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Theorem cayleyhamilton0 21498
 Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 21499 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 21500)! (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 2801 . . 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 21497 . 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 2810 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
2120a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
2221fveq1d 6651 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
2322oveq1d 7154 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
2423mpteq2dva 5128 . . . . . . . 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 7155 . . . . . . 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 2803 . . . . . 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 480 . . . . 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 7146 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝑛𝐸(𝑇𝑀)) = (𝑙𝐸(𝑇𝑀)))
29 fveq2 6649 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝐺𝑛) = (𝐺𝑙))
3028, 29oveq12d 7157 . . . . . . . . . 10 (𝑛 = 𝑙 → ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)) = ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3130cbvmptv 5136 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))) = (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3231oveq2i 7150 . . . . . . . 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 21475 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))) = 𝑍)
3432, 33syl5eq 2848 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍)
35 fveq2 6649 . . . . . . . 8 ((𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍 → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = (𝑈𝑍))
36 crngring 19306 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3736anim2i 619 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38373adant3 1129 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39 eqid 2801 . . . . . . . . . . . 12 (0g𝐴) = (0g𝐴)
401, 15, 3, 4, 39, 7m2cpminv0 21370 . . . . . . . . . . 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, 42eqtr4di 2854 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈𝑍) = 0 )
4443adantr 484 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑈𝑍) = 0 )
4535, 44sylan9eqr 2858 . . . . . . 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 686 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = 0 )
4746adantr 484 . . . . 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 2836 . . . 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 416 . . 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 3256 . 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∃wrex 3110  ifcif 4428   class class class wbr 5033   ↦ cmpt 5113  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  Fincfn 8496  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668   − cmin 10863  ℕcn 11629  ℕ0cn0 11889  ...cfz 12889  Basecbs 16479  .rcmulr 16562   ·𝑠 cvsca 16565  0gc0g 16709   Σg cgsu 16710  -gcsg 18101  .gcmg 18220  mulGrpcmgp 19236  1rcur 19248  Ringcrg 19294  CRingccrg 19295  Poly1cpl1 20810  coe1cco1 20811   Mat cmat 21016   matToPolyMat cmat2pmat 21313   cPolyMatToMat ccpmat2mat 21314   CharPlyMat cchpmat 21435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-addf 10609  ax-mulf 10610 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-xor 1503  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofr 7394  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-cur 7920  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-splice 14107  df-reverse 14116  df-s2 14205  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-starv 16576  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-unif 16584  df-hom 16585  df-cco 16586  df-0g 16711  df-gsum 16712  df-prds 16717  df-pws 16719  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-submnd 17953  df-efmnd 18030  df-grp 18102  df-minusg 18103  df-sbg 18104  df-mulg 18221  df-subg 18272  df-ghm 18352  df-gim 18395  df-cntz 18443  df-oppg 18470  df-symg 18492  df-pmtr 18566  df-psgn 18615  df-evpm 18616  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-srg 19253  df-ring 19296  df-cring 19297  df-oppr 19373  df-dvdsr 19391  df-unit 19392  df-invr 19422  df-dvr 19433  df-rnghom 19467  df-drng 19501  df-subrg 19530  df-lmod 19633  df-lss 19701  df-sra 19941  df-rgmod 19942  df-cnfld 20096  df-zring 20168  df-zrh 20201  df-dsmm 20425  df-frlm 20440  df-assa 20546  df-ascl 20548  df-psr 20598  df-mvr 20599  df-mpl 20600  df-opsr 20602  df-psr1 20813  df-vr1 20814  df-ply1 20815  df-coe1 20816  df-mamu 20995  df-mat 21017  df-mdet 21194  df-madu 21243  df-cpmat 21315  df-mat2pmat 21316  df-cpmat2mat 21317  df-decpmat 21372  df-pm2mp 21402  df-chpmat 21436 This theorem is referenced by:  cayleyhamilton  21499
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