aks4d1p9 - Mathbox for metakunt

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Theorem aks4d1p9 42588

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 12250	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12279	. . . . . . . . . 10 ⊢ 0 < 2
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
5		aks4d1p9.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
6		eluzelz 12793	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	7	zred 12628	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		0red 11142	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
10		3re 12256	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
12		3pos 12281	. . . . . . . . . . 11 ⊢ 0 < 3
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
14		eluzle 12796	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
16	9, 11, 8, 13, 15	ltletrd 11301	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
17		1red 11140	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
18		1lt2 12342	. . . . . . . . . . . 12 ⊢ 1 < 2
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
20	17, 19	ltned 11277	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 2)
21	20	necomd 2991	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 1)
22	2, 4, 8, 16, 21	relogbcld 42474	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
23	22	resqcld 14082	. . . . . . 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 42579	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
28	27	simpld 496	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
29		elfznn 13502	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ)
31	5, 24, 25, 26	aks4d1p8 42587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
32	30, 7, 31	3jca 1135	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1))
33		odzcl 16759	. . . . . . . . 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 12545	. . . . . . 7 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
36		flge 13759	. . . . . . 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 591	. . . . . 6 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) ↔ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
38	37	biimpd 231	. . . . 5 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) → ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
39	38	imp 408	. . . 4 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)))
40	30	nnzd 12545	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ ℤ)
41	40	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∈ ℤ)
42	7	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑁 ∈ ℤ)
43	34	nnnn0d 12493	. . . . . . . . . . 11 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
44	43	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℕ₀)
45	42, 44	zexpcld 14044	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑((od_ℤ‘𝑅)‘𝑁)) ∈ ℤ)
46		1zzd 12553	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ∈ ℤ)
47	45, 46	zsubcld 12633	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∈ ℤ)
48	5, 25	aks4d1lem1 42562	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
49	48	simpld 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
50	49	nnred 12184	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	49	nngt0d 12221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐵)
52	2, 4, 50, 51, 21	relogbcld 42474	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
53	52	flcld 13752	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
54		2cnd 12254	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
55	9, 4	gtned 11276	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
56	54, 55, 21	3jca 1135	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1))
57		logb1 26755	. . . . . . . . . . . . . . . . 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 12554	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
60	59	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℤ)
61	2	leidd 11711	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 2)
62		0lt1 11667	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 1
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 1)
64	49	nnge1d 12220	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 𝐵)
65	60, 61, 17, 63, 50, 51, 64	logblebd 42477	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝐵))
66	58, 65	eqbrtrrd 5099	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
67		0zd 12531	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
68		flge 13759	. . . . . . . . . . . . . . . 16 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
69	52, 67, 68	syl2anc 591	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
70	66, 69	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (⌊‘(2 log_b 𝐵)))
71	53, 70	jca 517	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
72		elnn0z 12532	. . . . . . . . . . . . 13 ⊢ ((⌊‘(2 log_b 𝐵)) ∈ ℕ₀ ↔ ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
73	71, 72	sylibr 236	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
74	7, 73	zexpcld 14044	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
75		fzfid 13930	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
76	7	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
77		elfznn 13502	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
78	77	nnnn0d 12493	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ₀)
79	78	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
80	76, 79	zexpcld 14044	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
81		1zzd 12553	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
82	80, 81	zsubcld 12633	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
83	75, 82	fprodzcl 15914	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
84	74, 83	zmulcld 12634	. . . . . . . . . 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 259	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℤ)
88	87	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝐴 ∈ ℤ)
89		iddvds 16233	. . . . . . . . . . 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 16761	. . . . . . . . . . 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 591	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ↔ ((od_ℤ‘𝑅)‘𝑁) ∥ ((od_ℤ‘𝑅)‘𝑁)))
93	90, 92	mpbird 259	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
94	93	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∥ ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1))
95	73	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
96	42, 95	zexpcld 14044	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
97		fzfid 13930	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
98	42	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
99	77	adantl 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ)
100	99	nnnn0d 12493	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101	98, 100	zexpcld 14044	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
102		1zzd 12553	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
103	101, 102	zsubcld 12633	. . . . . . . . . . 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 15914	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
105		fveq2 6831	. . . . . . . . . . . . 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 5085	. . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ⊆ (1...(⌊‘((2 log_b 𝑁)↑2))))
108	7	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
109		elfznn 13502	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑥 ∈ ℕ)
110	109	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ)
111	110	nnnn0d 12493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ₀)
112	108, 111	zexpcld 14044	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑥) ∈ ℤ)
113		1zzd 12553	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
114	112, 113	zsubcld 12633	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑥) − 1) ∈ ℤ)
115	114	fmpttd 7060	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)):(1...(⌊‘((2 log_b 𝑁)↑2)))⟶ℤ)
116	75, 107, 115	fprodfvdvdsd 16298	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (2 log_b 𝑁) ∈ ℝ)
119	118	resqcld 14082	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((2 log_b 𝑁)↑2) ∈ ℝ)
120	119	flcld 13752	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (⌊‘((2 log_b 𝑁)↑2)) ∈ ℤ)
121	35	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
122	34	nnge1d 12220	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ ((od_ℤ‘𝑅)‘𝑁))
123	122	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ≤ ((od_ℤ‘𝑅)‘𝑁))
124		simpr 486	. . . . . . . . . . . . 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 13464	. . . . . . . . . . . 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 3563	. . . . . . . . . . 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 2742	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑥 = ((od_ℤ‘𝑅)‘𝑁)) → 𝑥 = ((od_ℤ‘𝑅)‘𝑁))
129	128	oveq2d 7376	. . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . 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 6947	. . . . . . . . . . . 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 2742	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 6947	. . . . . . . . . . . . 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 15882	. . . . . . . . . . . 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 5088	. . . . . . . . . . 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 234	. . . . . . . . . 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 16263	. . . . . . . . 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 5103	. . . . . . . 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 16259	. . . . . . 7 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 𝑅 ∥ 𝐴)
145	144	ex 414	. . . . . 6 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2)) → 𝑅 ∥ 𝐴))
146	145	adantr 482	. . . . 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 408	. . . 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 694	. . 3 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → 𝑅 ∥ 𝐴)
149	27	simprd 497	. . . 4 ⊢ (𝜑 → ¬ 𝑅 ∥ 𝐴)
150	149	adantr 482	. . 3 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)) → ¬ 𝑅 ∥ 𝐴)
151	148, 150	pm2.65da 823	. 2 ⊢ (𝜑 → ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2))
152	34	nnred 12184	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℝ)
153	23, 152	ltnled 11288	. 2 ⊢ (𝜑 → (((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁) ↔ ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2)))
154	151, 153	mpbird 259	1 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 {crab 3393 class class class wbr 5075 ↦ cmpt 5156 ‘cfv 6489 (class class class)co 7360 infcinf 9348 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 · cmul 11038 < clt 11174 ≤ cle 11175 − cmin 11372 ℕcn 12169 2c2 12231 3c3 12232 5c5 12234 9c9 12238 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ...cfz 13456 ⌊cfl 13744 ⌈cceil 13745 ↑cexp 14018 ∏cprod 15863 ∥ cdvds 16216 gcd cgcd 16458 od_ℤcodz 16728 log_b clogb 26750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cc 10352 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-symdif 4184 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-disj 5043 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-ceil 13747 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-prod 15864 df-ef 16027 df-e 16028 df-sin 16029 df-cos 16030 df-pi 16032 df-dvds 16217 df-gcd 16459 df-lcm 16554 df-lcmf 16555 df-prm 16636 df-odz 16730 df-phi 16731 df-pc 16803 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-cmp 23374 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-ovol 25453 df-vol 25454 df-mbf 25608 df-itg1 25609 df-itg2 25610 df-ibl 25611 df-itg 25612 df-0p 25659 df-limc 25855 df-dv 25856 df-log 26542 df-cxp 26543 df-logb 26751

This theorem is referenced by: aks4d1 42589

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