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Theorem chtppilimlem1 27805

Description: Lemma for chtppilim 27807. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Hypotheses**
Ref	Expression
chtppilim.1	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
chtppilim.2	⊢ (𝜑 → 𝐴 < 1)
chtppilim.3	⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
chtppilim.4	⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))

**Assertion**
Ref	Expression
chtppilimlem1	⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Proof of Theorem chtppilimlem1 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chtppilim.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
2	1	rpred 13167	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	recnd 11340	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4	3	sqvald 14289	. . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
5	4	oveq1d 7435	. . 3 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))))
6		chtppilim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
7		2re 12420	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8		elicopnf 13579	. . . . . . . . . 10 ⊢ (2 ∈ ℝ → (𝑁 ∈ (2[,)+∞) ↔ (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁)))
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ (𝑁 ∈ (2[,)+∞) ↔ (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁))
10	6, 9	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁))
11	10	simpld 500	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
12		ppicl 27463	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (π‘𝑁) ∈ ℕ₀)
13	11, 12	syl 18	. . . . . 6 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ₀)
14	13	nn0red 12671	. . . . 5 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ)
15	14	recnd 11340	. . . 4 ⊢ (𝜑 → (π‘𝑁) ∈ ℂ)
16		0red 11314	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
17	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
18		2pos 12450	. . . . . . . . 9 ⊢ 0 < 2
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 2)
20	10	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ 𝑁)
21	16, 17, 11, 19, 20	ltletrd 11473	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	11, 21	elrpd 13164	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
23	22	relogcld 26956	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
24	23	recnd 11340	. . . 4 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
25	3, 3, 15, 24	mul4d 11525	. . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
26	5, 25	eqtrd 2795	. 2 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
27	2, 14	remulcld 11342	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) ∈ ℝ)
28	2, 23	remulcld 11342	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ)
29	27, 28	remulcld 11342	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) ∈ ℝ)
30	22, 2	rpcxpcld 27066	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ⁺)
31	30	rpred 13167	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ)
32		ppicl 27463	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
33	31, 32	syl 18	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
34	33	nn0red 12671	. . . . 5 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
35	14, 34	resubcld 11747	. . . 4 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) ∈ ℝ)
36	35, 28	remulcld 11342	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ∈ ℝ)
37		chtcl 27441	. . . 4 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) ∈ ℝ)
38	11, 37	syl 18	. . 3 ⊢ (𝜑 → (θ‘𝑁) ∈ ℝ)
39		1red 11312	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
40		1lt2 12518	. . . . . . . 8 ⊢ 1 < 2
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 2)
42	39, 17, 11, 41, 20	ltletrd 11473	. . . . . 6 ⊢ (𝜑 → 1 < 𝑁)
43	11, 42	rplogcld 26962	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
44	1, 43	rpmulcld 13183	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ⁺)
45	14, 31	resubcld 11747	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ∈ ℝ)
46		ppinncl 27506	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 2 ≤ 𝑁) → (π‘𝑁) ∈ ℕ)
47	10, 46	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ)
48	31, 47	nndivred 12395	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) ∈ ℝ)
49		chtppilim.4	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))
50	48, 39, 2, 49	ltsub13d 11925	. . . . . . 7 ⊢ (𝜑 → 𝐴 < (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
51	31	recnd 11340	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℂ)
52	47	nnrpd 13165	. . . . . . . . . 10 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 13176	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0))
54		divsubdir 12013	. . . . . . . . 9 ⊢ (((π‘𝑁) ∈ ℂ ∧ (𝑁↑_𝑐𝐴) ∈ ℂ ∧ ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0)) → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
55	15, 51, 53, 54	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
56		divid 12007	. . . . . . . . . 10 ⊢ (((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0) → ((π‘𝑁) / (π‘𝑁)) = 1)
57	53, 56	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) / (π‘𝑁)) = 1)
58	57	oveq1d 7435	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
59	55, 58	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
60	50, 59	breqtrrd 5132	. . . . . 6 ⊢ (𝜑 → 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)))
61	2, 45, 52	ltmuldivd 13214	. . . . . 6 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ↔ 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁))))
62	60, 61	mpbird 260	. . . . 5 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)))
63		ppiltx 27509	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ⁺ → (π‘(𝑁↑_𝑐𝐴)) < (𝑁↑_𝑐𝐴))
64	30, 63	syl 18	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) < (𝑁↑_𝑐𝐴))
65	34, 31, 14, 64	ltsub2dd 11932	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
66	27, 45, 35, 62, 65	lttrd 11474	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
67	27, 35, 44, 66	ltmul1dd 13222	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
68		fzfid 14119	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin)
69		inss1 4133	. . . . . 6 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))
70		ssfi 9191	. . . . . 6 ⊢ (((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin ∧ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))) → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
71	68, 69, 70	sylancl 598	. . . . 5 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
72		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ))
73	72	elin2d 4102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
74		prmnn 16854	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
75	74	nnrpd 13165	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ⁺)
76	73, 75	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
77	76	relogcld 26956	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
78	71, 77	fsumrecl 15905	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ∈ ℝ)
79	28	recnd 11340	. . . . . . 7 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℂ)
80		fsumconst 15961	. . . . . . 7 ⊢ ((((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin ∧ (𝐴 · (log‘𝑁)) ∈ ℂ) → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
81	71, 79, 80	syl2anc 596	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
82		ppifl 27492	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (π‘(⌊‘𝑁)) = (π‘𝑁))
83	11, 82	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘𝑁)) = (π‘𝑁))
84		ppifl 27492	. . . . . . . . . 10 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
85	31, 84	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
86	83, 85	oveq12d 7438	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
87	39, 11, 42	ltled 11461	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
88		chtppilim.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 1)
89		1re 11311	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
90		ltle 11401	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 → 𝐴 ≤ 1))
91	2, 89, 90	sylancl 598	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 1 → 𝐴 ≤ 1))
92	88, 91	mpd 16	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 1)
93	11, 87, 2, 39, 92	cxplead 27054	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ (𝑁↑_𝑐1))
94	11	recnd 11340	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
95	94	cxp1d 27039	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐1) = 𝑁)
96	93, 95	breqtrd 5130	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ 𝑁)
97		flword2 13956	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁↑_𝑐𝐴) ≤ 𝑁) → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
98	31, 11, 96, 97	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
99		ppidif 27495	. . . . . . . . 9 ⊢ ((⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))) → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
100	98, 99	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
101	86, 100	eqtr3d 2797	. . . . . . 7 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
102	101	oveq1d 7435	. . . . . 6 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
103	81, 102	eqtr4d 2798	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
104	28	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ∈ ℝ)
105	31	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ∈ ℝ)
106		reflcl 13939	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
107		peano2re 11486	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
108	31, 106, 107	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
109	108	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
110	76	rpred 13167	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
111		fllep1 13944	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
112	31, 111	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
113	112	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
114	72	elin1d 4101	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)))
115		elfzle1 13663	. . . . . . . . . 10 ⊢ (𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
116	114, 115	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
117	105, 109, 110, 113, 116	letrd 11470	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ 𝑝)
118	22	rpne0d 13172	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
119	94, 118, 3	cxpefd 27045	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑁))))
120	119	eqcomd 2766	. . . . . . . . 9 ⊢ (𝜑 → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
121	120	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
122	76	reeflogd 26957	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
123	117, 121, 122	3brtr4d 5136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝)))
124		efle 16291	. . . . . . . 8 ⊢ (((𝐴 · (log‘𝑁)) ∈ ℝ ∧ (log‘𝑝) ∈ ℝ) → ((𝐴 · (log‘𝑁)) ≤ (log‘𝑝) ↔ (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝))))
125	104, 77, 124	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((𝐴 · (log‘𝑁)) ≤ (log‘𝑝) ↔ (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝))))
126	123, 125	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ≤ (log‘𝑝))
127	71, 104, 77, 126	fsumle 15971	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
128	103, 127	eqbrtrrd 5128	. . . 4 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
129		fzfid 14119	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘𝑁)) ∈ Fin)
130		inss1 4133	. . . . . . 7 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))
131		ssfi 9191	. . . . . . 7 ⊢ (((1...(⌊‘𝑁)) ∈ Fin ∧ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))) → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
132	129, 130, 131	sylancl 598	. . . . . 6 ⊢ (𝜑 → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
133		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ))
134	133	elin2d 4102	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
135		prmuz2 16876	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
136	134, 135	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
137		eluz2b2 13051	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
138	136, 137	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (𝑝 ∈ ℕ ∧ 1 < 𝑝))
139	138	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℕ)
140	139	nnred 12353	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
141	138	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 1 < 𝑝)
142	140, 141	rplogcld 26962	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
143	142	rpred 13167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
144	142	rpge0d 13171	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
145	30	rpge0d 13171	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑁↑_𝑐𝐴))
146		flge0nn0 13963	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 0 ≤ (𝑁↑_𝑐𝐴)) → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
147	31, 145, 146	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
148		nn0p1nn 12648	. . . . . . . . 9 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
149	147, 148	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
150		nnuz 13007	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
151	149, 150	eleqtrdi 2870	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1))
152		fzss1 13700	. . . . . . 7 ⊢ (((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)))
153		ssrin 4138	. . . . . . 7 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)) → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
154	151, 152, 153	3syl 19	. . . . . 6 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
155	132, 143, 144, 154	fsumless 15968	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
156		chtval 27442	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
157	11, 156	syl 18	. . . . . 6 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
158		2eluzge1 13012	. . . . . . . 8 ⊢ 2 ∈ (ℤ_≥‘1)
159		ppisval2 27437	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ (ℤ_≥‘1)) → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
160	11, 158, 159	sylancl 598	. . . . . . 7 ⊢ (𝜑 → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
161	160	sumeq1d 15872	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
162	157, 161	eqtrd 2795	. . . . 5 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
163	155, 162	breqtrrd 5132	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ (θ‘𝑁))
164	36, 78, 38, 128, 163	letrd 11470	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ (θ‘𝑁))
165	29, 36, 38, 67, 164	ltletrd 11473	. 2 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (θ‘𝑁))
166	26, 165	eqbrtrd 5126	1 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ⊆ wss 3895 ∩ cin 4089 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 Fincfn 8976 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 + caddc 11206 · cmul 11208 +∞cpnf 11343 < clt 11346 ≤ cle 11347 − cmin 11544 / cdiv 11976 ℕcn 12338 2c2 12400 ℕ₀cn0 12609 ℤ_≥cuz 12968 ℝ⁺crp 13123 [,)cico 13481 [,]cicc 13482 ...cfz 13642 ⌊cfl 13933 ↑cexp 14207 ♯chash 14477 Σcsu 15858 expce 16232 ℙcprime 16851 logclog 26887 ↑_𝑐ccxp 26888 θccht 27423 πcppi 27426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-oadd 8483 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-xnn0 12683 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-sum 15859 df-ef 16238 df-sin 16240 df-cos 16241 df-pi 16243 df-dvds 16428 df-prm 16852 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-xms 24644 df-ms 24645 df-tms 24646 df-cncf 25204 df-limc 26191 df-dv 26192 df-log 26889 df-cxp 26890 df-cht 27429 df-ppi 27432

This theorem is used by: chtppilimlem2 27806

W3C validator