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Theorem cayleyhamiltonALT 22263
Description: Alternate proof of cayleyhamilton 22262, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22261 directly, but has the same structure as the proof of cayleyhamilton0 22261. In contrast to the proof of cayleyhamilton0 22261, 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 ๐ด = (๐‘ Mat ๐‘…)
cayleyhamilton.b ๐ต = (Baseโ€˜๐ด)
cayleyhamilton.0 0 = (0gโ€˜๐ด)
cayleyhamilton.c ๐ถ = (๐‘ CharPlyMat ๐‘…)
cayleyhamilton.k ๐พ = (coe1โ€˜(๐ถโ€˜๐‘€))
cayleyhamilton.m โˆ— = ( ยท๐‘  โ€˜๐ด)
cayleyhamilton.e โ†‘ = (.gโ€˜(mulGrpโ€˜๐ด))
Assertion
Ref Expression
cayleyhamiltonALT ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = 0 )
Distinct variable groups:   ๐ด,๐‘›   ๐ต,๐‘›   ๐ถ,๐‘›   ๐‘›,๐‘€   ๐‘›,๐‘   ๐‘…,๐‘›   โˆ— ,๐‘›   โ†‘ ,๐‘›
Allowed substitution hints:   ๐พ(๐‘›)   0 (๐‘›)

Proof of Theorem cayleyhamiltonALT
Dummy variables ๐‘ ๐‘š ๐‘  ๐‘ฅ ๐‘ฆ ๐‘™ ๐‘— are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cayleyhamilton.a . . . 4 ๐ด = (๐‘ Mat ๐‘…)
2 cayleyhamilton.b . . . 4 ๐ต = (Baseโ€˜๐ด)
3 eqid 2733 . . . 4 (Poly1โ€˜๐‘…) = (Poly1โ€˜๐‘…)
4 eqid 2733 . . . 4 (๐‘ Mat (Poly1โ€˜๐‘…)) = (๐‘ Mat (Poly1โ€˜๐‘…))
5 eqid 2733 . . . 4 (.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))) = (.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))
6 eqid 2733 . . . 4 (-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))) = (-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))
7 eqid 2733 . . . 4 (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))) = (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))
8 eqid 2733 . . . 4 (๐‘ matToPolyMat ๐‘…) = (๐‘ matToPolyMat ๐‘…)
9 cayleyhamilton.c . . . 4 ๐ถ = (๐‘ CharPlyMat ๐‘…)
10 eqid 2733 . . . 4 (๐ถโ€˜๐‘€) = (๐ถโ€˜๐‘€)
11 eqeq1 2737 . . . . . 6 (๐‘™ = ๐‘› โ†’ (๐‘™ = 0 โ†” ๐‘› = 0))
12 eqeq1 2737 . . . . . . 7 (๐‘™ = ๐‘› โ†’ (๐‘™ = (๐‘  + 1) โ†” ๐‘› = (๐‘  + 1)))
13 breq2 5113 . . . . . . . 8 (๐‘™ = ๐‘› โ†’ ((๐‘  + 1) < ๐‘™ โ†” (๐‘  + 1) < ๐‘›))
14 oveq1 7368 . . . . . . . . . . 11 (๐‘™ = ๐‘› โ†’ (๐‘™ โˆ’ 1) = (๐‘› โˆ’ 1))
1514fveq2d 6850 . . . . . . . . . 10 (๐‘™ = ๐‘› โ†’ (๐‘โ€˜(๐‘™ โˆ’ 1)) = (๐‘โ€˜(๐‘› โˆ’ 1)))
1615fveq2d 6850 . . . . . . . . 9 (๐‘™ = ๐‘› โ†’ ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1))) = ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1))))
17 fveq2 6846 . . . . . . . . . . 11 (๐‘™ = ๐‘› โ†’ (๐‘โ€˜๐‘™) = (๐‘โ€˜๐‘›))
1817fveq2d 6850 . . . . . . . . . 10 (๐‘™ = ๐‘› โ†’ ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™)) = ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›)))
1918oveq2d 7377 . . . . . . . . 9 (๐‘™ = ๐‘› โ†’ (((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))) = (((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›))))
2016, 19oveq12d 7379 . . . . . . . 8 (๐‘™ = ๐‘› โ†’ (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™)))) = (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›)))))
2113, 20ifbieq2d 4516 . . . . . . 7 (๐‘™ = ๐‘› โ†’ if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))) = if((๐‘  + 1) < ๐‘›, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›))))))
2212, 21ifbieq2d 4516 . . . . . 6 (๐‘™ = ๐‘› โ†’ if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™)))))) = if(๐‘› = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘›, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›)))))))
2311, 22ifbieq2d 4516 . . . . 5 (๐‘™ = ๐‘› โ†’ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))) = if(๐‘› = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘› = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘›, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›))))))))
2423cbvmptv 5222 . . . 4 (๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™)))))))) = (๐‘› โˆˆ โ„•0 โ†ฆ if(๐‘› = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘› = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘›, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘› โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘›))))))))
25 eqid 2733 . . . 4 (Baseโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))) = (Baseโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))
26 eqid 2733 . . . 4 (1rโ€˜๐ด) = (1rโ€˜๐ด)
27 cayleyhamilton.m . . . 4 โˆ— = ( ยท๐‘  โ€˜๐ด)
28 eqid 2733 . . . 4 (๐‘ cPolyMatToMat ๐‘…) = (๐‘ cPolyMatToMat ๐‘…)
29 cayleyhamilton.e . . . 4 โ†‘ = (.gโ€˜(mulGrpโ€˜๐ด))
30 eqid 2733 . . . 4 (.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = (.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))
311, 2, 3, 4, 5, 6, 7, 8, 9, 10, 24, 25, 26, 27, 28, 29, 30cayhamlem4 22260 . . 3 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ โˆƒ๐‘  โˆˆ โ„• โˆƒ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ ))(๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘ cPolyMatToMat ๐‘…)โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))))
32 eqid 2733 . . . . . . . . 9 (๐‘ ConstPolyMat ๐‘…) = (๐‘ ConstPolyMat ๐‘…)
3328, 32cpm2mfval 22121 . . . . . . . 8 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing) โ†’ (๐‘ cPolyMatToMat ๐‘…) = (๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0))))
3433eqcomd 2739 . . . . . . 7 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing) โ†’ (๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0))) = (๐‘ cPolyMatToMat ๐‘…))
35343adant3 1133 . . . . . 6 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0))) = (๐‘ cPolyMatToMat ๐‘…))
3635fveq1d 6848 . . . . 5 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) = ((๐‘ cPolyMatToMat ๐‘…)โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))))
3736eqeq2d 2744 . . . 4 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ ((๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) โ†” (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘ cPolyMatToMat ๐‘…)โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))))
38372rexbidv 3210 . . 3 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (โˆƒ๐‘  โˆˆ โ„• โˆƒ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ ))(๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) โ†” โˆƒ๐‘  โˆˆ โ„• โˆƒ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ ))(๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘ cPolyMatToMat ๐‘…)โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))))
3931, 38mpbird 257 . 2 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ โˆƒ๐‘  โˆˆ โ„• โˆƒ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ ))(๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))))
40 cayleyhamilton.k . . . . . . . . . . . . 13 ๐พ = (coe1โ€˜(๐ถโ€˜๐‘€))
4140eqcomi 2742 . . . . . . . . . . . 12 (coe1โ€˜(๐ถโ€˜๐‘€)) = ๐พ
4241a1i 11 . . . . . . . . . . 11 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง ๐‘› โˆˆ โ„•0) โ†’ (coe1โ€˜(๐ถโ€˜๐‘€)) = ๐พ)
4342fveq1d 6848 . . . . . . . . . 10 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง ๐‘› โˆˆ โ„•0) โ†’ ((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) = (๐พโ€˜๐‘›))
4443oveq1d 7376 . . . . . . . . 9 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง ๐‘› โˆˆ โ„•0) โ†’ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)) = ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))
4544mpteq2dva 5209 . . . . . . . 8 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€))) = (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€))))
4645oveq2d 7377 . . . . . . 7 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))))
4746eqeq1d 2735 . . . . . 6 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) โ†” (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))))
4847biimpa 478 . . . . 5 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))))
49 oveq1 7368 . . . . . . . . . . . 12 (๐‘› = ๐‘— โ†’ (๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)) = (๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)))
50 fveq2 6846 . . . . . . . . . . . 12 (๐‘› = ๐‘— โ†’ ((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›) = ((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—))
5149, 50oveq12d 7379 . . . . . . . . . . 11 (๐‘› = ๐‘— โ†’ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)) = ((๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—)))
5251cbvmptv 5222 . . . . . . . . . 10 (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))) = (๐‘— โˆˆ โ„•0 โ†ฆ ((๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—)))
5352oveq2i 7372 . . . . . . . . 9 ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))) = ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘— โˆˆ โ„•0 โ†ฆ ((๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—))))
5453a1i 11 . . . . . . . 8 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))) = ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘— โˆˆ โ„•0 โ†ฆ ((๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—)))))
551, 2, 3, 4, 5, 6, 7, 8, 24, 30cayhamlem1 22238 . . . . . . . 8 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘— โˆˆ โ„•0 โ†ฆ ((๐‘—(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘—)))) = (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))
5654, 55eqtrd 2773 . . . . . . 7 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))) = (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))
57 fveq2 6846 . . . . . . . 8 (((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))) = (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))))
58 crngring 19984 . . . . . . . . . . . . 13 (๐‘… โˆˆ CRing โ†’ ๐‘… โˆˆ Ring)
5958anim2i 618 . . . . . . . . . . . 12 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing) โ†’ (๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring))
60593adant3 1133 . . . . . . . . . . 11 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring))
6128, 32cpm2mfval 22121 . . . . . . . . . . . . . 14 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ (๐‘ cPolyMatToMat ๐‘…) = (๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0))))
6261eqcomd 2739 . . . . . . . . . . . . 13 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ (๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0))) = (๐‘ cPolyMatToMat ๐‘…))
6362fveq1d 6848 . . . . . . . . . . . 12 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = ((๐‘ cPolyMatToMat ๐‘…)โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))))
64 eqid 2733 . . . . . . . . . . . . 13 (0gโ€˜๐ด) = (0gโ€˜๐ด)
651, 28, 3, 4, 64, 7m2cpminv0 22133 . . . . . . . . . . . 12 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ((๐‘ cPolyMatToMat ๐‘…)โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = (0gโ€˜๐ด))
6663, 65eqtrd 2773 . . . . . . . . . . 11 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = (0gโ€˜๐ด))
6760, 66syl 17 . . . . . . . . . 10 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = (0gโ€˜๐ด))
68 cayleyhamilton.0 . . . . . . . . . 10 0 = (0gโ€˜๐ด)
6967, 68eqtr4di 2791 . . . . . . . . 9 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = 0 )
7069adantr 482 . . . . . . . 8 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜(0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) = 0 )
7157, 70sylan9eqr 2795 . . . . . . 7 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง ((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))) = (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) = 0 )
7256, 71mpdan 686 . . . . . 6 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) = 0 )
7372adantr 482 . . . . 5 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))) โ†’ ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) = 0 )
7448, 73eqtrd 2773 . . . 4 ((((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โˆง (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›)))))) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = 0 )
7574ex 414 . . 3 (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โˆง (๐‘  โˆˆ โ„• โˆง ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ )))) โ†’ ((๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = 0 ))
7675rexlimdvva 3202 . 2 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (โˆƒ๐‘  โˆˆ โ„• โˆƒ๐‘ โˆˆ (๐ต โ†‘m (0...๐‘ ))(๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ (((coe1โ€˜(๐ถโ€˜๐‘€))โ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = ((๐‘š โˆˆ (๐‘ ConstPolyMat ๐‘…) โ†ฆ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ((coe1โ€˜(๐‘ฅ๐‘š๐‘ฆ))โ€˜0)))โ€˜((๐‘ Mat (Poly1โ€˜๐‘…)) ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐‘›(.gโ€˜(mulGrpโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))))((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€))(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘™ โˆˆ โ„•0 โ†ฆ if(๐‘™ = 0, ((0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜0)))), if(๐‘™ = (๐‘  + 1), ((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘ )), if((๐‘  + 1) < ๐‘™, (0gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…))), (((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜(๐‘™ โˆ’ 1)))(-gโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))(((๐‘ matToPolyMat ๐‘…)โ€˜๐‘€)(.rโ€˜(๐‘ Mat (Poly1โ€˜๐‘…)))((๐‘ matToPolyMat ๐‘…)โ€˜(๐‘โ€˜๐‘™))))))))โ€˜๐‘›))))) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = 0 ))
7739, 76mpd 15 1 ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (๐ด ฮฃg (๐‘› โˆˆ โ„•0 โ†ฆ ((๐พโ€˜๐‘›) โˆ— (๐‘› โ†‘ ๐‘€)))) = 0 )
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 397   โˆง w3a 1088   = wceq 1542   โˆˆ wcel 2107  โˆƒwrex 3070  ifcif 4490   class class class wbr 5109   โ†ฆ cmpt 5192  โ€˜cfv 6500  (class class class)co 7361   โˆˆ cmpo 7363   โ†‘m cmap 8771  Fincfn 8889  0cc0 11059  1c1 11060   + caddc 11062   < clt 11197   โˆ’ cmin 11393  โ„•cn 12161  โ„•0cn0 12421  ...cfz 13433  Basecbs 17091  .rcmulr 17142   ยท๐‘  cvsca 17145  0gc0g 17329   ฮฃg cgsu 17330  -gcsg 18758  .gcmg 18880  mulGrpcmgp 19904  1rcur 19921  Ringcrg 19972  CRingccrg 19973  Poly1cpl1 21571  coe1cco1 21572   Mat cmat 21777   ConstPolyMat ccpmat 22075   matToPolyMat cmat2pmat 22076   cPolyMatToMat ccpmat2mat 22077   CharPlyMat cchpmat 22198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-addf 11138  ax-mulf 11139
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-ofr 7622  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-tpos 8161  df-cur 8202  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-sup 9386  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-xnn0 12494  df-z 12508  df-dec 12627  df-uz 12772  df-rp 12924  df-fz 13434  df-fzo 13577  df-seq 13916  df-exp 13977  df-hash 14240  df-word 14412  df-lsw 14460  df-concat 14468  df-s1 14493  df-substr 14538  df-pfx 14568  df-splice 14647  df-reverse 14656  df-s2 14746  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-starv 17156  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-hom 17165  df-cco 17166  df-0g 17331  df-gsum 17332  df-prds 17337  df-pws 17339  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-submnd 18610  df-efmnd 18687  df-grp 18759  df-minusg 18760  df-sbg 18761  df-mulg 18881  df-subg 18933  df-ghm 19014  df-gim 19057  df-cntz 19105  df-oppg 19132  df-symg 19157  df-pmtr 19232  df-psgn 19281  df-evpm 19282  df-cmn 19572  df-abl 19573  df-mgp 19905  df-ur 19922  df-srg 19926  df-ring 19974  df-cring 19975  df-oppr 20057  df-dvdsr 20078  df-unit 20079  df-invr 20109  df-dvr 20120  df-rnghom 20156  df-drng 20221  df-subrg 20262  df-lmod 20367  df-lss 20437  df-sra 20678  df-rgmod 20679  df-cnfld 20820  df-zring 20893  df-zrh 20927  df-dsmm 21161  df-frlm 21176  df-assa 21282  df-ascl 21284  df-psr 21334  df-mvr 21335  df-mpl 21336  df-opsr 21338  df-psr1 21574  df-vr1 21575  df-ply1 21576  df-coe1 21577  df-mamu 21756  df-mat 21778  df-mdet 21957  df-madu 22006  df-cpmat 22078  df-mat2pmat 22079  df-cpmat2mat 22080  df-decpmat 22135  df-pm2mp 22165  df-chpmat 22199
This theorem is referenced by: (None)
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