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Theorem cayleyhamilton0 20626
 Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 20627 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 20628)! (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 2621 . . 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 20625 . 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 2630 . . . . . . . . . . . 12 (coe1‘(𝐶𝑀)) = 𝐾
2120a1i 11 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶𝑀)) = 𝐾)
2221fveq1d 6155 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶𝑀))‘𝑛) = (𝐾𝑛))
2322oveq1d 6625 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶𝑀))‘𝑛) (𝑛 𝑀)) = ((𝐾𝑛) (𝑛 𝑀)))
2423mpteq2dva 4709 . . . . . . . 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 6626 . . . . . . 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 2623 . . . . . 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 501 . . . . 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 6617 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝑛𝐸(𝑇𝑀)) = (𝑙𝐸(𝑇𝑀)))
29 fveq2 6153 . . . . . . . . . . 11 (𝑛 = 𝑙 → (𝐺𝑛) = (𝐺𝑙))
3028, 29oveq12d 6628 . . . . . . . . . 10 (𝑛 = 𝑙 → ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)) = ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3130cbvmptv 4715 . . . . . . . . 9 (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))) = (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))
3231oveq2i 6621 . . . . . . . 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 20603 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇𝑀)) × (𝐺𝑙)))) = 𝑍)
3432, 33syl5eq 2667 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍)
35 fveq2 6153 . . . . . . . 8 ((𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛)))) = 𝑍 → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = (𝑈𝑍))
36 crngring 18490 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3736anim2i 592 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38373adant3 1079 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39 eqid 2621 . . . . . . . . . . . 12 (0g𝐴) = (0g𝐴)
401, 15, 3, 4, 39, 7m2cpminv0 20498 . . . . . . . . . . 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 2673 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈𝑍) = 0 )
4443adantr 481 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈𝑍) = 0 )
4535, 44sylan9eqr 2677 . . . . . . 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 701 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))) = 0 )
4746adantr 481 . . . . 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 2655 . . . 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 450 . . 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 3032 . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  ifcif 4063   class class class wbr 4618   ↦ cmpt 4678  ‘cfv 5852  (class class class)co 6610   ↑𝑚 cmap 7809  Fincfn 7907  0cc0 9888  1c1 9889   + caddc 9891   < clt 10026   − cmin 10218  ℕcn 10972  ℕ0cn0 11244  ...cfz 12276  Basecbs 15792  .rcmulr 15874   ·𝑠 cvsca 15877  0gc0g 16032   Σg cgsu 16033  -gcsg 17356  .gcmg 17472  mulGrpcmgp 18421  1rcur 18433  Ringcrg 18479  CRingccrg 18480  Poly1cpl1 19479  coe1cco1 19480   Mat cmat 20145   matToPolyMat cmat2pmat 20441   cPolyMatToMat ccpmat2mat 20442   CharPlyMat cchpmat 20563 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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-addf 9967  ax-mulf 9968 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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-ofr 6858  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-tpos 7304  df-cur 7345  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-xnn0 11316  df-z 11330  df-dec 11446  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-word 13246  df-lsw 13247  df-concat 13248  df-s1 13249  df-substr 13250  df-splice 13251  df-reverse 13252  df-s2 13538  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-0g 16034  df-gsum 16035  df-prds 16040  df-pws 16042  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-gim 17633  df-cntz 17682  df-oppg 17708  df-symg 17730  df-pmtr 17794  df-psgn 17843  df-evpm 17844  df-cmn 18127  df-abl 18128  df-mgp 18422  df-ur 18434  df-srg 18438  df-ring 18481  df-cring 18482  df-oppr 18555  df-dvdsr 18573  df-unit 18574  df-invr 18604  df-dvr 18615  df-rnghom 18647  df-drng 18681  df-subrg 18710  df-lmod 18797  df-lss 18865  df-sra 19104  df-rgmod 19105  df-assa 19244  df-ascl 19246  df-psr 19288  df-mvr 19289  df-mpl 19290  df-opsr 19292  df-psr1 19482  df-vr1 19483  df-ply1 19484  df-coe1 19485  df-cnfld 19679  df-zring 19751  df-zrh 19784  df-dsmm 20008  df-frlm 20023  df-mamu 20122  df-mat 20146  df-mdet 20323  df-madu 20372  df-cpmat 20443  df-mat2pmat 20444  df-cpmat2mat 20445  df-decpmat 20500  df-pm2mp 20530  df-chpmat 20564 This theorem is referenced by:  cayleyhamilton  20627
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