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Theorem cayleyhamilton0 21497

Description: The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 21498 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 21499)! (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 2821	. . 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 21496	. 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 2830	. . . . . . . . . . . 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 6672	. . . . . . . . . 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 7171	. . . . . . . . 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 5161	. . . . . . . 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 7172	. . . . . . 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 2823	. . . . . 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 479	. . . . 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 7163	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝑛𝐸(𝑇‘𝑀)) = (𝑙𝐸(𝑇‘𝑀)))
29		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑙 → (𝐺‘𝑛) = (𝐺‘𝑙))
30	28, 29	oveq12d 7174	. . . . . . . . . 10 ⊢ (𝑛 = 𝑙 → ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛)) = ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
31	30	cbvmptv 5169	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))) = (𝑙 ∈ ℕ₀ ↦ ((𝑙𝐸(𝑇‘𝑀)) × (𝐺‘𝑙)))
32	31	oveq2i 7167	. . . . . . . 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 21474	. . . . . . . 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	syl5eq 2868	. . . . . . 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 6670	. . . . . . . 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 19308	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
37	36	anim2i 618	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38	37	3adant3 1128	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
39		eqid 2821	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
40	1, 15, 3, 4, 39, 7	m2cpminv0 21369	. . . . . . . . . . 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	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑈‘𝑍) = 0 )
44	43	adantr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑈‘𝑍) = 0 )
45	35, 44	sylan9eqr 2878	. . . . . . 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 483	. . . . 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 2856	. . . 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 415	. . 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 3294	. 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ifcif 4467 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 Fincfn 8509 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 − cmin 10870 ℕcn 11638 ℕ₀cn0 11898 ...cfz 12893 Basecbs 16483 ._rcmulr 16566 ·_𝑠 cvsca 16569 0_gc0g 16713 Σ_g cgsu 16714 -_gcsg 18105 ._gcmg 18224 mulGrpcmgp 19239 1_rcur 19251 Ringcrg 19297 CRingccrg 19298 Poly₁cpl1 20345 coe₁cco1 20346 Mat cmat 21016 matToPolyMat cmat2pmat 21312 cPolyMatToMat ccpmat2mat 21313 CharPlyMat cchpmat 21434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-ofr 7410 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-tpos 7892 df-cur 7933 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-word 13863 df-lsw 13915 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-splice 14112 df-reverse 14121 df-s2 14210 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-0g 16715 df-gsum 16716 df-prds 16721 df-pws 16723 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-efmnd 18034 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-subg 18276 df-ghm 18356 df-gim 18399 df-cntz 18447 df-oppg 18474 df-symg 18496 df-pmtr 18570 df-psgn 18619 df-evpm 18620 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-srg 19256 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-rnghom 19467 df-drng 19504 df-subrg 19533 df-lmod 19636 df-lss 19704 df-sra 19944 df-rgmod 19945 df-assa 20085 df-ascl 20087 df-psr 20136 df-mvr 20137 df-mpl 20138 df-opsr 20140 df-psr1 20348 df-vr1 20349 df-ply1 20350 df-coe1 20351 df-cnfld 20546 df-zring 20618 df-zrh 20651 df-dsmm 20876 df-frlm 20891 df-mamu 20995 df-mat 21017 df-mdet 21194 df-madu 21243 df-cpmat 21314 df-mat2pmat 21315 df-cpmat2mat 21316 df-decpmat 21371 df-pm2mp 21401 df-chpmat 21435

This theorem is referenced by: cayleyhamilton 21498

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