aks4d1p3 - Mathbox for metakunt

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Theorem aks4d1p3 41605

Description: There exists a small enough number such that it does not divide 𝐴. (Contributed by metakunt, 27-Oct-2024.)

**Hypotheses**
Ref	Expression
aks4d1p3.1	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks4d1p3.2	⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
aks4d1p3.3	⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))

**Assertion**
Ref	Expression
aks4d1p3	⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)

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

Proof of Theorem aks4d1p3 Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks4d1p3.1	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
2		aks4d1p3.2	. . . . . 6 ⊢ 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1))
3		aks4d1p3.3	. . . . . 6 ⊢ 𝐵 = (⌈‘((2 log_b 𝑁)↑5))
4	1, 2, 3	aks4d1p1 41603	. . . . 5 ⊢ (𝜑 → 𝐴 < (2↑𝐵))
5	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → 𝐴 < (2↑𝐵))
6		2re 12316	. . . . . . . . 9 ⊢ 2 ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
8	3	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (⌈‘((2 log_b 𝑁)↑5)))
9		2pos 12345	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
11		eluzelz 12862	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	1, 11	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13	12	zred 12696	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
14		0red 11247	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ)
15		3re 12322	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
16	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ∈ ℝ)
17		3pos 12347	. . . . . . . . . . . . . . . 16 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 < 3)
19		eluzle 12865	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	1, 19	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 3 ≤ 𝑁)
21	14, 16, 13, 18, 20	ltletrd 11404	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑁)
22		1red 11245	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℝ)
23		1lt2 12413	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 2
24	23	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 < 2)
25	22, 24	ltned 11380	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≠ 2)
26	25	necomd 2986	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ≠ 1)
27	7, 10, 13, 21, 26	relogbcld 41499	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
28		5nn0 12522	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℕ₀
29	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 5 ∈ ℕ₀)
30	27, 29	reexpcld 14159	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ∈ ℝ)
31		ceilcl 13839	. . . . . . . . . . . 12 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
32	30, 31	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℤ)
33	8, 32	eqeltrd 2825	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℤ)
34	32	zred 12696	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌈‘((2 log_b 𝑁)↑5)) ∈ ℝ)
35	8, 34	eqeltrd 2825	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
36		7re 12335	. . . . . . . . . . . . . . 15 ⊢ 7 ∈ ℝ
37	36	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 7 ∈ ℝ)
38		7pos 12353	. . . . . . . . . . . . . . 15 ⊢ 0 < 7
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 7)
40	13, 20	3lexlogpow5ineq3 41584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 7 < ((2 log_b 𝑁)↑5))
41	14, 37, 30, 39, 40	lttrd 11405	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < ((2 log_b 𝑁)↑5))
42		ceilge 13842	. . . . . . . . . . . . . 14 ⊢ (((2 log_b 𝑁)↑5) ∈ ℝ → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
43	30, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2 log_b 𝑁)↑5) ≤ (⌈‘((2 log_b 𝑁)↑5)))
44	14, 30, 34, 41, 43	ltletrd 11404	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < (⌈‘((2 log_b 𝑁)↑5)))
45	44, 8	breqtrrd 5171	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐵)
46	14, 35, 45	ltled 11392	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ 𝐵)
47	33, 46	jca 510	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
48		elnn0z 12601	. . . . . . . . 9 ⊢ (𝐵 ∈ ℕ₀ ↔ (𝐵 ∈ ℤ ∧ 0 ≤ 𝐵))
49	47, 48	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
50	7, 49	reexpcld 14159	. . . . . . 7 ⊢ (𝜑 → (2↑𝐵) ∈ ℝ)
51	50	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (2↑𝐵) ∈ ℝ)
52		elfznn 13562	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ (1...𝐵) → 𝑞 ∈ ℕ)
53	52	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ (1...𝐵)) → 𝑞 ∈ ℕ)
54	53	nnzd 12615	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ (1...𝐵)) → 𝑞 ∈ ℤ)
55	54	ex 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝑞 ∈ (1...𝐵) → 𝑞 ∈ ℤ))
56	55	ssrdv 3978	. . . . . . . . 9 ⊢ (𝜑 → (1...𝐵) ⊆ ℤ)
57		fzfid 13970	. . . . . . . . 9 ⊢ (𝜑 → (1...𝐵) ∈ Fin)
58		lcmfcl 16598	. . . . . . . . 9 ⊢ (((1...𝐵) ⊆ ℤ ∧ (1...𝐵) ∈ Fin) → (lcm‘(1...𝐵)) ∈ ℕ₀)
59	56, 57, 58	syl2anc 582	. . . . . . . 8 ⊢ (𝜑 → (lcm‘(1...𝐵)) ∈ ℕ₀)
60	59	nn0red 12563	. . . . . . 7 ⊢ (𝜑 → (lcm‘(1...𝐵)) ∈ ℝ)
61	60	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (lcm‘(1...𝐵)) ∈ ℝ)
62	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)))
63		elnnz 12598	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
64	12, 21, 63	sylanbrc 581	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
65	7, 10, 35, 45, 26	relogbcld 41499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2 log_b 𝐵) ∈ ℝ)
66	65	flcld 13795	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℤ)
67	7, 10, 7, 10, 26	relogbcld 41499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ∈ ℝ)
68		0le1 11767	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 1
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ 1)
70	7	recnd 11272	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ∈ ℂ)
71	14, 10	gtned 11379	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 ≠ 0)
72		logbid1 26718	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 2) = 1)
73	70, 71, 26, 72	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 log_b 2) = 1)
74	73	eqcomd 2731	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 = (2 log_b 2))
75	69, 74	breqtrd 5169	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ (2 log_b 2))
76		2z 12624	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℤ
77	76	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ∈ ℤ)
78	7	leidd 11810	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 2)
79		2lt7 12432	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 < 7
80	79	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2 < 7)
81	7, 37, 80	ltled 11392	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 ≤ 7)
82	37, 30, 34, 40, 43	ltletrd 11404	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 7 < (⌈‘((2 log_b 𝑁)↑5)))
83	82, 8	breqtrrd 5171	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 7 < 𝐵)
84	37, 35, 83	ltled 11392	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 7 ≤ 𝐵)
85	7, 37, 35, 81, 84	letrd 11401	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 2 ≤ 𝐵)
86	77, 78, 7, 10, 35, 45, 85	logblebd 41502	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2 log_b 2) ≤ (2 log_b 𝐵))
87	14, 67, 65, 75, 86	letrd 11401	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (2 log_b 𝐵))
88		0zd 12600	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℤ)
89		flge 13802	. . . . . . . . . . . . . . 15 ⊢ (((2 log_b 𝐵) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
90	65, 88, 89	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0 ≤ (2 log_b 𝐵) ↔ 0 ≤ (⌊‘(2 log_b 𝐵))))
91	87, 90	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (⌊‘(2 log_b 𝐵)))
92	66, 91	jca 510	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
93		elnn0z 12601	. . . . . . . . . . . 12 ⊢ ((⌊‘(2 log_b 𝐵)) ∈ ℕ₀ ↔ ((⌊‘(2 log_b 𝐵)) ∈ ℤ ∧ 0 ≤ (⌊‘(2 log_b 𝐵))))
94	92, 93	sylibr 233	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘(2 log_b 𝐵)) ∈ ℕ₀)
95	64, 94	nnexpcld 14239	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑(⌊‘(2 log_b 𝐵))) ∈ ℕ)
96		fzfid 13970	. . . . . . . . . . 11 ⊢ (𝜑 → (1...(⌊‘((2 log_b 𝑁)↑2))) ∈ Fin)
97	12	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℤ)
98		elfznn 13562	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ ℕ)
99	98	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ)
100	99	nnnn0d 12562	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ ℕ₀)
101		zexpcl 14073	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℤ ∧ 𝑘 ∈ ℕ₀) → (𝑁↑𝑘) ∈ ℤ)
102	97, 100, 101	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℤ)
103		1zzd 12623	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℤ)
104	102, 103	zsubcld 12701	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℤ)
105		1cnd 11239	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℂ)
106	105	addridd 11444	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (1 + 0) = 1)
107	22	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ∈ ℝ)
108		1nn0 12518	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ₀
109	108	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ ℕ₀)
110	13, 109	reexpcld 14159	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁↑1) ∈ ℝ)
111	110	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑1) ∈ ℝ)
112	102	zred 12696	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑𝑘) ∈ ℝ)
113		1lt3 12415	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
114	113	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 1 < 3)
115	22, 16, 13, 114, 20	ltletrd 11404	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 𝑁)
116	13	recnd 11272	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℂ)
117	116	exp1d 14137	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁↑1) = 𝑁)
118	117	eqcomd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 = (𝑁↑1))
119	115, 118	breqtrd 5169	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 < (𝑁↑1))
120	119	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 < (𝑁↑1))
121	13	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑁 ∈ ℝ)
122	64	nnge1d 12290	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ≤ 𝑁)
123	122	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 ≤ 𝑁)
124		elfzuz 13529	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2))) → 𝑘 ∈ (ℤ_≥‘1))
125	124	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 𝑘 ∈ (ℤ_≥‘1))
126	121, 123, 125	leexp2ad 14248	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (𝑁↑1) ≤ (𝑁↑𝑘))
127	107, 111, 112, 120, 126	ltletrd 11404	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 1 < (𝑁↑𝑘))
128	106, 127	eqbrtrd 5165	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (1 + 0) < (𝑁↑𝑘))
129	14	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 0 ∈ ℝ)
130	107, 129, 112	ltaddsub2d 11845	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((1 + 0) < (𝑁↑𝑘) ↔ 0 < ((𝑁↑𝑘) − 1)))
131	128, 130	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → 0 < ((𝑁↑𝑘) − 1))
132	104, 131	jca 510	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → (((𝑁↑𝑘) − 1) ∈ ℤ ∧ 0 < ((𝑁↑𝑘) − 1)))
133		elnnz 12598	. . . . . . . . . . . 12 ⊢ (((𝑁↑𝑘) − 1) ∈ ℕ ↔ (((𝑁↑𝑘) − 1) ∈ ℤ ∧ 0 < ((𝑁↑𝑘) − 1)))
134	132, 133	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))) → ((𝑁↑𝑘) − 1) ∈ ℕ)
135	96, 134	fprodnncl 15931	. . . . . . . . . 10 ⊢ (𝜑 → ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1) ∈ ℕ)
136	95, 135	nnmulcld 12295	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁↑(⌊‘(2 log_b 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 log_b 𝑁)↑2)))((𝑁↑𝑘) − 1)) ∈ ℕ)
137	62, 136	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ)
138	137	nnred 12257	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
139	138	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → 𝐴 ∈ ℝ)
140	1, 2, 3	aks4d1p2 41604	. . . . . . 7 ⊢ (𝜑 → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
141	140	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (2↑𝐵) ≤ (lcm‘(1...𝐵)))
142	137	nnzd 12615	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
143	142	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → 𝐴 ∈ ℤ)
144	56	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (1...𝐵) ⊆ ℤ)
145		fzfid 13970	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (1...𝐵) ∈ Fin)
146		lcmfdvdsb 16613	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (1...𝐵) ⊆ ℤ ∧ (1...𝐵) ∈ Fin) → (∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴 ↔ (lcm‘(1...𝐵)) ∥ 𝐴))
147	143, 144, 145, 146	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴 ↔ (lcm‘(1...𝐵)) ∥ 𝐴))
148	147	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴 → (lcm‘(1...𝐵)) ∥ 𝐴))
149	148	syldbl2 839	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (lcm‘(1...𝐵)) ∥ 𝐴)
150	59	nn0zd 12614	. . . . . . . . 9 ⊢ (𝜑 → (lcm‘(1...𝐵)) ∈ ℤ)
151	150	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (lcm‘(1...𝐵)) ∈ ℤ)
152	137	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → 𝐴 ∈ ℕ)
153		dvdsle 16286	. . . . . . . 8 ⊢ (((lcm‘(1...𝐵)) ∈ ℤ ∧ 𝐴 ∈ ℕ) → ((lcm‘(1...𝐵)) ∥ 𝐴 → (lcm‘(1...𝐵)) ≤ 𝐴))
154	151, 152, 153	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → ((lcm‘(1...𝐵)) ∥ 𝐴 → (lcm‘(1...𝐵)) ≤ 𝐴))
155	149, 154	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (lcm‘(1...𝐵)) ≤ 𝐴)
156	51, 61, 139, 141, 155	letrd 11401	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → (2↑𝐵) ≤ 𝐴)
157	51, 139	lenltd 11390	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → ((2↑𝐵) ≤ 𝐴 ↔ ¬ 𝐴 < (2↑𝐵)))
158	156, 157	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → ¬ 𝐴 < (2↑𝐵))
159	5, 158	pm2.21dd 194	. . 3 ⊢ ((𝜑 ∧ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → ¬ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴)
160		simpr 483	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴) → ¬ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴)
161	159, 160	pm2.61dan 811	. 2 ⊢ (𝜑 → ¬ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴)
162		rexnal 3090	. 2 ⊢ (∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴 ↔ ¬ ∀𝑟 ∈ (1...𝐵)𝑟 ∥ 𝐴)
163	161, 162	sylibr 233	1 ⊢ (𝜑 → ∃𝑟 ∈ (1...𝐵) ¬ 𝑟 ∥ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ⊆ wss 3939 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 < clt 11278 ≤ cle 11279 − cmin 11474 ℕcn 12242 2c2 12297 3c3 12298 5c5 12300 7c7 12302 ℕ₀cn0 12502 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13516 ⌊cfl 13787 ⌈cceil 13788 ↑cexp 14058 ∏cprod 15881 ∥ cdvds 16230 lcmclcmf 16559 log_b clogb 26714

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-symdif 4237 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-ceil 13790 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-prod 15882 df-ef 16043 df-e 16044 df-sin 16045 df-cos 16046 df-pi 16048 df-dvds 16231 df-gcd 16469 df-lcm 16560 df-lcmf 16561 df-prm 16642 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-cmp 23309 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-ovol 25411 df-vol 25412 df-mbf 25566 df-itg1 25567 df-itg2 25568 df-ibl 25569 df-itg 25570 df-0p 25617 df-limc 25813 df-dv 25814 df-log 26508 df-cxp 26509 df-logb 26715

This theorem is referenced by: aks4d1p4 41606 aks4d1p5 41607 aks4d1p7 41610 aks4d1p8 41614

W3C validator