aks4d1p9 - Mathbox for metakunt

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Theorem aks4d1p9 42070

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 12338	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12367	. . . . . . . . . 10 ⊢ 0 < 2
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
5		aks4d1p9.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
6		eluzelz 12886	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	7	zred 12720	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		0red 11262	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
10		3re 12344	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
12		3pos 12369	. . . . . . . . . . 11 ⊢ 0 < 3
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
14		eluzle 12889	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
16	9, 11, 8, 13, 15	ltletrd 11419	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
17		1red 11260	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
18		1lt2 12435	. . . . . . . . . . . 12 ⊢ 1 < 2
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
20	17, 19	ltned 11395	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 2)
21	20	necomd 2994	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 1)
22	2, 4, 8, 16, 21	relogbcld 41955	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
23	22	resqcld 14162	. . . . . . 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 42061	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
28	27	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
29		elfznn 13590	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ)
31	5, 24, 25, 26	aks4d1p8 42069	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
32	30, 7, 31	3jca 1127	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1))
33		odzcl 16827	. . . . . . . . 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 12638	. . . . . . 7 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
36		flge 13842	. . . . . . 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 12638	. . . . . . . . 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 12585	. . . . . . . . . . 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 14125	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑((od_ℤ‘𝑅)‘𝑁)) ∈ ℤ)
46		1zzd 12646	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ∈ ℤ)
47	45, 46	zsubcld 12725	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∈ ℤ)
48	5, 25	aks4d1lem1 42044	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
49	48	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
50	49	nnred 12279	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	49	nngt0d 12313	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐵)
52	2, 4, 50, 51, 21	relogbcld 41955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
53	52	flcld 13835	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
54		2cnd 12342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
55	9, 4	gtned 11394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
56	54, 55, 21	3jca 1127	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1))
57		logb1 26827	. . . . . . . . . . . . . . . . 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 12647	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
60	59	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℤ)
61	2	leidd 11827	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 2)
62		0lt1 11783	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 1
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 1)
64	49	nnge1d 12312	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 𝐵)
65	60, 61, 17, 63, 50, 51, 64	logblebd 41958	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝐵))
66	58, 65	eqbrtrrd 5172	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
67		0zd 12623	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
68		flge 13842	. . . . . . . . . . . . . . . 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 12624	. . . . . . . . . . . . 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 14125	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
75		fzfid 14011	. . . . . . . . . . . 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 13590	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
78	77	nnnn0d 12585	. . . . . . . . . . . . . . 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 14125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
81		1zzd 12646	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
82	80, 81	zsubcld 12725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
83	75, 82	fprodzcl 15987	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
84	74, 83	zmulcld 12726	. . . . . . . . . 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 2824	. . . . . . . . . 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 16304	. . . . . . . . . . 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 16829	. . . . . . . . . . 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 14125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
97		fzfid 14011	. . . . . . . . . . 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 12585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101	98, 100	zexpcld 14125	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
102		1zzd 12646	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
103	101, 102	zsubcld 12725	. . . . . . . . . . 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 15987	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
105		fveq2 6907	. . . . . . . . . . . . 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 5158	. . . . . . . . . . . 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 4019	. . . . . . . . . . . . . 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 13590	. . . . . . . . . . . . . . . . . . 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 12585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ₀)
112	108, 111	zexpcld 14125	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑥) ∈ ℤ)
113		1zzd 12646	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
114	112, 113	zsubcld 12725	. . . . . . . . . . . . . . 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 16368	. . . . . . . . . . . . 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 14162	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((2 log_b 𝑁)↑2) ∈ ℝ)
120	119	flcld 13835	. . . . . . . . . . . . 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 12312	. . . . . . . . . . . . . 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 13552	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 15955	. . . . . . . . . . . 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 5161	. . . . . . . . . . 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 16333	. . . . . . . . 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 5176	. . . . . . . 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 16329	. . . . . . 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 12279	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℝ)
153	23, 152	ltnled 11406	. 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 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 infcinf 9479 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 2c2 12319 3c3 12320 5c5 12322 9c9 12326 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ...cfz 13544 ⌊cfl 13827 ⌈cceil 13828 ↑cexp 14099 ∏cprod 15936 ∥ cdvds 16287 gcd cgcd 16528 od_ℤcodz 16797 log_b clogb 26822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-ceil 13830 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-prod 15937 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-pi 16105 df-dvds 16288 df-gcd 16529 df-lcm 16624 df-lcmf 16625 df-prm 16706 df-odz 16799 df-phi 16800 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-limc 25916 df-dv 25917 df-log 26613 df-cxp 26614 df-logb 26823

This theorem is referenced by: aks4d1 42071

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