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Theorem chcoeffeq 21496
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 2823 . . 3 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11 eqid 2823 . . 3 ( ·𝑠𝑌) = ( ·𝑠𝑌)
12 eqid 2823 . . 3 (1r𝑌) = (1r𝑌)
13 eqid 2823 . . 3 (var1𝑅) = (var1𝑅)
14 eqid 2823 . . 3 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
15 eqid 2823 . . 3 (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16 chcoeffeq.w . . 3 𝑊 = (Base‘𝑌)
17 eqid 2823 . . 3 (Poly1𝐴) = (Poly1𝐴)
18 eqid 2823 . . 3 (var1𝐴) = (var1𝐴)
19 eqid 2823 . . 3 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
20 eqid 2823 . . 3 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
21 chcoeffeq.u . . 3 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22 eqid 2823 . . 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 21493 . 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 2823 . . . . . . 7 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25 eqid 2823 . . . . . . 7 (algSc‘𝑃) = (algSc‘𝑃)
26 chcoeffeq.c . . . . . . 7 𝐶 = (𝑁 CharPlyMat 𝑅)
27 chcoeffeq.k . . . . . . 7 𝐾 = (𝐶𝑀)
28 eqid 2823 . . . . . . 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 21483 . . . . . 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 2823 . . . . . . . 8 (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
331, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6cpmadurid 21477 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)))
3426fveq1i 6673 . . . . . . . . . . 11 (𝐶𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3527, 34eqtri 2846 . . . . . . . . . 10 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3635a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀))
3736eqcomd 2829 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
3837oveq1d 7173 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)) = (𝐾( ·𝑠𝑌)(1r𝑌)))
3933, 38eqtrd 2858 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)))
40 fveq2 6672 . . . . . . . . 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 487 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 487 . . . . . . . . . . . . 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 2839 . . . . . . . . . . . 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 21495 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 483 . . . . . . . . . . . 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 242 . . . . . . . . . . 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 422 . . . . . . . . . 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 415 . . . . . . 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 458 . . . 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 3276 . . 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 3276 . 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 398  w3a 1083   = wceq 1537  wcel 2114  wral 3140  wrex 3141  ifcif 4469   class class class wbr 5068  cmpt 5148  cfv 6357  (class class class)co 7158  m cmap 8408  Fincfn 8511  0cc0 10539  1c1 10540   + caddc 10542   < clt 10677  cmin 10872  cn 11640  0cn0 11900  ...cfz 12895  Basecbs 16485  .rcmulr 16568   ·𝑠 cvsca 16571  0gc0g 16715   Σg cgsu 16716  -gcsg 18107  .gcmg 18226  mulGrpcmgp 19241  1rcur 19253  CRingccrg 19300  algSccascl 20086  var1cv1 20346  Poly1cpl1 20347  coe1cco1 20348   Mat cmat 21018   maAdju cmadu 21243   ConstPolyMat ccpmat 21313   matToPolyMat cmat2pmat 21314   cPolyMatToMat ccpmat2mat 21315   pMatToMatPoly cpm2mp 21402   CharPlyMat cchpmat 21436
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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-addf 10618  ax-mulf 10619
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-tpos 7894  df-cur 7935  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-word 13865  df-lsw 13917  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-splice 14114  df-reverse 14123  df-s2 14212  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-0g 16717  df-gsum 16718  df-prds 16723  df-pws 16725  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-efmnd 18036  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-gim 18401  df-cntz 18449  df-oppg 18476  df-symg 18498  df-pmtr 18572  df-psgn 18621  df-evpm 18622  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-srg 19258  df-ring 19301  df-cring 19302  df-oppr 19375  df-dvdsr 19393  df-unit 19394  df-invr 19424  df-dvr 19435  df-rnghom 19469  df-drng 19506  df-subrg 19535  df-lmod 19638  df-lss 19706  df-sra 19946  df-rgmod 19947  df-assa 20087  df-ascl 20089  df-psr 20138  df-mvr 20139  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-vr1 20351  df-ply1 20352  df-coe1 20353  df-cnfld 20548  df-zring 20620  df-zrh 20653  df-dsmm 20878  df-frlm 20893  df-mamu 20997  df-mat 21019  df-mdet 21196  df-madu 21245  df-cpmat 21316  df-mat2pmat 21317  df-cpmat2mat 21318  df-decpmat 21373  df-pm2mp 21403  df-chpmat 21437
This theorem is referenced by:  cayhamlem3  21497
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