Theorem cayleyhamilton0 22804

Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22805 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 22806)! (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 = (0g‘𝐴)
cayleyhamilton0.1	⊢ 1 = (1r‘𝐴)
cayleyhamilton0.m	⊢ ∗ = ( ·𝑠 ‘𝐴)
cayleyhamilton0.e1	⊢ ↑ = (.g‘(mulGrp‘𝐴))
cayleyhamilton0.c	⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
cayleyhamilton0.k	⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
cayleyhamilton0.p	⊢ 𝑃 = (Poly1‘𝑅)
cayleyhamilton0.y	⊢ 𝑌 = (𝑁 Mat 𝑃)
cayleyhamilton0.r	⊢ × = (.r‘𝑌)
cayleyhamilton0.s	⊢ − = (-g‘𝑌)
cayleyhamilton0.z	⊢ 𝑍 = (0g‘𝑌)
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 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
cayleyhamilton0.u	⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)

Assertion

Ref	Expression
cayleyhamilton0	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (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.

Step	Hyp	Ref	Expression
1		cayleyhamilton0.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		cayleyhamilton0.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		cayleyhamilton0.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
4		cayleyhamilton0.y	. . 3 ⊢ 𝑌 = (𝑁 Mat 𝑃)
5		cayleyhamilton0.r	. . 3 ⊢ × = (.r‘𝑌)
6		cayleyhamilton0.s	. . 3 ⊢ − = (-g‘𝑌)
7		cayleyhamilton0.z	. . 3 ⊢ 𝑍 = (0g‘𝑌)
8		cayleyhamilton0.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton0.c	. . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2725	. . 3 ⊢ (𝐶‘𝑀) = (𝐶‘𝑀)
11		cayleyhamilton0.g	. . 3 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛))))))))
12		cayleyhamilton0.w	. . 3 ⊢ 𝑊 = (Base‘𝑌)
13		cayleyhamilton0.1	. . 3 ⊢ 1 = (1r‘𝐴)
14		cayleyhamilton0.m	. . 3 ⊢ ∗ = ( ·𝑠 ‘𝐴)
15		cayleyhamilton0.u	. . 3 ⊢ 𝑈 = (𝑁 cPolyMatToMat 𝑅)
16		cayleyhamilton0.e1	. . 3 ⊢ ↑ = (.g‘(mulGrp‘𝐴))
17		cayleyhamilton0.e2	. . 3 ⊢ 𝐸 = (.g‘(mulGrp‘𝑌))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	cayhamlem4 22803	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
19		cayleyhamilton0.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (coe1‘(𝐶‘𝑀))
20	19	eqcomi 2734	. . . . . . . . . . . 12 ⊢ (coe1‘(𝐶‘𝑀)) = 𝐾
21	20	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝐶‘𝑀)) = 𝐾)
22	21	fveq1d 6892	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝐶‘𝑀))‘𝑛) = (𝐾‘𝑛))
23	22	oveq1d 7428	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)) = ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))
24	23	mpteq2dva 5244	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀))) = (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀))))
25	24	oveq2d 7429	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))))
26	25	eqeq1d 2727	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) ↔ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))))
27	26	biimpa 475	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
28		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝑛𝐸(𝑇‘𝑀)) = (𝑙𝐸(𝑇‘𝑀)))
29		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝐺‘𝑛) = (𝐺‘𝑙))
30	28, 29	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)) = ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
31	30	cbvmptv 5257	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))) = (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
32	31	oveq2i 7424	. . . . . . . 8 ⊢ (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙))))
33	1, 2, 3, 4, 5, 6, 7, 8, 11, 17	cayhamlem1 22781	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑙 ∈ ℕ0 ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))) = 𝑍)
34	32, 33	eqtrid 2777	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍)
35		fveq2 6890	. . . . . . . 8 ⊢ ((𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍 → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = (𝑈‘𝑍))
36		crngring 20184	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
37	36	anim2i 615	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38	37	3adant3 1129	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39		eqid 2725	. . . . . . . . . . . 12 ⊢ (0g‘𝐴) = (0g‘𝐴)
40	1, 15, 3, 4, 39, 7	m2cpminv0 22676	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘𝑍) = (0g‘𝐴))
41	38, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘𝑍) = (0g‘𝐴))
42		cayleyhamilton0.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐴)
43	41, 42	eqtr4di 2783	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘𝑍) = 0 )
44	43	adantr 479	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑈‘𝑍) = 0 )
45	35, 44	sylan9eqr 2787	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = 0 )
46	34, 45	mpdan 685	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = 0 )
47	46	adantr 479	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = 0 )
48	27, 47	eqtrd 2765	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) ∧ (𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
49	48	ex 411	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → ((𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
50	49	rexlimdvva 3202	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑m (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
51	18, 50	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∃wrex 3060  ifcif 4525   class class class wbr 5144   ↦ cmpt 5227  ‘cfv 6543  (class class class)co 7413   ↑_m cmap 8838  Fincfn 8957  0cc0 11133  1c1 11134   + caddc 11136   < clt 11273   − cmin 11469  ℕcn 12237  ℕ₀cn0 12497  ...cfz 13511  Basecbs 17174  ._rcmulr 17228   ·_𝑠 cvsca 17231  0_gc0g 17415   Σ_g cgsu 17416  -_gcsg 18891  ._gcmg 19022  mulGrpcmgp 20073  1_rcur 20120  Ringcrg 20172  CRingccrg 20173  Poly₁cpl1 22099  coe₁cco1 22100   Mat cmat 22320   matToPolyMat cmat2pmat 22619   cPolyMatToMat ccpmat2mat 22620   CharPlyMat cchpmat 22741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-addf 11212  ax-mulf 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7679  df-ofr 7680  df-om 7866  df-1st 7987  df-2nd 7988  df-supp 8159  df-tpos 8225  df-cur 8266  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9381  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-xnn0 12570  df-z 12584  df-dec 12703  df-uz 12848  df-rp 13002  df-fz 13512  df-fzo 13655  df-seq 13994  df-exp 14054  df-hash 14317  df-word 14492  df-lsw 14540  df-concat 14548  df-s1 14573  df-substr 14618  df-pfx 14648  df-splice 14727  df-reverse 14736  df-s2 14826  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17417  df-gsum 17418  df-prds 17423  df-pws 17425  df-mre 17560  df-mrc 17561  df-acs 17563  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-submnd 18735  df-efmnd 18820  df-grp 18892  df-minusg 18893  df-sbg 18894  df-mulg 19023  df-subg 19077  df-ghm 19167  df-gim 19212  df-cntz 19267  df-oppg 19296  df-symg 19321  df-pmtr 19396  df-psgn 19445  df-evpm 19446  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-srg 20126  df-ring 20174  df-cring 20175  df-oppr 20272  df-dvdsr 20295  df-unit 20296  df-invr 20326  df-dvr 20339  df-rhm 20410  df-subrng 20482  df-subrg 20507  df-drng 20625  df-lmod 20744  df-lss 20815  df-sra 21057  df-rgmod 21058  df-cnfld 21279  df-zring 21372  df-zrh 21428  df-dsmm 21665  df-frlm 21680  df-assa 21786  df-ascl 21788  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-psr1 22102  df-vr1 22103  df-ply1 22104  df-coe1 22105  df-mamu 22304  df-mat 22321  df-mdet 22500  df-madu 22549  df-cpmat 22621  df-mat2pmat 22622  df-cpmat2mat 22623  df-decpmat 22678  df-pm2mp 22708  df-chpmat 22742

This theorem is referenced by:  cayleyhamilton  22805