aks4d1p9 - Mathbox for metakunt

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Theorem aks4d1p9 42081

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 12202	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12231	. . . . . . . . . 10 ⊢ 0 < 2
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
5		aks4d1p9.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
6		eluzelz 12745	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	7	zred 12580	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		0red 11118	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
10		3re 12208	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
12		3pos 12233	. . . . . . . . . . 11 ⊢ 0 < 3
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
14		eluzle 12748	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
16	9, 11, 8, 13, 15	ltletrd 11276	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
17		1red 11116	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
18		1lt2 12294	. . . . . . . . . . . 12 ⊢ 1 < 2
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
20	17, 19	ltned 11252	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 2)
21	20	necomd 2980	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 1)
22	2, 4, 8, 16, 21	relogbcld 41966	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
23	22	resqcld 14032	. . . . . . 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 42072	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
28	27	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
29		elfznn 13456	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ)
31	5, 24, 25, 26	aks4d1p8 42080	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
32	30, 7, 31	3jca 1128	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1))
33		odzcl 16705	. . . . . . . . 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 12498	. . . . . . 7 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
36		flge 13709	. . . . . . 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 12498	. . . . . . . . 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 12445	. . . . . . . . . . 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 13994	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑((od_ℤ‘𝑅)‘𝑁)) ∈ ℤ)
46		1zzd 12506	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ∈ ℤ)
47	45, 46	zsubcld 12585	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∈ ℤ)
48	5, 25	aks4d1lem1 42055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
49	48	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
50	49	nnred 12143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	49	nngt0d 12177	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐵)
52	2, 4, 50, 51, 21	relogbcld 41966	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
53	52	flcld 13702	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
54		2cnd 12206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
55	9, 4	gtned 11251	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
56	54, 55, 21	3jca 1128	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1))
57		logb1 26677	. . . . . . . . . . . . . . . . 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 12507	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
60	59	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℤ)
61	2	leidd 11686	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 2)
62		0lt1 11642	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 1
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 1)
64	49	nnge1d 12176	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 𝐵)
65	60, 61, 17, 63, 50, 51, 64	logblebd 41969	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝐵))
66	58, 65	eqbrtrrd 5116	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
67		0zd 12483	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
68		flge 13709	. . . . . . . . . . . . . . . 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 12484	. . . . . . . . . . . . 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 13994	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
75		fzfid 13880	. . . . . . . . . . . 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 13456	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
78	77	nnnn0d 12445	. . . . . . . . . . . . . . 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 13994	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
81		1zzd 12506	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
82	80, 81	zsubcld 12585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
83	75, 82	fprodzcl 15861	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
84	74, 83	zmulcld 12586	. . . . . . . . . 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 2813	. . . . . . . . . 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 16180	. . . . . . . . . . 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 16707	. . . . . . . . . . 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 13994	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
97		fzfid 13880	. . . . . . . . . . 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 12445	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101	98, 100	zexpcld 13994	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
102		1zzd 12506	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
103	101, 102	zsubcld 12585	. . . . . . . . . . 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 15861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
105		fveq2 6822	. . . . . . . . . . . . 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 5102	. . . . . . . . . . . 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 3959	. . . . . . . . . . . . . 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 13456	. . . . . . . . . . . . . . . . . . 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 12445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ₀)
112	108, 111	zexpcld 13994	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑥) ∈ ℤ)
113		1zzd 12506	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
114	112, 113	zsubcld 12585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑥) − 1) ∈ ℤ)
115	114	fmpttd 7049	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)):(1...(⌊‘((2 log_b 𝑁)↑2)))⟶ℤ)
116	75, 107, 115	fprodfvdvdsd 16245	. . . . . . . . . . . . 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 14032	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((2 log_b 𝑁)↑2) ∈ ℝ)
120	119	flcld 13702	. . . . . . . . . . . . 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 12176	. . . . . . . . . . . . . 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 13418	. . . . . . . . . . . 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 3578	. . . . . . . . . . 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 2730	. . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . 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 6937	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . 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 6937	. . . . . . . . . . . . 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 15829	. . . . . . . . . . . 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 5105	. . . . . . . . . . 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 16210	. . . . . . . . 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 5120	. . . . . . . 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 16206	. . . . . . 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 816	. 2 ⊢ (𝜑 → ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2))
152	34	nnred 12143	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℝ)
153	23, 152	ltnled 11263	. 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 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 {crab 3394 class class class wbr 5092 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 infcinf 9331 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 ℕcn 12128 2c2 12183 3c3 12184 5c5 12186 9c9 12190 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ...cfz 13410 ⌊cfl 13694 ⌈cceil 13695 ↑cexp 13968 ∏cprod 15810 ∥ cdvds 16163 gcd cgcd 16405 od_ℤcodz 16674 log_b clogb 26672

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4204 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-ceil 13697 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-prod 15811 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-lcm 16501 df-lcmf 16502 df-prm 16583 df-odz 16676 df-phi 16677 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-limc 25765 df-dv 25766 df-log 26463 df-cxp 26464 df-logb 26673

This theorem is referenced by: aks4d1 42082

W3C validator