fmtnofac1 - Mathbox for Alexander van der Vekens

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Theorem fmtnofac1 46238

Description: Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of Fermat Number/Euler's Result", 24-Jul-2021, https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result): "Let F_n be a Fermat number. Let m be divisor of F_n. Then m is in the form: k*2^(n+1)+1 where k is a positive integer." Here, however, k must be a nonnegative integer, because k must be 0 to represent 1 (which is a divisor of F_n ).

Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number F_n is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 46237. (Contributed by AV, 30-Jul-2021.)

**Assertion**
Ref	Expression
fmtnofac1	⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Distinct variable groups: 𝑘,𝑀 𝑘,𝑁

Proof of Theorem fmtnofac1 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn1uz2 12909	. . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		5prm 17042	. . . . . . 7 ⊢ 5 ∈ ℙ
3		dvdsprime 16624	. . . . . . 7 ⊢ ((5 ∈ ℙ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ 5 ↔ (𝑀 = 5 ∨ 𝑀 = 1)))
4	2, 3	mpan 689	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 ↔ (𝑀 = 5 ∨ 𝑀 = 1)))
5		1nn0 12488	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
6	5	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 5 → 1 ∈ ℕ₀)
7		simpl 484	. . . . . . . . 9 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → 𝑀 = 5)
8		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝑘 · 4) = (1 · 4))
9	8	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑘 · 4) + 1) = ((1 · 4) + 1))
10	9	adantl 483	. . . . . . . . 9 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → ((𝑘 · 4) + 1) = ((1 · 4) + 1))
11	7, 10	eqeq12d 2749	. . . . . . . 8 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → (𝑀 = ((𝑘 · 4) + 1) ↔ 5 = ((1 · 4) + 1)))
12		df-5 12278	. . . . . . . . . 10 ⊢ 5 = (4 + 1)
13		4cn 12297	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
14	13	mullidi 11219	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
15	14	eqcomi 2742	. . . . . . . . . . 11 ⊢ 4 = (1 · 4)
16	15	oveq1i 7419	. . . . . . . . . 10 ⊢ (4 + 1) = ((1 · 4) + 1)
17	12, 16	eqtri 2761	. . . . . . . . 9 ⊢ 5 = ((1 · 4) + 1)
18	17	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 5 → 5 = ((1 · 4) + 1))
19	6, 11, 18	rspcedvd 3615	. . . . . . 7 ⊢ (𝑀 = 5 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
20		0nn0 12487	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
21	20	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 1 → 0 ∈ ℕ₀)
22		simpl 484	. . . . . . . . 9 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → 𝑀 = 1)
23		oveq1 7416	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑘 · 4) = (0 · 4))
24	23	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((𝑘 · 4) + 1) = ((0 · 4) + 1))
25	24	adantl 483	. . . . . . . . 9 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → ((𝑘 · 4) + 1) = ((0 · 4) + 1))
26	22, 25	eqeq12d 2749	. . . . . . . 8 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → (𝑀 = ((𝑘 · 4) + 1) ↔ 1 = ((0 · 4) + 1)))
27	13	mul02i 11403	. . . . . . . . . . . 12 ⊢ (0 · 4) = 0
28	27	oveq1i 7419	. . . . . . . . . . 11 ⊢ ((0 · 4) + 1) = (0 + 1)
29		0p1e1 12334	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
30	28, 29	eqtri 2761	. . . . . . . . . 10 ⊢ ((0 · 4) + 1) = 1
31	30	eqcomi 2742	. . . . . . . . 9 ⊢ 1 = ((0 · 4) + 1)
32	31	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 1 → 1 = ((0 · 4) + 1))
33	21, 26, 32	rspcedvd 3615	. . . . . . 7 ⊢ (𝑀 = 1 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
34	19, 33	jaoi 856	. . . . . 6 ⊢ ((𝑀 = 5 ∨ 𝑀 = 1) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
35	4, 34	syl6bi 253	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1)))
36		fveq2 6892	. . . . . . . 8 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1))
37		fmtno1 46209	. . . . . . . 8 ⊢ (FermatNo‘1) = 5
38	36, 37	eqtrdi 2789	. . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = 5)
39	38	breq2d 5161	. . . . . 6 ⊢ (𝑁 = 1 → (𝑀 ∥ (FermatNo‘𝑁) ↔ 𝑀 ∥ 5))
40		oveq1 7416	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → (𝑁 + 1) = (1 + 1))
41		1p1e2 12337	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
42	40, 41	eqtrdi 2789	. . . . . . . . . . . 12 ⊢ (𝑁 = 1 → (𝑁 + 1) = 2)
43	42	oveq2d 7425	. . . . . . . . . . 11 ⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) = (2↑2))
44		sq2 14161	. . . . . . . . . . 11 ⊢ (2↑2) = 4
45	43, 44	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) = 4)
46	45	oveq2d 7425	. . . . . . . . 9 ⊢ (𝑁 = 1 → (𝑘 · (2↑(𝑁 + 1))) = (𝑘 · 4))
47	46	oveq1d 7424	. . . . . . . 8 ⊢ (𝑁 = 1 → ((𝑘 · (2↑(𝑁 + 1))) + 1) = ((𝑘 · 4) + 1))
48	47	eqeq2d 2744	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ 𝑀 = ((𝑘 · 4) + 1)))
49	48	rexbidv 3179	. . . . . 6 ⊢ (𝑁 = 1 → (∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1)))
50	39, 49	imbi12d 345	. . . . 5 ⊢ (𝑁 = 1 → ((𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)) ↔ (𝑀 ∥ 5 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))))
51	35, 50	imbitrrid 245	. . . 4 ⊢ (𝑁 = 1 → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
52		fmtnofac2 46237	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑛 ∈ ℕ₀ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1))
53		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
54		2nn0 12489	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ₀)
56	53, 55	nn0mulcld 12537	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 · 2) ∈ ℕ₀)
57	56	adantl 483	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · 2) ∈ ℕ₀)
58	57	adantr 482	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → (𝑛 · 2) ∈ ℕ₀)
59		simpr 486	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1))
60		oveq1 7416	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 · 2) → (𝑘 · (2↑(𝑁 + 1))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
61	60	oveq1d 7424	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 · 2) → ((𝑘 · (2↑(𝑁 + 1))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1))
62	59, 61	eqeqan12d 2747	. . . . . . . 8 ⊢ (((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) ∧ 𝑘 = (𝑛 · 2)) → (𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ ((𝑛 · (2↑(𝑁 + 2))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1)))
63		eluzge2nn0 12871	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
64	63	nn0cnd 12534	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℂ)
65		add1p1 12463	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) + 1) = (𝑁 + 2))
66	64, 65	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((𝑁 + 1) + 1) = (𝑁 + 2))
67	66	eqcomd 2739	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 + 2) = ((𝑁 + 1) + 1))
68	67	oveq2d 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 2)) = (2↑((𝑁 + 1) + 1)))
69		2cnd 12290	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℂ)
70		peano2nn0 12512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
71	63, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 + 1) ∈ ℕ₀)
72	69, 71	expp1d 14112	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑((𝑁 + 1) + 1)) = ((2↑(𝑁 + 1)) · 2))
73	54	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℕ₀)
74	73, 71	nn0expcld 14209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 1)) ∈ ℕ₀)
75	74	nn0cnd 12534	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 1)) ∈ ℂ)
76	75, 69	mulcomd 11235	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((2↑(𝑁 + 1)) · 2) = (2 · (2↑(𝑁 + 1))))
77	68, 72, 76	3eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 2)) = (2 · (2↑(𝑁 + 1))))
78	77	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (2↑(𝑁 + 2)) = (2 · (2↑(𝑁 + 1))))
79	78	oveq2d 7425	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = (𝑛 · (2 · (2↑(𝑁 + 1)))))
80		nn0cn 12482	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
81	80	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℂ)
82		2cnd 12290	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 2 ∈ ℂ)
83	75	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (2↑(𝑁 + 1)) ∈ ℂ)
84	81, 82, 83	mulassd 11237	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → ((𝑛 · 2) · (2↑(𝑁 + 1))) = (𝑛 · (2 · (2↑(𝑁 + 1)))))
85	79, 84	eqtr4d 2776	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
86	85	3ad2antl1 1186	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
87	86	adantr 482	. . . . . . . . 9 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
88	87	oveq1d 7424	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → ((𝑛 · (2↑(𝑁 + 2))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1))
89	58, 62, 88	rspcedvd 3615	. . . . . . 7 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
90	89	rexlimdva2 3158	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → (∃𝑛 ∈ ℕ₀ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
91	52, 90	mpd 15	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
92	91	3exp 1120	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
93	51, 92	jaoi 856	. . 3 ⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)) → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
94	1, 93	sylbi 216	. 2 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
95	94	3imp 1112	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℕcn 12212 2c2 12267 4c4 12269 5c5 12270 ℕ₀cn0 12472 ℤ_≥cuz 12822 ↑cexp 14027 ∥ cdvds 16197 ℙcprime 16608 FermatNocfmtno 46195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-ioo 13328 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-prod 15850 df-dvds 16198 df-gcd 16436 df-prm 16609 df-odz 16698 df-phi 16699 df-pc 16770 df-lgs 26798 df-fmtno 46196

This theorem is referenced by: (None)

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