chtppilimlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chtppilimlem1		Structured version Visualization version GIF version

Theorem chtppilimlem1 25162

Description: Lemma for chtppilim 25164. (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 11872	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	recnd 10068	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4	3	sqvald 13005	. . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
5	4	oveq1d 6665	. . 3 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))))
6		chtppilim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
7		2re 11090	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8		elicopnf 12269	. . . . . . . . . 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 208	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁))
11	10	simpld 475	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
12		ppicl 24857	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (π‘𝑁) ∈ ℕ₀)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ₀)
14	13	nn0red 11352	. . . . 5 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ)
15	14	recnd 10068	. . . 4 ⊢ (𝜑 → (π‘𝑁) ∈ ℂ)
16		0red 10041	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
17	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
18		2pos 11112	. . . . . . . . 9 ⊢ 0 < 2
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 2)
20	10	simprd 479	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ 𝑁)
21	16, 17, 11, 19, 20	ltletrd 10197	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	11, 21	elrpd 11869	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
23	22	relogcld 24369	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
24	23	recnd 10068	. . . 4 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
25	3, 3, 15, 24	mul4d 10248	. . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
26	5, 25	eqtrd 2656	. 2 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
27	2, 14	remulcld 10070	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) ∈ ℝ)
28	2, 23	remulcld 10070	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ)
29	27, 28	remulcld 10070	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) ∈ ℝ)
30	22, 2	rpcxpcld 24476	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ⁺)
31	30	rpred 11872	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ)
32		ppicl 24857	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
34	33	nn0red 11352	. . . . 5 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
35	14, 34	resubcld 10458	. . . 4 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) ∈ ℝ)
36	35, 28	remulcld 10070	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ∈ ℝ)
37		chtcl 24835	. . . 4 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) ∈ ℝ)
38	11, 37	syl 17	. . 3 ⊢ (𝜑 → (θ‘𝑁) ∈ ℝ)
39		1red 10055	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
40		1lt2 11194	. . . . . . . 8 ⊢ 1 < 2
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 2)
42	39, 17, 11, 41, 20	ltletrd 10197	. . . . . 6 ⊢ (𝜑 → 1 < 𝑁)
43	11, 42	rplogcld 24375	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
44	1, 43	rpmulcld 11888	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ⁺)
45	14, 31	resubcld 10458	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ∈ ℝ)
46		ppinncl 24900	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 2 ≤ 𝑁) → (π‘𝑁) ∈ ℕ)
47	10, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ)
48	31, 47	nndivred 11069	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) ∈ ℝ)
49		chtppilim.4	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))
50	48, 39, 2, 49	ltsub13d 10633	. . . . . . 7 ⊢ (𝜑 → 𝐴 < (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
51	31	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℂ)
52	47	nnrpd 11870	. . . . . . . . . 10 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 11881	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0))
54		divsubdir 10721	. . . . . . . . 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 1326	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
56		divid 10714	. . . . . . . . . 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 6665	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
59	55, 58	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
60	50, 59	breqtrrd 4681	. . . . . 6 ⊢ (𝜑 → 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)))
61	2, 45, 52	ltmuldivd 11919	. . . . . 6 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ↔ 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁))))
62	60, 61	mpbird 247	. . . . 5 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)))
63		ppiltx 24903	. . . . . . 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 10640	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
66	27, 45, 35, 62, 65	lttrd 10198	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
67	27, 35, 44, 66	ltmul1dd 11927	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
68		fzfid 12772	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin)
69		inss1 3833	. . . . . 6 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))
70		ssfi 8180	. . . . . 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 694	. . . . 5 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
72		inss2 3834	. . . . . . . 8 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ℙ
73		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ))
74	72, 73	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
75		prmnn 15388	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
76	75	nnrpd 11870	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ⁺)
77	74, 76	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
78	77	relogcld 24369	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
79	71, 78	fsumrecl 14465	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ∈ ℝ)
80	28	recnd 10068	. . . . . . 7 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℂ)
81		fsumconst 14522	. . . . . . 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))))
82	71, 80, 81	syl2anc 693	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((#‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
83		ppifl 24886	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (π‘(⌊‘𝑁)) = (π‘𝑁))
84	11, 83	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘𝑁)) = (π‘𝑁))
85		ppifl 24886	. . . . . . . . . 10 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
86	31, 85	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘(𝑁↑_𝑐𝐴))) = (π‘(𝑁↑_𝑐𝐴)))
87	84, 86	oveq12d 6668	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
88	39, 11, 42	ltled 10185	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
89		chtppilim.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 1)
90		1re 10039	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
91		ltle 10126	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 → 𝐴 ≤ 1))
92	2, 90, 91	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 1 → 𝐴 ≤ 1))
93	89, 92	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 1)
94	11, 88, 2, 39, 93	cxplead 24467	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ (𝑁↑_𝑐1))
95	11	recnd 10068	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
96	95	cxp1d 24452	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐1) = 𝑁)
97	94, 96	breqtrd 4679	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ 𝑁)
98		flword2 12614	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁↑_𝑐𝐴) ≤ 𝑁) → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
99	31, 11, 97, 98	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
100		ppidif 24889	. . . . . . . . 9 ⊢ ((⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))) → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (#‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
101	99, 100	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = (#‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
102	87, 101	eqtr3d 2658	. . . . . . 7 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) = (#‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
103	102	oveq1d 6665	. . . . . 6 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) = ((#‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
104	82, 103	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
105	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ∈ ℝ)
106	31	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ∈ ℝ)
107		reflcl 12597	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
108		peano2re 10209	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
109	31, 107, 108	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
110	109	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
111	77	rpred 11872	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
112		fllep1 12602	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
113	31, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
114	113	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
115	69, 73	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)))
116		elfzle1 12344	. . . . . . . . . 10 ⊢ (𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
117	115, 116	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ≤ 𝑝)
118	106, 110, 111, 114, 117	letrd 10194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ 𝑝)
119	22	rpne0d 11877	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
120	95, 119, 3	cxpefd 24458	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑁))))
121	120	eqcomd 2628	. . . . . . . . 9 ⊢ (𝜑 → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
122	121	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
123	77	reeflogd 24370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
124	118, 122, 123	3brtr4d 4685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝)))
125		efle 14848	. . . . . . . 8 ⊢ (((𝐴 · (log‘𝑁)) ∈ ℝ ∧ (log‘𝑝) ∈ ℝ) → ((𝐴 · (log‘𝑁)) ≤ (log‘𝑝) ↔ (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝))))
126	105, 78, 125	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((𝐴 · (log‘𝑁)) ≤ (log‘𝑝) ↔ (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝))))
127	124, 126	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ≤ (log‘𝑝))
128	71, 105, 78, 127	fsumle 14531	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
129	104, 128	eqbrtrrd 4677	. . . 4 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
130		fzfid 12772	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘𝑁)) ∈ Fin)
131		inss1 3833	. . . . . . 7 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))
132		ssfi 8180	. . . . . . 7 ⊢ (((1...(⌊‘𝑁)) ∈ Fin ∧ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))) → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
133	130, 131, 132	sylancl 694	. . . . . 6 ⊢ (𝜑 → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
134		inss2 3834	. . . . . . . . . . . . 13 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ ℙ
135		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ))
136	134, 135	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
137		prmuz2 15408	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
138	136, 137	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
139		eluz2b2 11761	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
140	138, 139	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (𝑝 ∈ ℕ ∧ 1 < 𝑝))
141	140	simpld 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℕ)
142	141	nnred 11035	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
143	140	simprd 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 1 < 𝑝)
144	142, 143	rplogcld 24375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
145	144	rpred 11872	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
146	144	rpge0d 11876	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
147	30	rpge0d 11876	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑁↑_𝑐𝐴))
148		flge0nn0 12621	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 0 ≤ (𝑁↑_𝑐𝐴)) → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
149	31, 147, 148	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
150		nn0p1nn 11332	. . . . . . . . 9 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
151	149, 150	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
152		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
153	151, 152	syl6eleq 2711	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1))
154		fzss1 12380	. . . . . . 7 ⊢ (((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)))
155		ssrin 3838	. . . . . . 7 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)) → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
156	153, 154, 155	3syl 18	. . . . . 6 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ ((1...(⌊‘𝑁)) ∩ ℙ))
157	133, 145, 146, 156	fsumless 14528	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
158		chtval 24836	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
159	11, 158	syl 17	. . . . . 6 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
160		2eluzge1 11734	. . . . . . . 8 ⊢ 2 ∈ (ℤ_≥‘1)
161		ppisval2 24831	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ (ℤ_≥‘1)) → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
162	11, 160, 161	sylancl 694	. . . . . . 7 ⊢ (𝜑 → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
163	162	sumeq1d 14431	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
164	159, 163	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
165	157, 164	breqtrrd 4681	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ (θ‘𝑁))
166	36, 79, 38, 129, 165	letrd 10194	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ (θ‘𝑁))
167	29, 36, 38, 67, 166	ltletrd 10197	. 2 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (θ‘𝑁))
168	26, 167	eqbrtrd 4675	1 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤ_≥cuz 11687 ℝ⁺crp 11832 [,)cico 12177 [,]cicc 12178 ...cfz 12326 ⌊cfl 12591 ↑cexp 12860 #chash 13117 Σcsu 14416 expce 14792 ℙcprime 15385 logclog 24301 ↑_𝑐ccxp 24302 θccht 24817 πcppi 24820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-prm 15386 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-cht 24823 df-ppi 24826

This theorem is referenced by: chtppilimlem2 25163

W3C validator