fmtnofac1 - Mathbox for Alexander van der Vekens

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Theorem fmtnofac1 46224

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 46223. (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 12905	. . 3 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ_≥‘2)))
2		5prm 17038	. . . . . . 7 ⊢ 5 ∈ ℙ
3		dvdsprime 16620	. . . . . . 7 ⊢ ((5 ∈ ℙ ∧ 𝑀 ∈ ℕ) → (𝑀 ∥ 5 ↔ (𝑀 = 5 ∨ 𝑀 = 1)))
4	2, 3	mpan 688	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 ↔ (𝑀 = 5 ∨ 𝑀 = 1)))
5		1nn0 12484	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
6	5	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 5 → 1 ∈ ℕ₀)
7		simpl 483	. . . . . . . . 9 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → 𝑀 = 5)
8		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝑘 · 4) = (1 · 4))
9	8	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑘 · 4) + 1) = ((1 · 4) + 1))
10	9	adantl 482	. . . . . . . . 9 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → ((𝑘 · 4) + 1) = ((1 · 4) + 1))
11	7, 10	eqeq12d 2748	. . . . . . . 8 ⊢ ((𝑀 = 5 ∧ 𝑘 = 1) → (𝑀 = ((𝑘 · 4) + 1) ↔ 5 = ((1 · 4) + 1)))
12		df-5 12274	. . . . . . . . . 10 ⊢ 5 = (4 + 1)
13		4cn 12293	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
14	13	mullidi 11215	. . . . . . . . . . . 12 ⊢ (1 · 4) = 4
15	14	eqcomi 2741	. . . . . . . . . . 11 ⊢ 4 = (1 · 4)
16	15	oveq1i 7415	. . . . . . . . . 10 ⊢ (4 + 1) = ((1 · 4) + 1)
17	12, 16	eqtri 2760	. . . . . . . . 9 ⊢ 5 = ((1 · 4) + 1)
18	17	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 5 → 5 = ((1 · 4) + 1))
19	6, 11, 18	rspcedvd 3614	. . . . . . 7 ⊢ (𝑀 = 5 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
20		0nn0 12483	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
21	20	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 1 → 0 ∈ ℕ₀)
22		simpl 483	. . . . . . . . 9 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → 𝑀 = 1)
23		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝑘 · 4) = (0 · 4))
24	23	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((𝑘 · 4) + 1) = ((0 · 4) + 1))
25	24	adantl 482	. . . . . . . . 9 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → ((𝑘 · 4) + 1) = ((0 · 4) + 1))
26	22, 25	eqeq12d 2748	. . . . . . . 8 ⊢ ((𝑀 = 1 ∧ 𝑘 = 0) → (𝑀 = ((𝑘 · 4) + 1) ↔ 1 = ((0 · 4) + 1)))
27	13	mul02i 11399	. . . . . . . . . . . 12 ⊢ (0 · 4) = 0
28	27	oveq1i 7415	. . . . . . . . . . 11 ⊢ ((0 · 4) + 1) = (0 + 1)
29		0p1e1 12330	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
30	28, 29	eqtri 2760	. . . . . . . . . 10 ⊢ ((0 · 4) + 1) = 1
31	30	eqcomi 2741	. . . . . . . . 9 ⊢ 1 = ((0 · 4) + 1)
32	31	a1i 11	. . . . . . . 8 ⊢ (𝑀 = 1 → 1 = ((0 · 4) + 1))
33	21, 26, 32	rspcedvd 3614	. . . . . . 7 ⊢ (𝑀 = 1 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
34	19, 33	jaoi 855	. . . . . 6 ⊢ ((𝑀 = 5 ∨ 𝑀 = 1) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1))
35	4, 34	syl6bi 252	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑀 ∥ 5 → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1)))
36		fveq2 6888	. . . . . . . 8 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = (FermatNo‘1))
37		fmtno1 46195	. . . . . . . 8 ⊢ (FermatNo‘1) = 5
38	36, 37	eqtrdi 2788	. . . . . . 7 ⊢ (𝑁 = 1 → (FermatNo‘𝑁) = 5)
39	38	breq2d 5159	. . . . . 6 ⊢ (𝑁 = 1 → (𝑀 ∥ (FermatNo‘𝑁) ↔ 𝑀 ∥ 5))
40		oveq1 7412	. . . . . . . . . . . . 13 ⊢ (𝑁 = 1 → (𝑁 + 1) = (1 + 1))
41		1p1e2 12333	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
42	40, 41	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑁 = 1 → (𝑁 + 1) = 2)
43	42	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) = (2↑2))
44		sq2 14157	. . . . . . . . . . 11 ⊢ (2↑2) = 4
45	43, 44	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑁 = 1 → (2↑(𝑁 + 1)) = 4)
46	45	oveq2d 7421	. . . . . . . . 9 ⊢ (𝑁 = 1 → (𝑘 · (2↑(𝑁 + 1))) = (𝑘 · 4))
47	46	oveq1d 7420	. . . . . . . 8 ⊢ (𝑁 = 1 → ((𝑘 · (2↑(𝑁 + 1))) + 1) = ((𝑘 · 4) + 1))
48	47	eqeq2d 2743	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ 𝑀 = ((𝑘 · 4) + 1)))
49	48	rexbidv 3178	. . . . . 6 ⊢ (𝑁 = 1 → (∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1) ↔ ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · 4) + 1)))
50	39, 49	imbi12d 344	. . . . 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 46223	. . . . . 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 12485	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
55	54	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 2 ∈ ℕ₀)
56	53, 55	nn0mulcld 12533	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 · 2) ∈ ℕ₀)
57	56	adantl 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · 2) ∈ ℕ₀)
58	57	adantr 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑀 = ((𝑛 · (2↑(𝑁 + 2))) + 1)) → (𝑛 · 2) ∈ ℕ₀)
59		simpr 485	. . . . . . . . 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 7412	. . . . . . . . . 10 ⊢ (𝑘 = (𝑛 · 2) → (𝑘 · (2↑(𝑁 + 1))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
61	60	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑘 = (𝑛 · 2) → ((𝑘 · (2↑(𝑁 + 1))) + 1) = (((𝑛 · 2) · (2↑(𝑁 + 1))) + 1))
62	59, 61	eqeqan12d 2746	. . . . . . . 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 12867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ₀)
64	63	nn0cnd 12530	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℂ)
65		add1p1 12459	. . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 + 2) = ((𝑁 + 1) + 1))
68	67	oveq2d 7421	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 2)) = (2↑((𝑁 + 1) + 1)))
69		2cnd 12286	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 2 ∈ ℂ)
70		peano2nn0 12508	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
71	63, 70	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 + 1) ∈ ℕ₀)
72	69, 71	expp1d 14108	. . . . . . . . . . . . . . 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 14205	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 1)) ∈ ℕ₀)
75	74	nn0cnd 12530	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 1)) ∈ ℂ)
76	75, 69	mulcomd 11231	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘2) → ((2↑(𝑁 + 1)) · 2) = (2 · (2↑(𝑁 + 1))))
77	68, 72, 76	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (2↑(𝑁 + 2)) = (2 · (2↑(𝑁 + 1))))
78	77	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (2↑(𝑁 + 2)) = (2 · (2↑(𝑁 + 1))))
79	78	oveq2d 7421	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = (𝑛 · (2 · (2↑(𝑁 + 1)))))
80		nn0cn 12478	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
81	80	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℂ)
82		2cnd 12286	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 2 ∈ ℂ)
83	75	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (2↑(𝑁 + 1)) ∈ ℂ)
84	81, 82, 83	mulassd 11233	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → ((𝑛 · 2) · (2↑(𝑁 + 1))) = (𝑛 · (2 · (2↑(𝑁 + 1)))))
85	79, 84	eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
86	85	3ad2antl1 1185	. . . . . . . . . 10 ⊢ (((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · (2↑(𝑁 + 2))) = ((𝑛 · 2) · (2↑(𝑁 + 1))))
87	86	adantr 481	. . . . . . . . 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 7420	. . . . . . . 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 3614	. . . . . . 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 3157	. . . . . 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 1119	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑀 ∈ ℕ → (𝑀 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ₀ 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
93	51, 92	jaoi 855	. . 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 1111	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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ℕcn 12208 2c2 12263 4c4 12265 5c5 12266 ℕ₀cn0 12468 ℤ_≥cuz 12818 ↑cexp 14023 ∥ cdvds 16193 ℙcprime 16604 FermatNocfmtno 46181

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846 df-dvds 16194 df-gcd 16432 df-prm 16605 df-odz 16694 df-phi 16695 df-pc 16766 df-lgs 26787 df-fmtno 46182

This theorem is referenced by: (None)

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