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Theorem chtppilimlem1 26057

Description: Lemma for chtppilim 26059. (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 12419	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	recnd 10658	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4	3	sqvald 13503	. . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
5	4	oveq1d 7150	. . 3 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))))
6		chtppilim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
7		2re 11699	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8		elicopnf 12823	. . . . . . . . . 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 498	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
12		ppicl 25716	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (π‘𝑁) ∈ ℕ₀)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ₀)
14	13	nn0red 11944	. . . . 5 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ)
15	14	recnd 10658	. . . 4 ⊢ (𝜑 → (π‘𝑁) ∈ ℂ)
16		0red 10633	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
17	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
18		2pos 11728	. . . . . . . . 9 ⊢ 0 < 2
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 2)
20	10	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ 𝑁)
21	16, 17, 11, 19, 20	ltletrd 10789	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	11, 21	elrpd 12416	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
23	22	relogcld 25214	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
24	23	recnd 10658	. . . 4 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
25	3, 3, 15, 24	mul4d 10841	. . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
26	5, 25	eqtrd 2833	. 2 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
27	2, 14	remulcld 10660	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) ∈ ℝ)
28	2, 23	remulcld 10660	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ)
29	27, 28	remulcld 10660	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) ∈ ℝ)
30	22, 2	rpcxpcld 25323	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ⁺)
31	30	rpred 12419	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ)
32		ppicl 25716	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
34	33	nn0red 11944	. . . . 5 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
35	14, 34	resubcld 11057	. . . 4 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) ∈ ℝ)
36	35, 28	remulcld 10660	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ∈ ℝ)
37		chtcl 25694	. . . 4 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) ∈ ℝ)
38	11, 37	syl 17	. . 3 ⊢ (𝜑 → (θ‘𝑁) ∈ ℝ)
39		1red 10631	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
40		1lt2 11796	. . . . . . . 8 ⊢ 1 < 2
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 2)
42	39, 17, 11, 41, 20	ltletrd 10789	. . . . . 6 ⊢ (𝜑 → 1 < 𝑁)
43	11, 42	rplogcld 25220	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
44	1, 43	rpmulcld 12435	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ⁺)
45	14, 31	resubcld 11057	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ∈ ℝ)
46		ppinncl 25759	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 2 ≤ 𝑁) → (π‘𝑁) ∈ ℕ)
47	10, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ)
48	31, 47	nndivred 11679	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) ∈ ℝ)
49		chtppilim.4	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))
50	48, 39, 2, 49	ltsub13d 11235	. . . . . . 7 ⊢ (𝜑 → 𝐴 < (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
51	31	recnd 10658	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℂ)
52	47	nnrpd 12417	. . . . . . . . . 10 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 12428	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0))
54		divsubdir 11323	. . . . . . . . 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 1368	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
56		divid 11316	. . . . . . . . . 10 ⊢ (((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0) → ((π‘𝑁) / (π‘𝑁)) = 1)
57	53, 56	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) / (π‘𝑁)) = 1)
58	57	oveq1d 7150	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
59	55, 58	eqtrd 2833	. . . . . . 7 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
60	50, 59	breqtrrd 5058	. . . . . 6 ⊢ (𝜑 → 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)))
61	2, 45, 52	ltmuldivd 12466	. . . . . 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 25762	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ⁺ → (π‘(𝑁↑_𝑐𝐴)) < (𝑁↑_𝑐𝐴))
64	30, 63	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) < (𝑁↑_𝑐𝐴))
65	34, 31, 14, 64	ltsub2dd 11242	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
66	27, 45, 35, 62, 65	lttrd 10790	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
67	27, 35, 44, 66	ltmul1dd 12474	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
68		fzfid 13336	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin)
69		inss1 4155	. . . . . 6 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))
70		ssfi 8722	. . . . . 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 589	. . . . 5 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
72		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ))
73	72	elin2d 4126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
74		prmnn 16008	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
75	74	nnrpd 12417	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ⁺)
76	73, 75	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
77	76	relogcld 25214	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
78	71, 77	fsumrecl 15083	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ∈ ℝ)
79	28	recnd 10658	. . . . . . 7 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℂ)
80		fsumconst 15137	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
82		ppifl 25745	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (π‘(⌊‘𝑁)) = (π‘𝑁))
83	11, 82	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘𝑁)) = (π‘𝑁))
84		ppifl 25745	. . . . . . . . . 10 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
85	31, 84	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
86	83, 85	oveq12d 7153	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
87	39, 11, 42	ltled 10777	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
88		chtppilim.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 1)
89		1re 10630	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
90		ltle 10718	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 → 𝐴 ≤ 1))
91	2, 89, 90	sylancl 589	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 1 → 𝐴 ≤ 1))
92	88, 91	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 1)
93	11, 87, 2, 39, 92	cxplead 25312	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ (𝑁↑_𝑐1))
94	11	recnd 10658	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
95	94	cxp1d 25297	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐1) = 𝑁)
96	93, 95	breqtrd 5056	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ 𝑁)
97		flword2 13178	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁↑_𝑐𝐴) ≤ 𝑁) → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
98	31, 11, 96, 97	syl3anc 1368	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
99		ppidif 25748	. . . . . . . . 9 ⊢ ((⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))) → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
100	98, 99	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
101	86, 100	eqtr3d 2835	. . . . . . 7 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
102	101	oveq1d 7150	. . . . . 6 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
103	81, 102	eqtr4d 2836	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
104	28	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ∈ ℝ)
105	31	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ∈ ℝ)
106		reflcl 13161	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
107		peano2re 10802	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
108	31, 106, 107	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
109	108	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
110	76	rpred 12419	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
111		fllep1 13166	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
112	31, 111	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
113	112	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
114	72	elin1d 4125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)))
115		elfzle1 12905	. . . . . . . . . 10 ⊢ (𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
116	114, 115	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
117	105, 109, 110, 113, 116	letrd 10786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ 𝑝)
118	22	rpne0d 12424	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
119	94, 118, 3	cxpefd 25303	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑁))))
120	119	eqcomd 2804	. . . . . . . . 9 ⊢ (𝜑 → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
121	120	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
122	76	reeflogd 25215	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
123	117, 121, 122	3brtr4d 5062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝)))
124		efle 15463	. . . . . . . 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 587	. . . . . . 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 15146	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
128	103, 127	eqbrtrrd 5054	. . . 4 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
129		fzfid 13336	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘𝑁)) ∈ Fin)
130		inss1 4155	. . . . . . 7 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))
131		ssfi 8722	. . . . . . 7 ⊢ (((1...(⌊‘𝑁)) ∈ Fin ∧ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))) → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
132	129, 130, 131	sylancl 589	. . . . . 6 ⊢ (𝜑 → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
133		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ))
134	133	elin2d 4126	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
135		prmuz2 16030	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
137		eluz2b2 12309	. . . . . . . . . . 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 498	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℕ)
140	139	nnred 11640	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
141	138	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 1 < 𝑝)
142	140, 141	rplogcld 25220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
143	142	rpred 12419	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
144	142	rpge0d 12423	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
145	30	rpge0d 12423	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑁↑_𝑐𝐴))
146		flge0nn0 13185	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 0 ≤ (𝑁↑_𝑐𝐴)) → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
147	31, 145, 146	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
148		nn0p1nn 11924	. . . . . . . . 9 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
149	147, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
150		nnuz 12269	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
151	149, 150	eleqtrdi 2900	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1))
152		fzss1 12941	. . . . . . 7 ⊢ (((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)))
153		ssrin 4160	. . . . . . 7 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)) → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
154	151, 152, 153	3syl 18	. . . . . 6 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
155	132, 143, 144, 154	fsumless 15143	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
156		chtval 25695	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
157	11, 156	syl 17	. . . . . 6 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
158		2eluzge1 12282	. . . . . . . 8 ⊢ 2 ∈ (ℤ_≥‘1)
159		ppisval2 25690	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ (ℤ_≥‘1)) → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
160	11, 158, 159	sylancl 589	. . . . . . 7 ⊢ (𝜑 → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
161	160	sumeq1d 15050	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
162	157, 161	eqtrd 2833	. . . . 5 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
163	155, 162	breqtrrd 5058	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ (θ‘𝑁))
164	36, 78, 38, 128, 163	letrd 10786	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ (θ‘𝑁))
165	29, 36, 38, 67, 164	ltletrd 10789	. 2 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (θ‘𝑁))
166	26, 165	eqbrtrd 5052	1 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 +∞cpnf 10661 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤ_≥cuz 12231 ℝ⁺crp 12377 [,)cico 12728 [,]cicc 12729 ...cfz 12885 ⌊cfl 13155 ↑cexp 13425 ♯chash 13686 Σcsu 15034 expce 15407 ℙcprime 16005 logclog 25146 ↑_𝑐ccxp 25147 θccht 25676 πcppi 25679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-dvds 15600 df-prm 16006 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 df-cht 25682 df-ppi 25685

This theorem is referenced by: chtppilimlem2 26058

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