aks4d1p9 - Mathbox for metakunt

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Theorem aks4d1p9 42089

Description: Show that the order is bound by the squared binary logarithm. (Contributed by metakunt, 14-Nov-2024.)

**Hypotheses**
Ref	Expression
aks4d1p9.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks4d1p9.2	⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
aks4d1p9.3	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
aks4d1p9.4	⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )

**Assertion**
Ref	Expression
aks4d1p9	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))

Distinct variable groups: 𝐴,𝑟 𝐵,𝑟 𝑘,𝑁 𝑁,𝑟 𝑅,𝑘 𝑅,𝑟 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑟) 𝐴(𝑘) 𝐵(𝑘)

Proof of Theorem aks4d1p9 Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 12340	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12369	. . . . . . . . . 10 ⊢ 0 < 2
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
5		aks4d1p9.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
6		eluzelz 12888	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	7	zred 12722	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		0red 11264	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
10		3re 12346	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
12		3pos 12371	. . . . . . . . . . 11 ⊢ 0 < 3
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
14		eluzle 12891	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
16	9, 11, 8, 13, 15	ltletrd 11421	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
17		1red 11262	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
18		1lt2 12437	. . . . . . . . . . . 12 ⊢ 1 < 2
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
20	17, 19	ltned 11397	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 2)
21	20	necomd 2996	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 1)
22	2, 4, 8, 16, 21	relogbcld 41974	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
23	22	resqcld 14165	. . . . . . 7 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) ∈ ℝ)
24		aks4d1p9.2	. . . . . . . . . . . . 13 ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
25		aks4d1p9.3	. . . . . . . . . . . . 13 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
26		aks4d1p9.4	. . . . . . . . . . . . 13 ⊢ 𝑅 = inf({𝑟 ∈ (1...𝐵) ∣ ¬ 𝑟 ∥ 𝐴}, ℝ, < )
27	5, 24, 25, 26	aks4d1p4 42080	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
28	27	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
29		elfznn 13593	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ)
31	5, 24, 25, 26	aks4d1p8 42088	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
32	30, 7, 31	3jca 1129	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1))
33		odzcl 16831	. . . . . . . . 9 ⊢ ((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
34	32, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ)
35	34	nnzd 12640	. . . . . . 7 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
36		flge 13845	. . . . . . 7 ⊢ ((((2 log_b 𝑁)↑2) ∈ ℝ ∧ ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ) → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) ↔ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
37	23, 35, 36	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) ↔ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
38	37	biimpd 229	. . . . 5 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) → ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
39	38	imp 406	. . . 4 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)))
40	30	nnzd 12640	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ℤ)
41	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∈ ℤ)
42	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑁 ∈ ℤ)
43	34	nnnn0d 12587	. . . . . . . . . . 11 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
45	42, 44	zexpcld 14128	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑((od_ℤ‘𝑅)‘𝑁)) ∈ ℤ)
46		1zzd 12648	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ∈ ℤ)
47	45, 46	zsubcld 12727	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∈ ℤ)
48	5, 25	aks4d1lem1 42063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
49	48	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
50	49	nnred 12281	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	49	nngt0d 12315	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐵)
52	2, 4, 50, 51, 21	relogbcld 41974	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
53	52	flcld 13838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
54		2cnd 12344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
55	9, 4	gtned 11396	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
56	54, 55, 21	3jca 1129	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1))
57		logb1 26812	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
58	56, 57	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) = 0)
59		2z 12649	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
60	59	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℤ)
61	2	leidd 11829	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 2)
62		0lt1 11785	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 1
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 1)
64	49	nnge1d 12314	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 𝐵)
65	60, 61, 17, 63, 50, 51, 64	logblebd 41977	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝐵))
66	58, 65	eqbrtrrd 5167	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
67		0zd 12625	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
68		flge 13845	. . . . . . . . . . . . . . . 16 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
69	52, 67, 68	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
70	66, 69	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (⌊‘(2 log_b 𝐵)))
71	53, 70	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
72		elnn0z 12626	. . . . . . . . . . . . 13 ⊢ ((⌊‘(2 log_b 𝐵)) ∈ ℕ₀ ↔ ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
73	71, 72	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
74	7, 73	zexpcld 14128	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
75		fzfid 14014	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
76	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
77		elfznn 13593	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
78	77	nnnn0d 12587	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ₀)
79	78	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
80	76, 79	zexpcld 14128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
81		1zzd 12648	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
82	80, 81	zsubcld 12727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
83	75, 82	fprodzcl 15990	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
84	74, 83	zmulcld 12728	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℤ)
85	24	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
86	85	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℤ ↔ ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℤ))
87	84, 86	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
88	87	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝐴 ∈ ℤ)
89		iddvds 16307	. . . . . . . . . . 11 ⊢ (((od_ℤ‘𝑅)‘𝑁) ∈ ℤ → ((od_ℤ‘𝑅)‘𝑁) ∥ ((od_ℤ‘𝑅)‘𝑁))
90	35, 89	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∥ ((od_ℤ‘𝑅)‘𝑁))
91		odzdvds 16833	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1) ∧ ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀) → (𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ↔ ((od_ℤ‘𝑅)‘𝑁) ∥ ((od_ℤ‘𝑅)‘𝑁)))
92	32, 43, 91	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ↔ ((od_ℤ‘𝑅)‘𝑁) ∥ ((od_ℤ‘𝑅)‘𝑁)))
93	90, 92	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
94	93	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
95	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
96	42, 95	zexpcld 14128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
97		fzfid 14014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
98	42	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
99	77	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ)
100	99	nnnn0d 12587	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101	98, 100	zexpcld 14128	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
102		1zzd 12648	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
103	101, 102	zsubcld 12727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
104	97, 103	fprodzcl 15990	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
105		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((od_ℤ‘𝑅)‘𝑁) → ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑧) = ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘((od_ℤ‘𝑅)‘𝑁)))
106	105	breq1d 5153	. . . . . . . . . . . 12 ⊢ (𝑧 = ((od_ℤ‘𝑅)‘𝑁) → (((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑧) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘) ↔ ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘((od_ℤ‘𝑅)‘𝑁)) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘)))
107		ssidd 4007	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ⊆ (1...(⌊‘((2 log_b 𝑁)↑2))))
108	7	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
109		elfznn 13593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑥 ∈ ℕ)
110	109	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ)
111	110	nnnn0d 12587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ₀)
112	108, 111	zexpcld 14128	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑥) ∈ ℤ)
113		1zzd 12648	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
114	112, 113	zsubcld 12727	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑥) − 1) ∈ ℤ)
115	114	fmpttd 7135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)):(1...(⌊‘((2 log_b 𝑁)↑2)))⟶ℤ)
116	75, 107, 115	fprodfvdvdsd 16371	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑧 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑧) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘))
117	116	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∀𝑧 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑧) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘))
118	22	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (2 log_b 𝑁) ∈ ℝ)
119	118	resqcld 14165	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((2 log_b 𝑁)↑2) ∈ ℝ)
120	119	flcld 13838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (⌊‘((2 log_b 𝑁)↑2)) ∈ ℤ)
121	35	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
122	34	nnge1d 12314	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ ((od_ℤ‘𝑅)‘𝑁))
123	122	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ≤ ((od_ℤ‘𝑅)‘𝑁))
124		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)))
125	46, 120, 121, 123, 124	elfzd 13555	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ∈ (1...(⌊‘((2 log_b 𝑁)↑2))))
126	106, 117, 125	rspcdva 3623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘((od_ℤ‘𝑅)‘𝑁)) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘))
127		eqidd 2738	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)) = (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)))
128		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑥 = ((od_ℤ‘𝑅)‘𝑁)) → 𝑥 = ((od_ℤ‘𝑅)‘𝑁))
129	128	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑥 = ((od_ℤ‘𝑅)‘𝑁)) → (𝑁↑𝑥) = (𝑁↑((od_ℤ‘𝑅)‘𝑁)))
130	129	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑥 = ((od_ℤ‘𝑅)‘𝑁)) → ((𝑁↑𝑥) − 1) = ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
131	127, 130, 125, 47	fvmptd 7023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘((od_ℤ‘𝑅)‘𝑁)) = ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
132		eqidd 2738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)) = (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)))
133		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) ∧ 𝑥 = 𝑘) → 𝑥 = 𝑘)
134	133	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) ∧ 𝑥 = 𝑘) → (𝑁↑𝑥) = (𝑁↑𝑘))
135	134	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) ∧ 𝑥 = 𝑘) → ((𝑁↑𝑥) − 1) = ((𝑁↑𝑘) − 1))
136		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))))
137	132, 135, 136, 103	fvmptd 7023	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘) = ((𝑁↑𝑘) − 1))
138	137	prodeq2dv 15958	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘) = ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
139	131, 138	breq12d 5156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘((od_ℤ‘𝑅)‘𝑁)) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1))‘𝑘) ↔ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
140	126, 139	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∥ ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
141	47, 96, 104, 140	dvdsmultr2d 16336	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∥ ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
142	24	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
143	141, 142	breqtrrd 5171	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∥ 𝐴)
144	41, 47, 88, 94, 143	dvdstrd 16332	. . . . . . 7 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∥ 𝐴)
145	144	ex 412	. . . . . 6 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)) → 𝑅 ∥ 𝐴))
146	145	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → (((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)) → 𝑅 ∥ 𝐴))
147	146	imp 406	. . . 4 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∥ 𝐴)
148	39, 147	mpdan 687	. . 3 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → 𝑅 ∥ 𝐴)
149	27	simprd 495	. . . 4 ⊢ (𝜑 → ¬ 𝑅 ∥ 𝐴)
150	149	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → ¬ 𝑅 ∥ 𝐴)
151	148, 150	pm2.65da 817	. 2 ⊢ (𝜑 → ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2))
152	34	nnred 12281	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℝ)
153	23, 152	ltnled 11408	. 2 ⊢ (𝜑 → (((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁) ↔ ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)))
154	151, 153	mpbird 257	1 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 2c2 12321 3c3 12322 5c5 12324 9c9 12328 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 ⌊cfl 13830 ⌈cceil 13831 ↑cexp 14102 ∏cprod 15939 ∥ cdvds 16290 gcd cgcd 16531 od_ℤcodz 16800 log_b clogb 26807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-ceil 13833 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-prod 15940 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-pi 16108 df-dvds 16291 df-gcd 16532 df-lcm 16627 df-lcmf 16628 df-prm 16709 df-odz 16802 df-phi 16803 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-logb 26808

This theorem is referenced by: aks4d1 42090

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