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Theorem cayleyhamiltonALT 20615
Description: Alternate proof of cayleyhamilton 20614, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20613 directly, but has the same structure as the proof of cayleyhamilton0 20613. In contrast to the proof of cayleyhamilton0 20613, 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 2621 . . . 4 (Poly1𝑅) = (Poly1𝑅)
4 eqid 2621 . . . 4 (𝑁 Mat (Poly1𝑅)) = (𝑁 Mat (Poly1𝑅))
5 eqid 2621 . . . 4 (.r‘(𝑁 Mat (Poly1𝑅))) = (.r‘(𝑁 Mat (Poly1𝑅)))
6 eqid 2621 . . . 4 (-g‘(𝑁 Mat (Poly1𝑅))) = (-g‘(𝑁 Mat (Poly1𝑅)))
7 eqid 2621 . . . 4 (0g‘(𝑁 Mat (Poly1𝑅))) = (0g‘(𝑁 Mat (Poly1𝑅)))
8 eqid 2621 . . . 4 (𝑁 matToPolyMat 𝑅) = (𝑁 matToPolyMat 𝑅)
9 cayleyhamilton.c . . . 4 𝐶 = (𝑁 CharPlyMat 𝑅)
10 eqid 2621 . . . 4 (𝐶𝑀) = (𝐶𝑀)
11 eqeq1 2625 . . . . . 6 (𝑙 = 𝑛 → (𝑙 = 0 ↔ 𝑛 = 0))
12 eqeq1 2625 . . . . . . 7 (𝑙 = 𝑛 → (𝑙 = (𝑠 + 1) ↔ 𝑛 = (𝑠 + 1)))
13 breq2 4617 . . . . . . . 8 (𝑙 = 𝑛 → ((𝑠 + 1) < 𝑙 ↔ (𝑠 + 1) < 𝑛))
14 oveq1 6611 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑙 − 1) = (𝑛 − 1))
1514fveq2d 6152 . . . . . . . . . 10 (𝑙 = 𝑛 → (𝑏‘(𝑙 − 1)) = (𝑏‘(𝑛 − 1)))
1615fveq2d 6152 . . . . . . . . 9 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑙 − 1))) = ((𝑁 matToPolyMat 𝑅)‘(𝑏‘(𝑛 − 1))))
17 fveq2 6148 . . . . . . . . . . 11 (𝑙 = 𝑛 → (𝑏𝑙) = (𝑏𝑛))
1817fveq2d 6152 . . . . . . . . . 10 (𝑙 = 𝑛 → ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑙)) = ((𝑁 matToPolyMat 𝑅)‘(𝑏𝑛)))
1918oveq2d 6620 . . . . . . . . 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 6622 . . . . . . . 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 4083 . . . . . . 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 4083 . . . . . 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 4083 . . . . 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 4710 . . . 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 2621 . . . 4 (Base‘(𝑁 Mat (Poly1𝑅))) = (Base‘(𝑁 Mat (Poly1𝑅)))
26 eqid 2621 . . . 4 (1r𝐴) = (1r𝐴)
27 cayleyhamilton.m . . . 4 = ( ·𝑠𝐴)
28 eqid 2621 . . . 4 (𝑁 cPolyMatToMat 𝑅) = (𝑁 cPolyMatToMat 𝑅)
29 cayleyhamilton.e . . . 4 = (.g‘(mulGrp‘𝐴))
30 eqid 2621 . . . 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 20612 . . 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 2621 . . . . . . . . 9 (𝑁 ConstPolyMat 𝑅) = (𝑁 ConstPolyMat 𝑅)
3328, 32cpm2mfval 20473 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
3433eqcomd 2627 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
35343adant3 1079 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
3635fveq1d 6150 . . . . 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 2631 . . . 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 3050 . . 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 2630 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
4241a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
4342fveq1d 6150 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
4443oveq1d 6619 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
4544mpteq2dva 4704 . . . . . . . 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 6620 . . . . . . 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 2623 . . . . . 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 501 . . . . 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 6611 . . . . . . . . . . . 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 6148 . . . . . . . . . . . 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 6622 . . . . . . . . . . 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 4710 . . . . . . . . . 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 6615 . . . . . . . . 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 20590 . . . . . . . 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 2655 . . . . . . 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 6148 . . . . . . . 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 18479 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5958anim2i 592 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
60593adant3 1079 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
6128, 32cpm2mfval 20473 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 cPolyMatToMat 𝑅) = (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
6261eqcomd 2627 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))) = (𝑁 cPolyMatToMat 𝑅))
6362fveq1d 6150 . . . . . . . . . . . 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 2621 . . . . . . . . . . . . 13 (0g𝐴) = (0g𝐴)
651, 28, 3, 4, 64, 7m2cpminv0 20485 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑁 cPolyMatToMat 𝑅)‘(0g‘(𝑁 Mat (Poly1𝑅)))) = (0g𝐴))
6663, 65eqtrd 2655 . . . . . . . . . . 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 2673 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑚 ∈ (𝑁 ConstPolyMat 𝑅) ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))‘(0g‘(𝑁 Mat (Poly1𝑅)))) = 0 )
7069adantr 481 . . . . . . . 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 2677 . . . . . . 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 701 . . . . . 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 481 . . . . 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 2655 . . . 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 450 . . 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 3031 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wrex 2908  ifcif 4058   class class class wbr 4613  cmpt 4673  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Fincfn 7899  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  cmin 10210  cn 10964  0cn0 11236  ...cfz 12268  Basecbs 15781  .rcmulr 15863   ·𝑠 cvsca 15866  0gc0g 16021   Σg cgsu 16022  -gcsg 17345  .gcmg 17461  mulGrpcmgp 18410  1rcur 18422  Ringcrg 18468  CRingccrg 18469  Poly1cpl1 19466  coe1cco1 19467   Mat cmat 20132   ConstPolyMat ccpmat 20427   matToPolyMat cmat2pmat 20428   cPolyMatToMat ccpmat2mat 20429   CharPlyMat cchpmat 20550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-tpos 7297  df-cur 7338  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-reverse 13244  df-s2 13530  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-0g 16023  df-gsum 16024  df-prds 16029  df-pws 16031  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-ghm 17579  df-gim 17622  df-cntz 17671  df-oppg 17697  df-symg 17719  df-pmtr 17783  df-psgn 17832  df-evpm 17833  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-srg 18427  df-ring 18470  df-cring 18471  df-oppr 18544  df-dvdsr 18562  df-unit 18563  df-invr 18593  df-dvr 18604  df-rnghom 18636  df-drng 18670  df-subrg 18699  df-lmod 18786  df-lss 18852  df-sra 19091  df-rgmod 19092  df-assa 19231  df-ascl 19233  df-psr 19275  df-mvr 19276  df-mpl 19277  df-opsr 19279  df-psr1 19469  df-vr1 19470  df-ply1 19471  df-coe1 19472  df-cnfld 19666  df-zring 19738  df-zrh 19771  df-dsmm 19995  df-frlm 20010  df-mamu 20109  df-mat 20133  df-mdet 20310  df-madu 20359  df-cpmat 20430  df-mat2pmat 20431  df-cpmat2mat 20432  df-decpmat 20487  df-pm2mp 20517  df-chpmat 20551
This theorem is referenced by: (None)
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