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Theorem cayleyhamiltonALT 20916
Description: Alternate proof of cayleyhamilton 20915, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20914 directly, but has the same structure as the proof of cayleyhamilton0 20914. In contrast to the proof of cayleyhamilton0 20914, 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 2771 . . . 4 (Poly1𝑅) = (Poly1𝑅)
4 eqid 2771 . . . 4 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
5 eqid 2771 . . . 4 (.r‘(𝑁 Mat (Poly1𝑅))) = (.r‘(𝑁 Mat (Poly1𝑅)))
6 eqid 2771 . . . 4 (-g‘(𝑁 Mat (Poly1𝑅))) = (-g‘(𝑁 Mat (Poly1𝑅)))
7 eqid 2771 . . . 4 (0g‘(𝑁 Mat (Poly1𝑅))) = (0g‘(𝑁 Mat (Poly1𝑅)))
8 eqid 2771 . . . 4 (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9 cayleyhamilton.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
10 eqid 2771 . . . 4 (𝐶𝑀) = (𝐶𝑀)
11 eqeq1 2775 . . . . . 6 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12 eqeq1 2775 . . . . . . 7 (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13 breq2 4790 . . . . . . . 8 (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14 oveq1 6800 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
1514fveq2d 6336 . . . . . . . . . 10 (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
1615fveq2d 6336 . . . . . . . . 9 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17 fveq2 6332 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑏𝑙) = (𝑏𝑛))
1817fveq2d 6336 . . . . . . . . . 10 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))
1918oveq2d 6809 . . . . . . . . 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 6811 . . . . . . . 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 4250 . . . . . . 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 4250 . . . . . 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 4250 . . . . 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 4884 . . . 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 2771 . . . 4 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
26 eqid 2771 . . . 4 (1r𝐴) = (1r𝐴)
27 cayleyhamilton.m . . . 4 = ( ·𝑠𝐴)
28 eqid 2771 . . . 4 (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29 cayleyhamilton.e . . . 4 = (.g‘(mulGrp‘𝐴))
30 eqid 2771 . . . 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 20913 . . 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 2771 . . . . . . . . 9 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
3328, 32cpm2mfval 20774 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
3433eqcomd 2777 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35343adant3 1126 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
3635fveq1d 6334 . . . . 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 2781 . . . 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 3205 . . 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 247 . 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 2780 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
4241a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
4342fveq1d 6334 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
4443oveq1d 6808 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
4544mpteq2dva 4878 . . . . . . . 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 6809 . . . . . . 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 2773 . . . . . 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 462 . . . . 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 6800 . . . . . . . . . . . 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 6332 . . . . . . . . . . . 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 6811 . . . . . . . . . . 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 4884 . . . . . . . . . 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 6804 . . . . . . . . 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 20891 . . . . . . . 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 2805 . . . . . . 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 6332 . . . . . . . 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 18766 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5958anim2i 603 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60593adant3 1126 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6128, 32cpm2mfval 20774 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
6261eqcomd 2777 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
6362fveq1d 6334 . . . . . . . . . . . 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 2771 . . . . . . . . . . . . 13 (0g𝐴) = (0g𝐴)
651, 28, 3, 4, 64, 7m2cpminv0 20786 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
6663, 65eqtrd 2805 . . . . . . . . . . 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, 68syl6eqr 2823 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = 0 )
7069adantr 466 . . . . . . . 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 2827 . . . . . . 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 667 . . . . . 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 466 . . . . 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 2805 . . . 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 397 . . 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 3186 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wrex 3062  ifcif 4225   class class class wbr 4786  cmpt 4863  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009  Fincfn 8109  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276  cmin 10468  cn 11222  0cn0 11494  ...cfz 12533  Basecbs 16064  .rcmulr 16150   ·𝑠 cvsca 16153  0gc0g 16308   Σg cgsu 16309  -gcsg 17632  .gcmg 17748  mulGrpcmgp 18697  1rcur 18709  Ringcrg 18755  CRingccrg 18756  Poly1cpl1 19762  coe1cco1 19763   Mat cmat 20430   ConstPolyMat ccpmat 20728   matToPolyMat cmat2pmat 20729   cPolyMatToMat ccpmat2mat 20730   CharPlyMat cchpmat 20851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-xor 1613  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-ot 4325  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-ofr 7045  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-tpos 7504  df-cur 7545  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-sup 8504  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-xnn0 11566  df-z 11580  df-dec 11696  df-uz 11889  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-splice 13500  df-reverse 13501  df-s2 13802  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-0g 16310  df-gsum 16311  df-prds 16316  df-pws 16318  df-mre 16454  df-mrc 16455  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-subg 17799  df-ghm 17866  df-gim 17909  df-cntz 17957  df-oppg 17983  df-symg 18005  df-pmtr 18069  df-psgn 18118  df-evpm 18119  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-srg 18714  df-ring 18757  df-cring 18758  df-oppr 18831  df-dvdsr 18849  df-unit 18850  df-invr 18880  df-dvr 18891  df-rnghom 18925  df-drng 18959  df-subrg 18988  df-lmod 19075  df-lss 19143  df-sra 19387  df-rgmod 19388  df-assa 19527  df-ascl 19529  df-psr 19571  df-mvr 19572  df-mpl 19573  df-opsr 19575  df-psr1 19765  df-vr1 19766  df-ply1 19767  df-coe1 19768  df-cnfld 19962  df-zring 20034  df-zrh 20067  df-dsmm 20293  df-frlm 20308  df-mamu 20407  df-mat 20431  df-mdet 20609  df-madu 20658  df-cpmat 20731  df-mat2pmat 20732  df-cpmat2mat 20733  df-decpmat 20788  df-pm2mp 20818  df-chpmat 20852
This theorem is referenced by: (None)
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