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Theorem chebbnd1lem1 26972

Description: Lemma for chebbnd1 26975: show a lower bound on π(𝑥) at even integers using similar techniques to those used to prove bpos 26796. (Note that the expression 𝐾 is actually equal to 2 · 𝑁, but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 26787, which shows that each term in the expansion ((2 · 𝑁)C𝑁) = ∏𝑝 ∈ ℙ (𝑝↑(𝑝 pCnt ((2 · 𝑁)C𝑁))) is at most 2 · 𝑁, so that the sum really only has nonzero elements up to 2 · 𝑁, and since each term is at most 2 · 𝑁, after taking logs we get the inequality π(2 · 𝑁) · log(2 · 𝑁) ≤ log((2 · 𝑁)C𝑁), and bclbnd 26783 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)

**Hypothesis**
Ref	Expression
chebbnd1lem1.1	⊢ 𝐾 = if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁))

**Assertion**
Ref	Expression
chebbnd1lem1	⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((4↑𝑁) / 𝑁)) < ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))))

Proof of Theorem chebbnd1lem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4nn 12295	. . . . . 6 ⊢ 4 ∈ ℕ
2		eluznn 12902	. . . . . . . 8 ⊢ ((4 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘4)) → 𝑁 ∈ ℕ)
3	1, 2	mpan 689	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℕ)
4	3	nnnn0d 12532	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℕ₀)
5		nnexpcl 14040	. . . . . 6 ⊢ ((4 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (4↑𝑁) ∈ ℕ)
6	1, 4, 5	sylancr 588	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (4↑𝑁) ∈ ℕ)
7	6	nnrpd 13014	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (4↑𝑁) ∈ ℝ⁺)
8	3	nnrpd 13014	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℝ⁺)
9	7, 8	rpdivcld 13033	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((4↑𝑁) / 𝑁) ∈ ℝ⁺)
10	9	relogcld 26131	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((4↑𝑁) / 𝑁)) ∈ ℝ)
11		fzctr 13613	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0...(2 · 𝑁)))
12	4, 11	syl 17	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ (0...(2 · 𝑁)))
13		bccl2 14283	. . . . 5 ⊢ (𝑁 ∈ (0...(2 · 𝑁)) → ((2 · 𝑁)C𝑁) ∈ ℕ)
14	12, 13	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℕ)
15	14	nnrpd 13014	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℝ⁺)
16	15	relogcld 26131	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((2 · 𝑁)C𝑁)) ∈ ℝ)
17		2z 12594	. . . . . . 7 ⊢ 2 ∈ ℤ
18		eluzelz 12832	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℤ)
19		zmulcl 12611	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 · 𝑁) ∈ ℤ)
20	17, 18, 19	sylancr 588	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 · 𝑁) ∈ ℤ)
21	20	zred 12666	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 · 𝑁) ∈ ℝ)
22		ppicl 26635	. . . . 5 ⊢ ((2 · 𝑁) ∈ ℝ → (π‘(2 · 𝑁)) ∈ ℕ₀)
23	21, 22	syl 17	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘(2 · 𝑁)) ∈ ℕ₀)
24	23	nn0red 12533	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘(2 · 𝑁)) ∈ ℝ)
25		2nn 12285	. . . . . 6 ⊢ 2 ∈ ℕ
26		nnmulcl 12236	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 · 𝑁) ∈ ℕ)
27	25, 3, 26	sylancr 588	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 · 𝑁) ∈ ℕ)
28	27	nnrpd 13014	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 · 𝑁) ∈ ℝ⁺)
29	28	relogcld 26131	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘(2 · 𝑁)) ∈ ℝ)
30	24, 29	remulcld 11244	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))) ∈ ℝ)
31		bclbnd 26783	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁))
32		logltb 26108	. . . 4 ⊢ ((((4↑𝑁) / 𝑁) ∈ ℝ⁺ ∧ ((2 · 𝑁)C𝑁) ∈ ℝ⁺) → (((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁) ↔ (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁))))
33	9, 15, 32	syl2anc 585	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁) ↔ (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁))))
34	31, 33	mpbid 231	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((4↑𝑁) / 𝑁)) < (log‘((2 · 𝑁)C𝑁)))
35		chebbnd1lem1.1	. . . . . . . 8 ⊢ 𝐾 = if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁))
36	27, 14	ifcld 4575	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ∈ ℕ)
37	35, 36	eqeltrid 2838	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝐾 ∈ ℕ)
38	37	nnred 12227	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝐾 ∈ ℝ)
39		ppicl 26635	. . . . . 6 ⊢ (𝐾 ∈ ℝ → (π‘𝐾) ∈ ℕ₀)
40	38, 39	syl 17	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘𝐾) ∈ ℕ₀)
41	40	nn0red 12533	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘𝐾) ∈ ℝ)
42	41, 29	remulcld 11244	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) ∈ ℝ)
43		fzfid 13938	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (1...𝐾) ∈ Fin)
44		inss1 4229	. . . . . 6 ⊢ ((1...𝐾) ∩ ℙ) ⊆ (1...𝐾)
45		ssfi 9173	. . . . . 6 ⊢ (((1...𝐾) ∈ Fin ∧ ((1...𝐾) ∩ ℙ) ⊆ (1...𝐾)) → ((1...𝐾) ∩ ℙ) ∈ Fin)
46	43, 44, 45	sylancl 587	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((1...𝐾) ∩ ℙ) ∈ Fin)
47	37	nnzd 12585	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝐾 ∈ ℤ)
48	14	nnzd 12585	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℤ)
49	14	nnred 12227	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((2 · 𝑁)C𝑁) ∈ ℝ)
50		min2 13169	. . . . . . . . . . . 12 ⊢ (((2 · 𝑁) ∈ ℝ ∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ ((2 · 𝑁)C𝑁))
51	21, 49, 50	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ ((2 · 𝑁)C𝑁))
52	35, 51	eqbrtrid 5184	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝐾 ≤ ((2 · 𝑁)C𝑁))
53		eluz2 12828	. . . . . . . . . 10 ⊢ (((2 · 𝑁)C𝑁) ∈ (ℤ_≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ ((2 · 𝑁)C𝑁) ∈ ℤ ∧ 𝐾 ≤ ((2 · 𝑁)C𝑁)))
54	47, 48, 52, 53	syl3anbrc 1344	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((2 · 𝑁)C𝑁) ∈ (ℤ_≥‘𝐾))
55		fzss2 13541	. . . . . . . . 9 ⊢ (((2 · 𝑁)C𝑁) ∈ (ℤ_≥‘𝐾) → (1...𝐾) ⊆ (1...((2 · 𝑁)C𝑁)))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (1...𝐾) ⊆ (1...((2 · 𝑁)C𝑁)))
57	56	ssrind 4236	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((1...𝐾) ∩ ℙ) ⊆ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ))
58	57	sselda 3983	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ))
59		simpr 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ))
60	59	elin1d 4199	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ (1...((2 · 𝑁)C𝑁)))
61		elfznn 13530	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...((2 · 𝑁)C𝑁)) → 𝑘 ∈ ℕ)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℕ)
63	59	elin2d 4200	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℙ)
64	14	adantr 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → ((2 · 𝑁)C𝑁) ∈ ℕ)
65	63, 64	pccld 16783	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀)
66	62, 65	nnexpcld 14208	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℕ)
67	66	nnrpd 13014	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ⁺)
68	67	relogcld 26131	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ)
69	58, 68	syldan 592	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ)
70	29	adantr 482	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(2 · 𝑁)) ∈ ℝ)
71		elinel2 4197	. . . . . . . 8 ⊢ (𝑘 ∈ ((1...𝐾) ∩ ℙ) → 𝑘 ∈ ℙ)
72		bposlem1 26787	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ ℙ) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))
73	3, 71, 72	syl2an 597	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))
74	58, 67	syldan 592	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ⁺)
75	74	reeflogd 26132	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) = (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))
76	28	adantr 482	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (2 · 𝑁) ∈ ℝ⁺)
77	76	reeflogd 26132	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (exp‘(log‘(2 · 𝑁))) = (2 · 𝑁))
78	73, 75, 77	3brtr4d 5181	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 · 𝑁))))
79		efle 16061	. . . . . . 7 ⊢ (((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℝ ∧ (log‘(2 · 𝑁)) ∈ ℝ) → ((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁)) ↔ (exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 · 𝑁)))))
80	69, 70, 79	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → ((log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁)) ↔ (exp‘(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))) ≤ (exp‘(log‘(2 · 𝑁)))))
81	78, 80	mpbird 257	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ (log‘(2 · 𝑁)))
82	46, 69, 70, 81	fsumle 15745	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ≤ Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)))
83	68	recnd 11242	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℂ)
84	58, 83	syldan 592	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...𝐾) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) ∈ ℂ)
85		eldifn 4128	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) → ¬ 𝑘 ∈ ((1...𝐾) ∩ ℙ))
86	85	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ¬ 𝑘 ∈ ((1...𝐾) ∩ ℙ))
87		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)))
88	87	eldifad 3961	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ))
89	88	elin1d 4199	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ (1...((2 · 𝑁)C𝑁)))
90	89, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ ℕ)
91	90	adantrr 716	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ℕ)
92	91	nnred 12227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ℝ)
93	88, 66	syldan 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℕ)
94	93	nnred 12227	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ)
95	94	adantrr 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ∈ ℝ)
96	21	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (2 · 𝑁) ∈ ℝ)
97	91	nncnd 12228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ℂ)
98	97	exp1d 14106	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑1) = 𝑘)
99	91	nnge1d 12260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 1 ≤ 𝑘)
100		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)
101		nnuz 12865	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ_≥‘1)
102	100, 101	eleqtrdi 2844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ (ℤ_≥‘1))
103	92, 99, 102	leexp2ad 14217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑1) ≤ (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))
104	98, 103	eqbrtrrd 5173	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))))
105	3	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑁 ∈ ℕ)
106	88	elin2d 4200	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ ℙ)
107	106	adantrr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ℙ)
108	105, 107, 72	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁))
109	92, 95, 96, 104, 108	letrd 11371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ (2 · 𝑁))
110		elfzle2 13505	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...((2 · 𝑁)C𝑁)) → 𝑘 ≤ ((2 · 𝑁)C𝑁))
111	89, 110	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ≤ ((2 · 𝑁)C𝑁))
112	111	adantrr 716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ ((2 · 𝑁)C𝑁))
113	49	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → ((2 · 𝑁)C𝑁) ∈ ℝ)
114		lemin 13171	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ ℝ ∧ (2 · 𝑁) ∈ ℝ ∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → (𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ↔ (𝑘 ≤ (2 · 𝑁) ∧ 𝑘 ≤ ((2 · 𝑁)C𝑁))))
115	92, 96, 113, 114	syl3anc 1372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ↔ (𝑘 ≤ (2 · 𝑁) ∧ 𝑘 ≤ ((2 · 𝑁)C𝑁))))
116	109, 112, 115	mpbir2and 712	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)))
117	116, 35	breqtrrdi 5191	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ≤ 𝐾)
118	37	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝐾 ∈ ℕ)
119	118	nnzd 12585	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝐾 ∈ ℤ)
120		fznn 13569	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℤ → (𝑘 ∈ (1...𝐾) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝐾)))
121	119, 120	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → (𝑘 ∈ (1...𝐾) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ 𝐾)))
122	91, 117, 121	mpbir2and 712	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ (1...𝐾))
123	122, 107	elind 4195	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ (𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ)) ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)) → 𝑘 ∈ ((1...𝐾) ∩ ℙ))
124	123	expr 458	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ → 𝑘 ∈ ((1...𝐾) ∩ ℙ)))
125	86, 124	mtod 197	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ¬ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ)
126	88, 65	syldan 592	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀)
127		elnn0 12474	. . . . . . . . . . . . 13 ⊢ ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ₀ ↔ ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ ∨ (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0))
128	126, 127	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → ((𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ ∨ (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0))
129	128	ord 863	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (¬ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℕ → (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0))
130	125, 129	mpd 15	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) = 0)
131	130	oveq2d 7425	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) = (𝑘↑0))
132	90	nncnd 12228	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → 𝑘 ∈ ℂ)
133	132	exp0d 14105	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑0) = 1)
134	131, 133	eqtrd 2773	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁))) = 1)
135	134	fveq2d 6896	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = (log‘1))
136		log1 26094	. . . . . . 7 ⊢ (log‘1) = 0
137	135, 136	eqtrdi 2789	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ (((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∖ ((1...𝐾) ∩ ℙ))) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = 0)
138		fzfid 13938	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (1...((2 · 𝑁)C𝑁)) ∈ Fin)
139		inss1 4229	. . . . . . 7 ⊢ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ⊆ (1...((2 · 𝑁)C𝑁))
140		ssfi 9173	. . . . . . 7 ⊢ (((1...((2 · 𝑁)C𝑁)) ∈ Fin ∧ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ⊆ (1...((2 · 𝑁)C𝑁))) → ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∈ Fin)
141	138, 139, 140	sylancl 587	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((1...((2 · 𝑁)C𝑁)) ∩ ℙ) ∈ Fin)
142	57, 84, 137, 141	fsumss 15671	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))))
143	62	nnrpd 13014	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → 𝑘 ∈ ℝ⁺)
144	65	nn0zd 12584	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ)
145		relogexp 26104	. . . . . . 7 ⊢ ((𝑘 ∈ ℝ⁺ ∧ (𝑘 pCnt ((2 · 𝑁)C𝑁)) ∈ ℤ) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = ((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)))
146	143, 144, 145	syl2anc 585	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘4) ∧ 𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)) → (log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = ((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)))
147	146	sumeq2dv 15649	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)))
148		pclogsum 26718	. . . . . 6 ⊢ (((2 · 𝑁)C𝑁) ∈ ℕ → Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)) = (log‘((2 · 𝑁)C𝑁)))
149	14, 148	syl 17	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...((2 · 𝑁)C𝑁)) ∩ ℙ)((𝑘 pCnt ((2 · 𝑁)C𝑁)) · (log‘𝑘)) = (log‘((2 · 𝑁)C𝑁)))
150	142, 147, 149	3eqtrd 2777	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(𝑘↑(𝑘 pCnt ((2 · 𝑁)C𝑁)))) = (log‘((2 · 𝑁)C𝑁)))
151	29	recnd 11242	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘(2 · 𝑁)) ∈ ℂ)
152		fsumconst 15736	. . . . . 6 ⊢ ((((1...𝐾) ∩ ℙ) ∈ Fin ∧ (log‘(2 · 𝑁)) ∈ ℂ) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((♯‘((1...𝐾) ∩ ℙ)) · (log‘(2 · 𝑁))))
153	46, 151, 152	syl2anc 585	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((♯‘((1...𝐾) ∩ ℙ)) · (log‘(2 · 𝑁))))
154		2eluzge1 12878	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘1)
155		ppival2g 26633	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 2 ∈ (ℤ_≥‘1)) → (π‘𝐾) = (♯‘((1...𝐾) ∩ ℙ)))
156	47, 154, 155	sylancl 587	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘𝐾) = (♯‘((1...𝐾) ∩ ℙ)))
157	156	oveq1d 7424	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) = ((♯‘((1...𝐾) ∩ ℙ)) · (log‘(2 · 𝑁))))
158	153, 157	eqtr4d 2776	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → Σ𝑘 ∈ ((1...𝐾) ∩ ℙ)(log‘(2 · 𝑁)) = ((π‘𝐾) · (log‘(2 · 𝑁))))
159	82, 150, 158	3brtr3d 5180	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((2 · 𝑁)C𝑁)) ≤ ((π‘𝐾) · (log‘(2 · 𝑁))))
160		min1 13168	. . . . . . 7 ⊢ (((2 · 𝑁) ∈ ℝ ∧ ((2 · 𝑁)C𝑁) ∈ ℝ) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ (2 · 𝑁))
161	21, 49, 160	syl2anc 585	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → if((2 · 𝑁) ≤ ((2 · 𝑁)C𝑁), (2 · 𝑁), ((2 · 𝑁)C𝑁)) ≤ (2 · 𝑁))
162	35, 161	eqbrtrid 5184	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝐾 ≤ (2 · 𝑁))
163		ppiwordi 26666	. . . . 5 ⊢ ((𝐾 ∈ ℝ ∧ (2 · 𝑁) ∈ ℝ ∧ 𝐾 ≤ (2 · 𝑁)) → (π‘𝐾) ≤ (π‘(2 · 𝑁)))
164	38, 21, 162, 163	syl3anc 1372	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (π‘𝐾) ≤ (π‘(2 · 𝑁)))
165		1red 11215	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 1 ∈ ℝ)
166		2re 12286	. . . . . . . 8 ⊢ 2 ∈ ℝ
167	166	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 2 ∈ ℝ)
168		1lt2 12383	. . . . . . . 8 ⊢ 1 < 2
169	168	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 1 < 2)
170		2t1e2 12375	. . . . . . . 8 ⊢ (2 · 1) = 2
171	3	nnge1d 12260	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 1 ≤ 𝑁)
172		eluzelre 12833	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 𝑁 ∈ ℝ)
173		2pos 12315	. . . . . . . . . . . 12 ⊢ 0 < 2
174	166, 173	pm3.2i 472	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ ∧ 0 < 2)
175	174	a1i 11	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 ∈ ℝ ∧ 0 < 2))
176		lemul2 12067	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (1 ≤ 𝑁 ↔ (2 · 1) ≤ (2 · 𝑁)))
177	165, 172, 175, 176	syl3anc 1372	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (1 ≤ 𝑁 ↔ (2 · 1) ≤ (2 · 𝑁)))
178	171, 177	mpbid 231	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (2 · 1) ≤ (2 · 𝑁))
179	170, 178	eqbrtrrid 5185	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 2 ≤ (2 · 𝑁))
180	165, 167, 21, 169, 179	ltletrd 11374	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘4) → 1 < (2 · 𝑁))
181	21, 180	rplogcld 26137	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘(2 · 𝑁)) ∈ ℝ⁺)
182	41, 24, 181	lemul1d 13059	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((π‘𝐾) ≤ (π‘(2 · 𝑁)) ↔ ((π‘𝐾) · (log‘(2 · 𝑁))) ≤ ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁)))))
183	164, 182	mpbid 231	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘4) → ((π‘𝐾) · (log‘(2 · 𝑁))) ≤ ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))))
184	16, 42, 30, 159, 183	letrd 11371	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((2 · 𝑁)C𝑁)) ≤ ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))))
185	10, 16, 30, 34, 184	ltletrd 11374	1 ⊢ (𝑁 ∈ (ℤ_≥‘4) → (log‘((4↑𝑁) / 𝑁)) < ((π‘(2 · 𝑁)) · (log‘(2 · 𝑁))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ifcif 4529 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 · cmul 11115 < clt 11248 ≤ cle 11249 / cdiv 11871 ℕcn 12212 2c2 12267 4c4 12269 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℝ⁺crp 12974 ...cfz 13484 ↑cexp 14027 Ccbc 14262 ♯chash 14290 Σcsu 15632 expce 16005 ℙcprime 16608 pCnt cpc 16769 logclog 26063 πcppi 26598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-ppi 26604

This theorem is referenced by: chebbnd1lem3 26974

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