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Theorem cayleyhamilton0 22143

Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22144 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 22145)! (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 = (0_g‘𝐴)
cayleyhamilton0.1	⊢ 1 = (1_r‘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 = (0_g‘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 Σ_g (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 = (0_g‘𝑌)
8		cayleyhamilton0.t	. . 3 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		cayleyhamilton0.c	. . 3 ⊢ 𝐶 = (𝑁 CharPlyMat 𝑅)
10		eqid 2737	. . 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 = (1_r‘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 22142	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
19		cayleyhamilton0.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (coe₁‘(𝐶‘𝑀))
20	19	eqcomi 2746	. . . . . . . . . . . 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 6831	. . . . . . . . . 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 7356	. . . . . . . . 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 5196	. . . . . . . 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 7357	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))))
26	25	eqeq1d 2739	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) ↔ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))))
27	26	biimpa 478	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))
28		oveq1 7348	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝑛𝐸(𝑇‘𝑀)) = (𝑙𝐸(𝑇‘𝑀)))
29		fveq2 6829	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝐺‘𝑛) = (𝐺‘𝑙))
30	28, 29	oveq12d 7359	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)) = ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
31	30	cbvmptv 5209	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))) = (𝑙 ∈ ℕ₀ ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
32	31	oveq2i 7352	. . . . . . . 8 ⊢ (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = (𝑌 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙))))
33	1, 2, 3, 4, 5, 6, 7, 8, 11, 17	cayhamlem1 22120	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑙 ∈ ℕ₀ ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))) = 𝑍)
34	32, 33	eqtrid 2789	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍)
35		fveq2 6829	. . . . . . . 8 ⊢ ((𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍 → (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = (𝑈‘𝑍))
36		crngring 19889	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
37	36	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38	37	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39		eqid 2737	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
40	1, 15, 3, 4, 39, 7	m2cpminv0 22015	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑈‘𝑍) = (0_g‘𝐴))
41	38, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘𝑍) = (0_g‘𝐴))
42		cayleyhamilton0.0	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝐴)
43	41, 42	eqtr4di 2795	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘𝑍) = 0 )
44	43	adantr 482	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈‘𝑍) = 0 )
45	35, 44	sylan9eqr 2799	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))) = 𝑍) → (𝑈‘(𝑌 Σ_g (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 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = 0 )
47	46	adantr 482	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) = 0 )
48	27, 47	eqtrd 2777	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) ∧ (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )
49	48	ex 414	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) → (𝐴 Σ_g (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 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘(𝐶‘𝑀))‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 ))
51	18, 50	mpd 15	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ifcif 4477 class class class wbr 5096 ↦ cmpt 5179 ‘cfv 6483 (class class class)co 7341 ↑_m cmap 8690 Fincfn 8808 0cc0 10976 1c1 10977 + caddc 10979 < clt 11114 − cmin 11310 ℕcn 12078 ℕ₀cn0 12338 ...cfz 13344 Basecbs 17009 ._rcmulr 17060 ·_𝑠 cvsca 17063 0_gc0g 17247 Σ_g cgsu 17248 -_gcsg 18675 ._gcmg 18796 mulGrpcmgp 19814 1_rcur 19831 Ringcrg 19877 CRingccrg 19878 Poly₁cpl1 21453 coe₁cco1 21454 Mat cmat 21659 matToPolyMat cmat2pmat 21958 cPolyMatToMat ccpmat2mat 21959 CharPlyMat cchpmat 22080

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5233 ax-sep 5247 ax-nul 5254 ax-pow 5312 ax-pr 5376 ax-un 7654 ax-cnex 11032 ax-resscn 11033 ax-1cn 11034 ax-icn 11035 ax-addcl 11036 ax-addrcl 11037 ax-mulcl 11038 ax-mulrcl 11039 ax-mulcom 11040 ax-addass 11041 ax-mulass 11042 ax-distr 11043 ax-i2m1 11044 ax-1ne0 11045 ax-1rid 11046 ax-rnegex 11047 ax-rrecex 11048 ax-cnre 11049 ax-pre-lttri 11050 ax-pre-lttrn 11051 ax-pre-ltadd 11052 ax-pre-mulgt0 11053 ax-addf 11055 ax-mulf 11056

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-xor 1510 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3731 df-csb 3847 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3920 df-nul 4274 df-if 4478 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4857 df-int 4899 df-iun 4947 df-iin 4948 df-br 5097 df-opab 5159 df-mpt 5180 df-tr 5214 df-id 5522 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5579 df-se 5580 df-we 5581 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6242 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6435 df-fun 6485 df-fn 6486 df-f 6487 df-f1 6488 df-fo 6489 df-f1o 6490 df-fv 6491 df-isom 6492 df-riota 7297 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7599 df-ofr 7600 df-om 7785 df-1st 7903 df-2nd 7904 df-supp 8052 df-tpos 8116 df-cur 8157 df-frecs 8171 df-wrecs 8202 df-recs 8276 df-rdg 8315 df-1o 8371 df-2o 8372 df-er 8573 df-map 8692 df-pm 8693 df-ixp 8761 df-en 8809 df-dom 8810 df-sdom 8811 df-fin 8812 df-fsupp 9231 df-sup 9303 df-oi 9371 df-card 9800 df-pnf 11116 df-mnf 11117 df-xr 11118 df-ltxr 11119 df-le 11120 df-sub 11312 df-neg 11313 df-div 11738 df-nn 12079 df-2 12141 df-3 12142 df-4 12143 df-5 12144 df-6 12145 df-7 12146 df-8 12147 df-9 12148 df-n0 12339 df-xnn0 12411 df-z 12425 df-dec 12543 df-uz 12688 df-rp 12836 df-fz 13345 df-fzo 13488 df-seq 13827 df-exp 13888 df-hash 14150 df-word 14322 df-lsw 14370 df-concat 14378 df-s1 14403 df-substr 14452 df-pfx 14482 df-splice 14561 df-reverse 14570 df-s2 14660 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-starv 17074 df-sca 17075 df-vsca 17076 df-ip 17077 df-tset 17078 df-ple 17079 df-ds 17081 df-unif 17082 df-hom 17083 df-cco 17084 df-0g 17249 df-gsum 17250 df-prds 17255 df-pws 17257 df-mre 17392 df-mrc 17393 df-acs 17395 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-mhm 18527 df-submnd 18528 df-efmnd 18604 df-grp 18676 df-minusg 18677 df-sbg 18678 df-mulg 18797 df-subg 18848 df-ghm 18928 df-gim 18971 df-cntz 19019 df-oppg 19046 df-symg 19071 df-pmtr 19146 df-psgn 19195 df-evpm 19196 df-cmn 19483 df-abl 19484 df-mgp 19815 df-ur 19832 df-srg 19836 df-ring 19879 df-cring 19880 df-oppr 19956 df-dvdsr 19977 df-unit 19978 df-invr 20008 df-dvr 20019 df-rnghom 20053 df-drng 20094 df-subrg 20126 df-lmod 20230 df-lss 20299 df-sra 20539 df-rgmod 20540 df-cnfld 20703 df-zring 20776 df-zrh 20810 df-dsmm 21044 df-frlm 21059 df-assa 21165 df-ascl 21167 df-psr 21217 df-mvr 21218 df-mpl 21219 df-opsr 21221 df-psr1 21456 df-vr1 21457 df-ply1 21458 df-coe1 21459 df-mamu 21638 df-mat 21660 df-mdet 21839 df-madu 21888 df-cpmat 21960 df-mat2pmat 21961 df-cpmat2mat 21962 df-decpmat 22017 df-pm2mp 22047 df-chpmat 22081

This theorem is referenced by: cayleyhamilton 22144

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