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Theorem cayleyhamiltonALT 22713
Description: Alternate proof of cayleyhamilton 22712, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22711 directly, but has the same structure as the proof of cayleyhamilton0 22711. In contrast to the proof of cayleyhamilton0 22711, 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 2731 . . . 4 (Poly1𝑅) = (Poly1𝑅)
4 eqid 2731 . . . 4 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
5 eqid 2731 . . . 4 (.r‘(𝑁 Mat (Poly1𝑅))) = (.r‘(𝑁 Mat (Poly1𝑅)))
6 eqid 2731 . . . 4 (-g‘(𝑁 Mat (Poly1𝑅))) = (-g‘(𝑁 Mat (Poly1𝑅)))
7 eqid 2731 . . . 4 (0g‘(𝑁 Mat (Poly1𝑅))) = (0g‘(𝑁 Mat (Poly1𝑅)))
8 eqid 2731 . . . 4 (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9 cayleyhamilton.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
10 eqid 2731 . . . 4 (𝐶𝑀) = (𝐶𝑀)
11 eqeq1 2735 . . . . . 6 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12 eqeq1 2735 . . . . . . 7 (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13 breq2 5152 . . . . . . . 8 (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14 oveq1 7419 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
1514fveq2d 6895 . . . . . . . . . 10 (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
1615fveq2d 6895 . . . . . . . . 9 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17 fveq2 6891 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑏𝑙) = (𝑏𝑛))
1817fveq2d 6895 . . . . . . . . . 10 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))
1918oveq2d 7428 . . . . . . . . 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 7430 . . . . . . . 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 4554 . . . . . . 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 4554 . . . . . 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 4554 . . . . 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 5261 . . . 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 2731 . . . 4 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
26 eqid 2731 . . . 4 (1r𝐴) = (1r𝐴)
27 cayleyhamilton.m . . . 4 = ( ·𝑠𝐴)
28 eqid 2731 . . . 4 (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29 cayleyhamilton.e . . . 4 = (.g‘(mulGrp‘𝐴))
30 eqid 2731 . . . 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 22710 . . 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 2731 . . . . . . . . 9 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
3328, 32cpm2mfval 22571 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
3433eqcomd 2737 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35343adant3 1131 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
3635fveq1d 6893 . . . . 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 2742 . . . 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 3218 . . 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 257 . 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 2740 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
4241a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
4342fveq1d 6893 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
4443oveq1d 7427 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
4544mpteq2dva 5248 . . . . . . . 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 7428 . . . . . . 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 2733 . . . . . 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 476 . . . . 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 7419 . . . . . . . . . . . 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 6891 . . . . . . . . . . . 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 7430 . . . . . . . . . . 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 5261 . . . . . . . . . 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 7423 . . . . . . . . 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 22688 . . . . . . . 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 2771 . . . . . . 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 6891 . . . . . . . 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 20146 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5958anim2i 616 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60593adant3 1131 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6128, 32cpm2mfval 22571 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
6261eqcomd 2737 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
6362fveq1d 6893 . . . . . . . . . . . 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 2731 . . . . . . . . . . . . 13 (0g𝐴) = (0g𝐴)
651, 28, 3, 4, 64, 7m2cpminv0 22583 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
6663, 65eqtrd 2771 . . . . . . . . . . 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 2789 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = 0 )
7069adantr 480 . . . . . . . 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 2793 . . . . . . 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 684 . . . . . 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 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)))))) → ((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 2771 . . . 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 412 . . 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 3210 . 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 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069  ifcif 4528   class class class wbr 5148  cmpt 5231  cfv 6543  (class class class)co 7412  cmpo 7414  m cmap 8826  Fincfn 8945  0cc0 11116  1c1 11117   + caddc 11119   < clt 11255  cmin 11451  cn 12219  0cn0 12479  ...cfz 13491  Basecbs 17151  .rcmulr 17205   ·𝑠 cvsca 17208  0gc0g 17392   Σg cgsu 17393  -gcsg 18863  .gcmg 18993  mulGrpcmgp 20035  1rcur 20082  Ringcrg 20134  CRingccrg 20135  Poly1cpl1 22020  coe1cco1 22021   Mat cmat 22227   ConstPolyMat ccpmat 22525   matToPolyMat cmat2pmat 22526   cPolyMatToMat ccpmat2mat 22527   CharPlyMat cchpmat 22648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-addf 11195  ax-mulf 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-tpos 8217  df-cur 8258  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-pm 8829  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9368  df-sup 9443  df-oi 9511  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-xnn0 12552  df-z 12566  df-dec 12685  df-uz 12830  df-rp 12982  df-fz 13492  df-fzo 13635  df-seq 13974  df-exp 14035  df-hash 14298  df-word 14472  df-lsw 14520  df-concat 14528  df-s1 14553  df-substr 14598  df-pfx 14628  df-splice 14707  df-reverse 14716  df-s2 14806  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-starv 17219  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-unif 17227  df-hom 17228  df-cco 17229  df-0g 17394  df-gsum 17395  df-prds 17400  df-pws 17402  df-mre 17537  df-mrc 17538  df-acs 17540  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-mhm 18711  df-submnd 18712  df-efmnd 18792  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18994  df-subg 19046  df-ghm 19135  df-gim 19180  df-cntz 19229  df-oppg 19258  df-symg 19283  df-pmtr 19358  df-psgn 19407  df-evpm 19408  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-srg 20088  df-ring 20136  df-cring 20137  df-oppr 20232  df-dvdsr 20255  df-unit 20256  df-invr 20286  df-dvr 20299  df-rhm 20370  df-subrng 20442  df-subrg 20467  df-drng 20585  df-lmod 20704  df-lss 20775  df-sra 21019  df-rgmod 21020  df-cnfld 21234  df-zring 21307  df-zrh 21363  df-dsmm 21597  df-frlm 21612  df-assa 21718  df-ascl 21720  df-psr 21772  df-mvr 21773  df-mpl 21774  df-opsr 21776  df-psr1 22023  df-vr1 22024  df-ply1 22025  df-coe1 22026  df-mamu 22206  df-mat 22228  df-mdet 22407  df-madu 22456  df-cpmat 22528  df-mat2pmat 22529  df-cpmat2mat 22530  df-decpmat 22585  df-pm2mp 22615  df-chpmat 22649
This theorem is referenced by: (None)
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