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Theorem chcoeffeq 20457
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 2609 . . 3 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
11 eqid 2609 . . 3 ( ·𝑠𝑌) = ( ·𝑠𝑌)
12 eqid 2609 . . 3 (1r𝑌) = (1r𝑌)
13 eqid 2609 . . 3 (var1𝑅) = (var1𝑅)
14 eqid 2609 . . 3 (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) = (((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀))
15 eqid 2609 . . 3 (𝑁 maAdju 𝑃) = (𝑁 maAdju 𝑃)
16 chcoeffeq.w . . 3 𝑊 = (Base‘𝑌)
17 eqid 2609 . . 3 (Poly1𝐴) = (Poly1𝐴)
18 eqid 2609 . . 3 (var1𝐴) = (var1𝐴)
19 eqid 2609 . . 3 ( ·𝑠 ‘(Poly1𝐴)) = ( ·𝑠 ‘(Poly1𝐴))
20 eqid 2609 . . 3 (.g‘(mulGrp‘(Poly1𝐴))) = (.g‘(mulGrp‘(Poly1𝐴)))
21 chcoeffeq.u . . 3 𝑈 = (𝑁 cPolyMatToMat 𝑅)
22 eqid 2609 . . 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 20454 . 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 2609 . . . . . . 7 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
25 eqid 2609 . . . . . . 7 (algSc‘𝑃) = (algSc‘𝑃)
26 chcoeffeq.c . . . . . . 7 𝐶 = (𝑁 CharPlyMat 𝑅)
27 chcoeffeq.k . . . . . . 7 𝐾 = (𝐶𝑀)
28 eqid 2609 . . . . . . 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 20444 . . . . . 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 2609 . . . . . . . 8 (𝑁 CharPlyMat 𝑅) = (𝑁 CharPlyMat 𝑅)
331, 2, 32, 3, 4, 13, 5, 7, 11, 12, 14, 15, 6cpmadurid 20438 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)))
3426fveq1i 6088 . . . . . . . . . . 11 (𝐶𝑀) = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3527, 34eqtri 2631 . . . . . . . . . 10 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀)
3635a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐾 = ((𝑁 CharPlyMat 𝑅)‘𝑀))
3736eqcomd 2615 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑁 CharPlyMat 𝑅)‘𝑀) = 𝐾)
3837oveq1d 6541 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (((𝑁 CharPlyMat 𝑅)‘𝑀)( ·𝑠𝑌)(1r𝑌)) = (𝐾( ·𝑠𝑌)(1r𝑌)))
3933, 38eqtrd 2643 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)) × ((𝑁 maAdju 𝑃)‘(((var1𝑅)( ·𝑠𝑌)(1r𝑌)) (𝑇𝑀)))) = (𝐾( ·𝑠𝑌)(1r𝑌)))
40 fveq2 6087 . . . . . . . . 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 475 . . . . . . . . . . . . . 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 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)‘((((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 475 . . . . . . . . . . . . 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 2624 . . . . . . . . . . . 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 20456 . . . . . . . . . . . . . 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 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)))))) → (((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 479 . . . . . . . . . . . 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 228 . . . . . . . . . . 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 627 . . . . . . . . . 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 92 . . . . . . . . 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 33 . . . . . . . 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 448 . . . . . . 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 92 . . . . . 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 46 . . . . 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 647 . . . 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 2999 . . 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 2999 . 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 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  ifcif 4035   class class class wbr 4577  cmpt 4637  cfv 5789  (class class class)co 6526  𝑚 cmap 7721  Fincfn 7818  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cmin 10117  cn 10869  0cn0 11141  ...cfz 12154  Basecbs 15643  .rcmulr 15717   ·𝑠 cvsca 15720  0gc0g 15871   Σg cgsu 15872  -gcsg 17195  .gcmg 17311  mulGrpcmgp 18260  1rcur 18272  CRingccrg 18319  algSccascl 19080  var1cv1 19315  Poly1cpl1 19316  coe1cco1 19317   Mat cmat 19979   maAdju cmadu 20204   ConstPolyMat ccpmat 20274   matToPolyMat cmat2pmat 20275   cPolyMatToMat ccpmat2mat 20276   pMatToMatPoly cpm2mp 20363   CharPlyMat cchpmat 20397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-xor 1456  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-ot 4133  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-tpos 7216  df-cur 7257  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-sup 8208  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-rp 11667  df-fz 12155  df-fzo 12292  df-seq 12621  df-exp 12680  df-hash 12937  df-word 13102  df-lsw 13103  df-concat 13104  df-s1 13105  df-substr 13106  df-splice 13107  df-reverse 13108  df-s2 13392  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-mulr 15730  df-starv 15731  df-sca 15732  df-vsca 15733  df-ip 15734  df-tset 15735  df-ple 15736  df-ds 15739  df-unif 15740  df-hom 15741  df-cco 15742  df-0g 15873  df-gsum 15874  df-prds 15879  df-pws 15881  df-mre 16017  df-mrc 16018  df-acs 16020  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-mhm 17106  df-submnd 17107  df-grp 17196  df-minusg 17197  df-sbg 17198  df-mulg 17312  df-subg 17362  df-ghm 17429  df-gim 17472  df-cntz 17521  df-oppg 17547  df-symg 17569  df-pmtr 17633  df-psgn 17682  df-evpm 17683  df-cmn 17966  df-abl 17967  df-mgp 18261  df-ur 18273  df-srg 18277  df-ring 18320  df-cring 18321  df-oppr 18394  df-dvdsr 18412  df-unit 18413  df-invr 18443  df-dvr 18454  df-rnghom 18486  df-drng 18520  df-subrg 18549  df-lmod 18636  df-lss 18702  df-sra 18941  df-rgmod 18942  df-assa 19081  df-ascl 19083  df-psr 19125  df-mvr 19126  df-mpl 19127  df-opsr 19129  df-psr1 19319  df-vr1 19320  df-ply1 19321  df-coe1 19322  df-cnfld 19516  df-zring 19586  df-zrh 19618  df-dsmm 19842  df-frlm 19857  df-mamu 19956  df-mat 19980  df-mdet 20157  df-madu 20206  df-cpmat 20277  df-mat2pmat 20278  df-cpmat2mat 20279  df-decpmat 20334  df-pm2mp 20364  df-chpmat 20398
This theorem is referenced by:  cayhamlem3  20458
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