 22388

W3C validator