aks4d1p9 - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > aks4d1p9		Structured version Visualization version GIF version

Theorem aks4d1p9 41469

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 12290	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
2	1	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℝ)
3		2pos 12319	. . . . . . . . . 10 ⊢ 0 < 2
4	3	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 < 2)
5		aks4d1p9.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
6		eluzelz 12836	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
7	5, 6	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
8	7	zred 12670	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ)
9		0red 11221	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
10		3re 12296	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 3 ∈ ℝ)
12		3pos 12321	. . . . . . . . . . 11 ⊢ 0 < 3
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 3)
14		eluzle 12839	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
15	5, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 3 ≤ 𝑁)
16	9, 11, 8, 13, 15	ltletrd 11378	. . . . . . . . 9 ⊢ (𝜑 → 0 < 𝑁)
17		1red 11219	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
18		1lt2 12387	. . . . . . . . . . . 12 ⊢ 1 < 2
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
20	17, 19	ltned 11354	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 2)
21	20	necomd 2990	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 1)
22	2, 4, 8, 16, 21	relogbcld 41354	. . . . . . . 8 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
23	22	resqcld 14095	. . . . . . 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 41460	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ (1...𝐵) ∧ ¬ 𝑅 ∥ 𝐴))
28	27	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (1...𝐵))
29		elfznn 13536	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (1...𝐵) → 𝑅 ∈ ℕ)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ ℕ)
31	5, 24, 25, 26	aks4d1p8 41468	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
32	30, 7, 31	3jca 1125	. . . . . . . . 9 ⊢ (𝜑 → (𝑅 ∈ ℕ ∧ 𝑁 ∈ ℤ ∧ (𝑁 gcd 𝑅) = 1))
33		odzcl 16735	. . . . . . . . 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 12589	. . . . . . 7 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℤ)
36		flge 13776	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → (((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2) ↔ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))))
38	37	biimpd 228	. . . . 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 12589	. . . . . . . . 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 12536	. . . . . . . . . . 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 14058	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑((od_ℤ‘𝑅)‘𝑁)) ∈ ℤ)
46		1zzd 12597	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → 1 ∈ ℤ)
47	45, 46	zsubcld 12675	. . . . . . . 8 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((𝑁↑((od_ℤ‘𝑅)‘𝑁)) − 1) ∈ ℤ)
48	5, 25	aks4d1lem1 41443	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 ∈ ℕ ∧ 9 < 𝐵))
49	48	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
50	49	nnred 12231	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ)
51	49	nngt0d 12265	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐵)
52	2, 4, 50, 51, 21	relogbcld 41354	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
53	52	flcld 13769	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
54		2cnd 12294	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
55	9, 4	gtned 11353	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
56	54, 55, 21	3jca 1125	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1))
57		logb1 26656	. . . . . . . . . . . . . . . . 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 12598	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℤ
60	59	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ∈ ℤ)
61	2	leidd 11784	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 2)
62		0lt1 11740	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 1
63	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 1)
64	49	nnge1d 12264	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ 𝐵)
65	60, 61, 17, 63, 50, 51, 64	logblebd 41357	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝐵))
66	58, 65	eqbrtrrd 5165	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
67		0zd 12574	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℤ)
68		flge 13776	. . . . . . . . . . . . . . . 16 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
69	52, 67, 68	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
70	66, 69	mpbid 231	. . . . . . . . . . . . . 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 12575	. . . . . . . . . . . . 13 ⊢ ((⌊‘(2 log_b 𝐵)) ∈ ℕ₀ ↔ ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
73	71, 72	sylibr 233	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
74	7, 73	zexpcld 14058	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
75		fzfid 13944	. . . . . . . . . . . 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 13536	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
78	77	nnnn0d 12536	. . . . . . . . . . . . . . 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 14058	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
81		1zzd 12597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
82	80, 81	zsubcld 12675	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
83	75, 82	fprodzcl 15904	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
84	74, 83	zmulcld 12676	. . . . . . . . . 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 2812	. . . . . . . . . 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 16220	. . . . . . . . . . 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 16737	. . . . . . . . . . 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 583	. . . . . . . . . 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 14058	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℤ)
97		fzfid 13944	. . . . . . . . . . 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 12536	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101	98, 100	zexpcld 14058	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
102		1zzd 12597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
103	101, 102	zsubcld 12675	. . . . . . . . . . 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 15904	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℤ)
105		fveq2 6885	. . . . . . . . . . . . 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 5151	. . . . . . . . . . . 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 4000	. . . . . . . . . . . . . 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 13536	. . . . . . . . . . . . . . . . . . 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 12536	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑥 ∈ ℕ₀)
112	108, 111	zexpcld 14058	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑥) ∈ ℤ)
113		1zzd 12597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
114	112, 113	zsubcld 12675	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑥) − 1) ∈ ℤ)
115	114	fmpttd 7110	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) ↦ ((𝑁↑𝑥) − 1)):(1...(⌊‘((2 log_b 𝑁)↑2)))⟶ℤ)
116	75, 107, 115	fprodfvdvdsd 16284	. . . . . . . . . . . . 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 14095	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((od_ℤ‘𝑅)‘𝑁) ≤ (⌊‘((2 log_b 𝑁)↑2))) → ((2 log_b 𝑁)↑2) ∈ ℝ)
120	119	flcld 13769	. . . . . . . . . . . . 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 12264	. . . . . . . . . . . . . 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 13498	. . . . . . . . . . . 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 3607	. . . . . . . . . . 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 2727	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . 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 6999	. . . . . . . . . . . 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 2727	. . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . 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 6999	. . . . . . . . . . . . 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 15873	. . . . . . . . . . . 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 5154	. . . . . . . . . . 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 231	. . . . . . . . . 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 16249	. . . . . . . . 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 5169	. . . . . . . 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 16245	. . . . . . 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 684	. . 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 814	. 2 ⊢ (𝜑 → ¬ ((od_ℤ‘𝑅)‘𝑁) ≤ ((2 log_b 𝑁)↑2))
152	34	nnred 12231	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑅)‘𝑁) ∈ ℝ)
153	23, 152	ltnled 11365	. 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 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∀wral 3055 {crab 3426 class class class wbr 5141 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 infcinf 9438 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 · cmul 11117 < clt 11252 ≤ cle 11253 − cmin 11448 ℕcn 12216 2c2 12271 3c3 12272 5c5 12274 9c9 12278 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13490 ⌊cfl 13761 ⌈cceil 13762 ↑cexp 14032 ∏cprod 15855 ∥ cdvds 16204 gcd cgcd 16442 od_ℤcodz 16705 log_b clogb 26651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-symdif 4237 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-ceil 13764 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-prod 15856 df-ef 16017 df-e 16018 df-sin 16019 df-cos 16020 df-pi 16022 df-dvds 16205 df-gcd 16443 df-lcm 16534 df-lcmf 16535 df-prm 16616 df-odz 16707 df-phi 16708 df-pc 16779 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-ovol 25348 df-vol 25349 df-mbf 25503 df-itg1 25504 df-itg2 25505 df-ibl 25506 df-itg 25507 df-0p 25554 df-limc 25750 df-dv 25751 df-log 26445 df-cxp 26446 df-logb 26652

This theorem is referenced by: aks4d1 41470

W3C validator