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

Theorem chcoeffeq 20891
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 2758 . . 3 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11 eqid 2758 . . 3 ( ·𝑠𝑌) = ( ·𝑠𝑌)
12 eqid 2758 . . 3 (1r𝑌) = (1r𝑌)
13 eqid 2758 . . 3 (var1𝑅) = (var1𝑅)
14 eqid 2758 . . 3 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
15 eqid 2758 . . 3 (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16 chcoeffeq.w . . 3 𝑊 = (Base‘𝑌)
17 eqid 2758 . . 3 (Poly1𝐴) = (Poly1𝐴)
18 eqid 2758 . . 3 (var1𝐴) = (var1𝐴)
19 eqid 2758 . . 3 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
20 eqid 2758 . . 3 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
21 chcoeffeq.u . . 3 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22 eqid 2758 . . 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 20888 . 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 2758 . . . . . . 7 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25 eqid 2758 . . . . . . 7 (algSc‘𝑃) = (algSc‘𝑃)
26 chcoeffeq.c . . . . . . 7 𝐶 = (𝑁 CharPlyMat 𝑅)
27 chcoeffeq.k . . . . . . 7 𝐾 = (𝐶𝑀)
28 eqid 2758 . . . . . . 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 20878 . . . . . 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 2758 . . . . . . . 8 (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
331, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6cpmadurid 20872 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)))
3426fveq1i 6351 . . . . . . . . . . 11 (𝐶𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3527, 34eqtri 2780 . . . . . . . . . 10 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3635a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀))
3736eqcomd 2764 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
3837oveq1d 6826 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)) = (𝐾( ·𝑠𝑌)(1r𝑌)))
3933, 38eqtrd 2792 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)))
40 fveq2 6350 . . . . . . . . 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 472 . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 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 20890 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 230 . . . . . . . . . . 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 631 . . . . . . . . . 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 449 . . . . . . 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 651 . . . 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 3153 . . 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 3153 . 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 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  wrex 3049  ifcif 4228   class class class wbr 4802  cmpt 4879  cfv 6047  (class class class)co 6811  𝑚 cmap 8021  Fincfn 8119  0cc0 10126  1c1 10127   + caddc 10129   < clt 10264  cmin 10456  cn 11210  0cn0 11482  ...cfz 12517  Basecbs 16057  .rcmulr 16142   ·𝑠 cvsca 16145  0gc0g 16300   Σg cgsu 16301  -gcsg 17623  .gcmg 17739  mulGrpcmgp 18687  1rcur 18699  CRingccrg 18746  algSccascl 19511  var1cv1 19746  Poly1cpl1 19747  coe1cco1 19748   Mat cmat 20413   maAdju cmadu 20638   ConstPolyMat ccpmat 20708   matToPolyMat cmat2pmat 20709   cPolyMatToMat ccpmat2mat 20710   pMatToMatPoly cpm2mp 20797   CharPlyMat cchpmat 20831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-addf 10205  ax-mulf 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-xor 1612  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-ot 4328  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-of 7060  df-ofr 7061  df-om 7229  df-1st 7331  df-2nd 7332  df-supp 7462  df-tpos 7519  df-cur 7560  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-pm 8024  df-ixp 8073  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fsupp 8439  df-sup 8511  df-oi 8578  df-card 8953  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-xnn0 11554  df-z 11568  df-dec 11684  df-uz 11878  df-rp 12024  df-fz 12518  df-fzo 12658  df-seq 12994  df-exp 13053  df-hash 13310  df-word 13483  df-lsw 13484  df-concat 13485  df-s1 13486  df-substr 13487  df-splice 13488  df-reverse 13489  df-s2 13791  df-struct 16059  df-ndx 16060  df-slot 16061  df-base 16063  df-sets 16064  df-ress 16065  df-plusg 16154  df-mulr 16155  df-starv 16156  df-sca 16157  df-vsca 16158  df-ip 16159  df-tset 16160  df-ple 16161  df-ds 16164  df-unif 16165  df-hom 16166  df-cco 16167  df-0g 16302  df-gsum 16303  df-prds 16308  df-pws 16310  df-mre 16446  df-mrc 16447  df-acs 16449  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-mhm 17534  df-submnd 17535  df-grp 17624  df-minusg 17625  df-sbg 17626  df-mulg 17740  df-subg 17790  df-ghm 17857  df-gim 17900  df-cntz 17948  df-oppg 17974  df-symg 17996  df-pmtr 18060  df-psgn 18109  df-evpm 18110  df-cmn 18393  df-abl 18394  df-mgp 18688  df-ur 18700  df-srg 18704  df-ring 18747  df-cring 18748  df-oppr 18821  df-dvdsr 18839  df-unit 18840  df-invr 18870  df-dvr 18881  df-rnghom 18915  df-drng 18949  df-subrg 18978  df-lmod 19065  df-lss 19133  df-sra 19372  df-rgmod 19373  df-assa 19512  df-ascl 19514  df-psr 19556  df-mvr 19557  df-mpl 19558  df-opsr 19560  df-psr1 19750  df-vr1 19751  df-ply1 19752  df-coe1 19753  df-cnfld 19947  df-zring 20019  df-zrh 20052  df-dsmm 20276  df-frlm 20291  df-mamu 20390  df-mat 20414  df-mdet 20591  df-madu 20640  df-cpmat 20711  df-mat2pmat 20712  df-cpmat2mat 20713  df-decpmat 20768  df-pm2mp 20798  df-chpmat 20832
This theorem is referenced by:  cayhamlem3  20892
  Copyright terms: Public domain W3C validator