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Theorem cayleyhamiltonALT 21537
 Description: Alternate proof of cayleyhamilton 21536, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 21535 directly, but has the same structure as the proof of cayleyhamilton0 21535. In contrast to the proof of cayleyhamilton0 21535, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cayleyhamilton.a 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton.b 𝐵 = (Base‘𝐴)
cayleyhamilton.0 0 = (0g𝐴)
cayleyhamilton.c 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton.k 𝐾 = (coe1‘(𝐶𝑀))
cayleyhamilton.m = ( ·𝑠𝐴)
cayleyhamilton.e = (.g‘(mulGrp‘𝐴))
Assertion
Ref Expression
cayleyhamiltonALT ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝐶,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   ,𝑛   ,𝑛
Allowed substitution hints:   𝐾(𝑛)   0 (𝑛)

Proof of Theorem cayleyhamiltonALT
Dummy variables 𝑏 𝑚 𝑠 𝑥 𝑦 𝑙 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cayleyhamilton.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 cayleyhamilton.b . . . 4 𝐵 = (Base‘𝐴)
3 eqid 2798 . . . 4 (Poly1𝑅) = (Poly1𝑅)
4 eqid 2798 . . . 4 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
5 eqid 2798 . . . 4 (.r‘(𝑁 Mat (Poly1𝑅))) = (.r‘(𝑁 Mat (Poly1𝑅)))
6 eqid 2798 . . . 4 (-g‘(𝑁 Mat (Poly1𝑅))) = (-g‘(𝑁 Mat (Poly1𝑅)))
7 eqid 2798 . . . 4 (0g‘(𝑁 Mat (Poly1𝑅))) = (0g‘(𝑁 Mat (Poly1𝑅)))
8 eqid 2798 . . . 4 (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9 cayleyhamilton.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
10 eqid 2798 . . . 4 (𝐶𝑀) = (𝐶𝑀)
11 eqeq1 2802 . . . . . 6 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12 eqeq1 2802 . . . . . . 7 (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13 breq2 5038 . . . . . . . 8 (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14 oveq1 7152 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
1514fveq2d 6659 . . . . . . . . . 10 (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
1615fveq2d 6659 . . . . . . . . 9 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17 fveq2 6655 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑏𝑙) = (𝑏𝑛))
1817fveq2d 6659 . . . . . . . . . 10 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))
1918oveq2d 7161 . . . . . . . . 9 (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))) = (((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛))))
2016, 19oveq12d 7163 . . . . . . . 8 (𝑙 = 𝑛 → (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)))) = (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))))
2113, 20ifbieq2d 4453 . . . . . . 7 (𝑙 = 𝑛 → if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))) = if((𝑠 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛))))))
2212, 21ifbieq2d 4453 . . . . . 6 (𝑙 = 𝑛 → if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)))))) = if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))))))
2311, 22ifbieq2d 4453 . . . . 5 (𝑙 = 𝑛 → if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))) = if(𝑛 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛))))))))
2423cbvmptv 5137 . . . 4 (𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)))))))) = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛))))))))
25 eqid 2798 . . . 4 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
26 eqid 2798 . . . 4 (1r𝐴) = (1r𝐴)
27 cayleyhamilton.m . . . 4 = ( ·𝑠𝐴)
28 eqid 2798 . . . 4 (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29 cayleyhamilton.e . . . 4 = (.g‘(mulGrp‘𝐴))
30 eqid 2798 . . . 4 (.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅)))) = (.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))
311, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 26, 27, 28, 29, 30cayhamlem4 21534 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))))
32 eqid 2798 . . . . . . . . 9 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
3328, 32cpm2mfval 21395 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
3433eqcomd 2804 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35343adant3 1129 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
3635fveq1d 6657 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))))
3736eqeq2d 2809 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))))
38372rexbidv 3260 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) ↔ ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑁 cPolyMatToMat 𝑅)‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))))
3931, 38mpbird 260 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))))
40 cayleyhamilton.k . . . . . . . . . . . . 13 𝐾 = (coe1‘(𝐶𝑀))
4140eqcomi 2807 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
4241a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
4342fveq1d 6657 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
4443oveq1d 7160 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
4544mpteq2dva 5129 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀))) = (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀))))
4645oveq2d 7161 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))))
4746eqeq1d 2800 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))))
4847biimpa 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))))
49 oveq1 7152 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀)) = (𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀)))
50 fveq2 6655 . . . . . . . . . . . 12 (𝑛 = 𝑗 → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗))
5149, 50oveq12d 7163 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)) = ((𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗)))
5251cbvmptv 5137 . . . . . . . . . 10 (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))) = (𝑗 ∈ ℕ0 ↦ ((𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗)))
5352oveq2i 7156 . . . . . . . . 9 ((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))) = ((𝑁 Mat (Poly1𝑅)) Σg (𝑗 ∈ ℕ0 ↦ ((𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗))))
5453a1i 11 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))) = ((𝑁 Mat (Poly1𝑅)) Σg (𝑗 ∈ ℕ0 ↦ ((𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗)))))
551, 2, 3, 4, 5, 6, 7, 8, 24, 30cayhamlem1 21512 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑁 Mat (Poly1𝑅)) Σg (𝑗 ∈ ℕ0 ↦ ((𝑗(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑗)))) = (0g‘(𝑁 Mat (Poly1𝑅))))
5654, 55eqtrd 2833 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))) = (0g‘(𝑁 Mat (Poly1𝑅))))
57 fveq2 6655 . . . . . . . 8 (((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))) = (0g‘(𝑁 Mat (Poly1𝑅))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))))
58 crngring 19323 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5958anim2i 619 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60593adant3 1129 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6128, 32cpm2mfval 21395 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
6261eqcomd 2804 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
6362fveq1d 6657 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = ((𝑁 cPolyMatToMat 𝑅)‘(0g‘(𝑁 Mat (Poly1𝑅)))))
64 eqid 2798 . . . . . . . . . . . . 13 (0g𝐴) = (0g𝐴)
651, 28, 3, 4, 64, 7m2cpminv0 21407 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
6663, 65eqtrd 2833 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
6760, 66syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
68 cayleyhamilton.0 . . . . . . . . . 10 0 = (0g𝐴)
6967, 68eqtr4di 2851 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = 0 )
7069adantr 484 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = 0 )
7157, 70sylan9eqr 2855 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ ((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))) = (0g‘(𝑁 Mat (Poly1𝑅)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) = 0 )
7256, 71mpdan 686 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) = 0 )
7372adantr 484 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) = 0 )
7448, 73eqtrd 2833 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
7574ex 416 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 ))
7675rexlimdvva 3254 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘((𝑁 Mat (Poly1𝑅)) Σg (𝑛 ∈ ℕ0 ↦ ((𝑛(.g‘(mulGrp‘(𝑁 Mat (Poly1𝑅))))((𝑁 matToPolyMat 𝑅)‘𝑀))(.r‘(𝑁 Mat (Poly1𝑅)))((𝑙 ∈ ℕ0 ↦ if(𝑙 = 0, ((0g‘(𝑁 Mat (Poly1𝑅)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏‘0)))), if(𝑙 = (𝑠 + 1), ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑠)), if((𝑠 + 1) < 𝑙, (0g‘(𝑁 Mat (Poly1𝑅))), (((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1)))(-g‘(𝑁 Mat (Poly1𝑅)))(((𝑁 matToPolyMat 𝑅)‘𝑀)(.r‘(𝑁 Mat (Poly1𝑅)))((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙))))))))‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 ))
7739, 76mpd 15 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ifcif 4428   class class class wbr 5034   ↦ cmpt 5114  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147   ↑m cmap 8407  Fincfn 8510  0cc0 10544  1c1 10545   + caddc 10547   < clt 10682   − cmin 10877  ℕcn 11643  ℕ0cn0 11903  ...cfz 12905  Basecbs 16495  .rcmulr 16578   ·𝑠 cvsca 16581  0gc0g 16725   Σg cgsu 16726  -gcsg 18117  .gcmg 18237  mulGrpcmgp 19253  1rcur 19265  Ringcrg 19311  CRingccrg 19312  Poly1cpl1 20847  coe1cco1 20848   Mat cmat 21053   ConstPolyMat ccpmat 21349   matToPolyMat cmat2pmat 21350   cPolyMatToMat ccpmat2mat 21351   CharPlyMat cchpmat 21472 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-addf 10623  ax-mulf 10624 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-ofr 7401  df-om 7574  df-1st 7684  df-2nd 7685  df-supp 7827  df-tpos 7893  df-cur 7934  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-ixp 8463  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-4 11708  df-5 11709  df-6 11710  df-7 11711  df-8 11712  df-9 11713  df-n0 11904  df-xnn0 11976  df-z 11990  df-dec 12107  df-uz 12252  df-rp 12398  df-fz 12906  df-fzo 13049  df-seq 13385  df-exp 13446  df-hash 13707  df-word 13878  df-lsw 13926  df-concat 13934  df-s1 13961  df-substr 14014  df-pfx 14044  df-splice 14123  df-reverse 14132  df-s2 14221  df-struct 16497  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-mulr 16591  df-starv 16592  df-sca 16593  df-vsca 16594  df-ip 16595  df-tset 16596  df-ple 16597  df-ds 16599  df-unif 16600  df-hom 16601  df-cco 16602  df-0g 16727  df-gsum 16728  df-prds 16733  df-pws 16735  df-mre 16869  df-mrc 16870  df-acs 16872  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-mhm 17968  df-submnd 17969  df-efmnd 18046  df-grp 18118  df-minusg 18119  df-sbg 18120  df-mulg 18238  df-subg 18289  df-ghm 18369  df-gim 18412  df-cntz 18460  df-oppg 18487  df-symg 18509  df-pmtr 18583  df-psgn 18632  df-evpm 18633  df-cmn 18921  df-abl 18922  df-mgp 19254  df-ur 19266  df-srg 19270  df-ring 19313  df-cring 19314  df-oppr 19390  df-dvdsr 19408  df-unit 19409  df-invr 19439  df-dvr 19450  df-rnghom 19484  df-drng 19518  df-subrg 19547  df-lmod 19650  df-lss 19718  df-sra 19958  df-rgmod 19959  df-cnfld 20113  df-zring 20185  df-zrh 20219  df-dsmm 20443  df-frlm 20458  df-assa 20564  df-ascl 20566  df-psr 20617  df-mvr 20618  df-mpl 20619  df-opsr 20621  df-psr1 20850  df-vr1 20851  df-ply1 20852  df-coe1 20853  df-mamu 21032  df-mat 21054  df-mdet 21231  df-madu 21280  df-cpmat 21352  df-mat2pmat 21353  df-cpmat2mat 21354  df-decpmat 21409  df-pm2mp 21439  df-chpmat 21473 This theorem is referenced by: (None)
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