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Theorem chcoeffeq 21109
 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 = (Poly1𝑅)
chcoeffeq.y 𝑌 = (𝑁 Mat 𝑃)
chcoeffeq.r × = (.r𝑌)
chcoeffeq.s = (-g𝑌)
chcoeffeq.0 0 = (0g𝑌)
chcoeffeq.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chcoeffeq.c 𝐶 = (𝑁 CharPlyMat 𝑅)
chcoeffeq.k 𝐾 = (𝐶𝑀)
chcoeffeq.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
chcoeffeq.w 𝑊 = (Base‘𝑌)
chcoeffeq.1 1 = (1r𝐴)
chcoeffeq.m = ( ·𝑠𝐴)
chcoeffeq.u 𝑈 = (𝑁 cPolyMatToMat 𝑅)
Assertion
Ref Expression
chcoeffeq ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘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
StepHypRef Expression
1 chcoeffeq.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 chcoeffeq.b . . 3 𝐵 = (Base‘𝐴)
3 chcoeffeq.p . . 3 𝑃 = (Poly1𝑅)
4 chcoeffeq.y . . 3 𝑌 = (𝑁 Mat 𝑃)
5 chcoeffeq.t . . 3 𝑇 = (𝑁 matToPolyMat 𝑅)
6 chcoeffeq.r . . 3 × = (.r𝑌)
7 chcoeffeq.s . . 3 = (-g𝑌)
8 chcoeffeq.0 . . 3 0 = (0g𝑌)
9 chcoeffeq.g . . 3 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
10 eqid 2778 . . 3 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11 eqid 2778 . . 3 ( ·𝑠𝑌) = ( ·𝑠𝑌)
12 eqid 2778 . . 3 (1r𝑌) = (1r𝑌)
13 eqid 2778 . . 3 (var1𝑅) = (var1𝑅)
14 eqid 2778 . . 3 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
15 eqid 2778 . . 3 (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16 chcoeffeq.w . . 3 𝑊 = (Base‘𝑌)
17 eqid 2778 . . 3 (Poly1𝐴) = (Poly1𝐴)
18 eqid 2778 . . 3 (var1𝐴) = (var1𝐴)
19 eqid 2778 . . 3 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
20 eqid 2778 . . 3 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
21 chcoeffeq.u . . 3 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22 eqid 2778 . . 3 (𝑁 pMatToMatPoly 𝑅) = (𝑁 pMatToMatPoly 𝑅)
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22cpmadumatpoly 21106 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
24 eqid 2778 . . . . . . 7 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25 eqid 2778 . . . . . . 7 (algSc‘𝑃) = (algSc‘𝑃)
26 chcoeffeq.c . . . . . . 7 𝐶 = (𝑁 CharPlyMat 𝑅)
27 chcoeffeq.k . . . . . . 7 𝐾 = (𝐶𝑀)
28 eqid 2778 . . . . . . 7 (𝐾( ·𝑠𝑌)(1r𝑌)) = (𝐾( ·𝑠𝑌)(1r𝑌))
29 chcoeffeq.1 . . . . . . 7 1 = (1r𝐴)
30 chcoeffeq.m . . . . . . 7 = ( ·𝑠𝐴)
311, 2, 3, 4, 13, 24, 11, 12, 25, 26, 27, 28, 29, 30, 5, 16, 17, 18, 19, 20, 22cpmidpmat 21096 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
32 eqid 2778 . . . . . . . 8 (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
331, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6cpmadurid 21090 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)))
3426fveq1i 6449 . . . . . . . . . . 11 (𝐶𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3527, 34eqtri 2802 . . . . . . . . . 10 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3635a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀))
3736eqcomd 2784 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
3837oveq1d 6939 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)) = (𝐾( ·𝑠𝑌)(1r𝑌)))
3933, 38eqtrd 2814 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)))
40 fveq2 6448 . . . . . . . . 9 (((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))))
41 simpr 479 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
4241adantr 474 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
43 simpr 479 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))))
4442, 43eqeq12d 2793 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) ↔ ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))))
451, 2, 3, 4, 6, 7, 8, 5, 26, 27, 9, 16, 29, 30, 21chcoeffeqlem 21108 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
4645adantr 474 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
4746adantr 474 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
4844, 47sylbid 232 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) ∧ ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴)))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
4948exp31 412 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))))
5049com24 95 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))))
5140, 50syl5 34 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))))
5251ex 403 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))))))
5352com24 95 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (((𝑁 pMatToMatPoly 𝑅)‘(𝐾( ·𝑠𝑌)(1r𝑌))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → (((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))))))
5431, 39, 53mp2d 49 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))))
5554impl 449 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
5655reximdva 3198 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) → (∃𝑏 ∈ (𝐵𝑚 (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
5756reximdva 3198 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))((𝑁 pMatToMatPoly 𝑅)‘((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
5823, 57mpd 15 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  ifcif 4307   class class class wbr 4888   ↦ cmpt 4967  ‘cfv 6137  (class class class)co 6924   ↑𝑚 cmap 8142  Fincfn 8243  0cc0 10274  1c1 10275   + caddc 10277   < clt 10413   − cmin 10608  ℕcn 11379  ℕ0cn0 11647  ...cfz 12648  Basecbs 16266  .rcmulr 16350   ·𝑠 cvsca 16353  0gc0g 16497   Σg cgsu 16498  -gcsg 17822  .gcmg 17938  mulGrpcmgp 18887  1rcur 18899  CRingccrg 18946  algSccascl 19719  var1cv1 19953  Poly1cpl1 19954  coe1cco1 19955   Mat cmat 20628   maAdju cmadu 20854   ConstPolyMat ccpmat 20926   matToPolyMat cmat2pmat 20927   cPolyMatToMat ccpmat2mat 20928   pMatToMatPoly cpm2mp 21015   CharPlyMat cchpmat 21049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-inf2 8837  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-addf 10353  ax-mulf 10354 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-xor 1583  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-ot 4407  df-uni 4674  df-int 4713  df-iun 4757  df-iin 4758  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-se 5317  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-isom 6146  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176  df-ofr 7177  df-om 7346  df-1st 7447  df-2nd 7448  df-supp 7579  df-tpos 7636  df-cur 7677  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-fsupp 8566  df-sup 8638  df-oi 8706  df-card 9100  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11036  df-nn 11380  df-2 11443  df-3 11444  df-4 11445  df-5 11446  df-6 11447  df-7 11448  df-8 11449  df-9 11450  df-n0 11648  df-xnn0 11720  df-z 11734  df-dec 11851  df-uz 11998  df-rp 12143  df-fz 12649  df-fzo 12790  df-seq 13125  df-exp 13184  df-hash 13442  df-word 13606  df-lsw 13659  df-concat 13667  df-s1 13692  df-substr 13737  df-pfx 13786  df-splice 13893  df-reverse 13911  df-s2 14005  df-struct 16268  df-ndx 16269  df-slot 16270  df-base 16272  df-sets 16273  df-ress 16274  df-plusg 16362  df-mulr 16363  df-starv 16364  df-sca 16365  df-vsca 16366  df-ip 16367  df-tset 16368  df-ple 16369  df-ds 16371  df-unif 16372  df-hom 16373  df-cco 16374  df-0g 16499  df-gsum 16500  df-prds 16505  df-pws 16507  df-mre 16643  df-mrc 16644  df-acs 16646  df-mgm 17639  df-sgrp 17681  df-mnd 17692  df-mhm 17732  df-submnd 17733  df-grp 17823  df-minusg 17824  df-sbg 17825  df-mulg 17939  df-subg 17986  df-ghm 18053  df-gim 18096  df-cntz 18144  df-oppg 18170  df-symg 18192  df-pmtr 18256  df-psgn 18305  df-evpm 18306  df-cmn 18592  df-abl 18593  df-mgp 18888  df-ur 18900  df-srg 18904  df-ring 18947  df-cring 18948  df-oppr 19021  df-dvdsr 19039  df-unit 19040  df-invr 19070  df-dvr 19081  df-rnghom 19115  df-drng 19152  df-subrg 19181  df-lmod 19268  df-lss 19336  df-sra 19580  df-rgmod 19581  df-assa 19720  df-ascl 19722  df-psr 19764  df-mvr 19765  df-mpl 19766  df-opsr 19768  df-psr1 19957  df-vr1 19958  df-ply1 19959  df-coe1 19960  df-cnfld 20154  df-zring 20226  df-zrh 20259  df-dsmm 20486  df-frlm 20501  df-mamu 20605  df-mat 20629  df-mdet 20807  df-madu 20856  df-cpmat 20929  df-mat2pmat 20930  df-cpmat2mat 20931  df-decpmat 20986  df-pm2mp 21016  df-chpmat 21050 This theorem is referenced by:  cayhamlem3  21110
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