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Theorem cayleyhamilton0 21216
Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 21217 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 21218)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
cayleyhamilton0.a 𝐴 = (𝑁 Mat 𝑅)
cayleyhamilton0.b 𝐵 = (Base‘𝐴)
cayleyhamilton0.0 0 = (0g𝐴)
cayleyhamilton0.1 1 = (1r𝐴)
cayleyhamilton0.m = ( ·𝑠𝐴)
cayleyhamilton0.e1 = (.g‘(mulGrp‘𝐴))
cayleyhamilton0.c 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton0.k 𝐾 = (coe1‘(𝐶𝑀))
cayleyhamilton0.p 𝑃 = (Poly1𝑅)
cayleyhamilton0.y 𝑌 = (𝑁 Mat 𝑃)
cayleyhamilton0.r × = (.r𝑌)
cayleyhamilton0.s = (-g𝑌)
cayleyhamilton0.z 𝑍 = (0g𝑌)
cayleyhamilton0.w 𝑊 = (Base‘𝑌)
cayleyhamilton0.e2 𝐸 = (.g‘(mulGrp‘𝑌))
cayleyhamilton0.t 𝑇 = (𝑁 matToPolyMat 𝑅)
cayleyhamilton0.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
cayleyhamilton0.u 𝑈 = (𝑁 cPolyMatToMat 𝑅)
Assertion
Ref Expression
cayleyhamilton0 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
Distinct variable groups:   𝐴,𝑏,𝑛,𝑠   𝐵,𝑏,𝑛,𝑠   𝐶,𝑛   𝑛,𝐸   𝑛,𝐺   𝐾,𝑏,𝑠   𝑀,𝑏,𝑛,𝑠   𝑁,𝑏,𝑛,𝑠   𝑃,𝑏,𝑛,𝑠   𝑅,𝑏,𝑛,𝑠   𝑇,𝑏,𝑛,𝑠   𝑈,𝑛   𝑛,𝑊   𝑌,𝑏,𝑛,𝑠   𝑛,𝑍   ,𝑏,𝑛,𝑠   ,𝑏,𝑛,𝑠   0 ,𝑏,𝑠   1 ,𝑛   × ,𝑛   ,𝑏,𝑛,𝑠
Allowed substitution hints:   𝐶(𝑠,𝑏)   × (𝑠,𝑏)   𝑈(𝑠,𝑏)   1 (𝑠,𝑏)   𝐸(𝑠,𝑏)   𝐺(𝑠,𝑏)   𝐾(𝑛)   𝑊(𝑠,𝑏)   0 (𝑛)   𝑍(𝑠,𝑏)

Proof of Theorem cayleyhamilton0
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 cayleyhamilton0.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 cayleyhamilton0.b . . 3 𝐵 = (Base‘𝐴)
3 cayleyhamilton0.p . . 3 𝑃 = (Poly1𝑅)
4 cayleyhamilton0.y . . 3 𝑌 = (𝑁 Mat 𝑃)
5 cayleyhamilton0.r . . 3 × = (.r𝑌)
6 cayleyhamilton0.s . . 3 = (-g𝑌)
7 cayleyhamilton0.z . . 3 𝑍 = (0g𝑌)
8 cayleyhamilton0.t . . 3 𝑇 = (𝑁 matToPolyMat 𝑅)
9 cayleyhamilton0.c . . 3 𝐶 = (𝑁 CharPlyMat 𝑅)
10 eqid 2771 . . 3 (𝐶𝑀) = (𝐶𝑀)
11 cayleyhamilton0.g . . 3 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
12 cayleyhamilton0.w . . 3 𝑊 = (Base‘𝑌)
13 cayleyhamilton0.1 . . 3 1 = (1r𝐴)
14 cayleyhamilton0.m . . 3 = ( ·𝑠𝐴)
15 cayleyhamilton0.u . . 3 𝑈 = (𝑁 cPolyMatToMat 𝑅)
16 cayleyhamilton0.e1 . . 3 = (.g‘(mulGrp‘𝐴))
17 cayleyhamilton0.e2 . . 3 𝐸 = (.g‘(mulGrp‘𝑌))
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cayhamlem4 21215 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
19 cayleyhamilton0.k . . . . . . . . . . . . 13 𝐾 = (coe1‘(𝐶𝑀))
2019eqcomi 2780 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
2120a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
2221fveq1d 6498 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
2322oveq1d 6989 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
2423mpteq2dva 5018 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀))) = (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀))))
2524oveq2d 6990 . . . . . . 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)))))
2625eqeq1d 2773 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))))
2726biimpa 469 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
28 oveq1 6981 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝑛𝐸(𝑇𝑀)) = (𝑙𝐸(𝑇𝑀)))
29 fveq2 6496 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝐺𝑛) = (𝐺𝑙))
3028, 29oveq12d 6992 . . . . . . . . . 10 (𝑛 = 𝑙 → ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)) = ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3130cbvmptv 5024 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))) = (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3231oveq2i 6985 . . . . . . . 8 (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙))))
331, 2, 3, 4, 5, 6, 7, 8, 11, 17cayhamlem1 21193 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))) = 𝑍)
3432, 33syl5eq 2819 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍)
35 fveq2 6496 . . . . . . . 8 ((𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍 → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = (𝑈𝑍))
36 crngring 19043 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3736anim2i 608 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38373adant3 1113 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39 eqid 2771 . . . . . . . . . . . 12 (0g𝐴) = (0g𝐴)
401, 15, 3, 4, 39, 7m2cpminv0 21088 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈𝑍) = (0g𝐴))
4138, 40syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈𝑍) = (0g𝐴))
42 cayleyhamilton0.0 . . . . . . . . . 10 0 = (0g𝐴)
4341, 42syl6eqr 2825 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈𝑍) = 0 )
4443adantr 473 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝑍) = 0 )
4535, 44sylan9eqr 2829 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = 0 )
4634, 45mpdan 675 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = 0 )
4746adantr 473 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = 0 )
4827, 47eqtrd 2807 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
4948ex 405 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 ))
5049rexlimdvva 3232 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 ))
5118, 50mpd 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 387  w3a 1069   = wceq 1508  wcel 2051  wrex 3082  ifcif 4344   class class class wbr 4925  cmpt 5004  cfv 6185  (class class class)co 6974  𝑚 cmap 8204  Fincfn 8304  0cc0 10333  1c1 10334   + caddc 10336   < clt 10472  cmin 10668  cn 11437  0cn0 11705  ...cfz 12706  Basecbs 16337  .rcmulr 16420   ·𝑠 cvsca 16423  0gc0g 16567   Σg cgsu 16568  -gcsg 17905  .gcmg 18023  mulGrpcmgp 18974  1rcur 18986  Ringcrg 19032  CRingccrg 19033  Poly1cpl1 20063  coe1cco1 20064   Mat cmat 20735   matToPolyMat cmat2pmat 21031   cPolyMatToMat ccpmat2mat 21032   CharPlyMat cchpmat 21153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-addf 10412  ax-mulf 10413
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-xor 1490  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-ot 4444  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-of 7225  df-ofr 7226  df-om 7395  df-1st 7499  df-2nd 7500  df-supp 7632  df-tpos 7693  df-cur 7734  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-map 8206  df-pm 8207  df-ixp 8258  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-fsupp 8627  df-sup 8699  df-oi 8767  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-2 11501  df-3 11502  df-4 11503  df-5 11504  df-6 11505  df-7 11506  df-8 11507  df-9 11508  df-n0 11706  df-xnn0 11778  df-z 11792  df-dec 11910  df-uz 12057  df-rp 12203  df-fz 12707  df-fzo 12848  df-seq 13183  df-exp 13243  df-hash 13504  df-word 13671  df-lsw 13724  df-concat 13732  df-s1 13757  df-substr 13802  df-pfx 13851  df-splice 13958  df-reverse 13976  df-s2 14070  df-struct 16339  df-ndx 16340  df-slot 16341  df-base 16343  df-sets 16344  df-ress 16345  df-plusg 16432  df-mulr 16433  df-starv 16434  df-sca 16435  df-vsca 16436  df-ip 16437  df-tset 16438  df-ple 16439  df-ds 16441  df-unif 16442  df-hom 16443  df-cco 16444  df-0g 16569  df-gsum 16570  df-prds 16575  df-pws 16577  df-mre 16727  df-mrc 16728  df-acs 16730  df-mgm 17722  df-sgrp 17764  df-mnd 17775  df-mhm 17815  df-submnd 17816  df-grp 17906  df-minusg 17907  df-sbg 17908  df-mulg 18024  df-subg 18072  df-ghm 18139  df-gim 18182  df-cntz 18230  df-oppg 18257  df-symg 18279  df-pmtr 18343  df-psgn 18392  df-evpm 18393  df-cmn 18680  df-abl 18681  df-mgp 18975  df-ur 18987  df-srg 18991  df-ring 19034  df-cring 19035  df-oppr 19108  df-dvdsr 19126  df-unit 19127  df-invr 19157  df-dvr 19168  df-rnghom 19202  df-drng 19239  df-subrg 19268  df-lmod 19370  df-lss 19438  df-sra 19678  df-rgmod 19679  df-assa 19818  df-ascl 19820  df-psr 19862  df-mvr 19863  df-mpl 19864  df-opsr 19866  df-psr1 20066  df-vr1 20067  df-ply1 20068  df-coe1 20069  df-cnfld 20263  df-zring 20335  df-zrh 20368  df-dsmm 20593  df-frlm 20608  df-mamu 20712  df-mat 20736  df-mdet 20913  df-madu 20962  df-cpmat 21033  df-mat2pmat 21034  df-cpmat2mat 21035  df-decpmat 21090  df-pm2mp 21120  df-chpmat 21154
This theorem is referenced by:  cayleyhamilton  21217
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