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 27412

Description: Lemma for chtppilim 27414. (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 12936	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	recnd 11147	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4	3	sqvald 14052	. . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
5	4	oveq1d 7367	. . 3 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))))
6		chtppilim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
7		2re 12206	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8		elicopnf 13347	. . . . . . . . . 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 218	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁))
11	10	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
12		ppicl 27069	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (π‘𝑁) ∈ ℕ₀)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ₀)
14	13	nn0red 12450	. . . . 5 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ)
15	14	recnd 11147	. . . 4 ⊢ (𝜑 → (π‘𝑁) ∈ ℂ)
16		0red 11122	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
17	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
18		2pos 12235	. . . . . . . . 9 ⊢ 0 < 2
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 2)
20	10	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ 𝑁)
21	16, 17, 11, 19, 20	ltletrd 11280	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	11, 21	elrpd 12933	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
23	22	relogcld 26560	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
24	23	recnd 11147	. . . 4 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
25	3, 3, 15, 24	mul4d 11332	. . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
26	5, 25	eqtrd 2768	. 2 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
27	2, 14	remulcld 11149	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) ∈ ℝ)
28	2, 23	remulcld 11149	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ)
29	27, 28	remulcld 11149	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) ∈ ℝ)
30	22, 2	rpcxpcld 26670	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ⁺)
31	30	rpred 12936	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ)
32		ppicl 27069	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
34	33	nn0red 12450	. . . . 5 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
35	14, 34	resubcld 11552	. . . 4 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) ∈ ℝ)
36	35, 28	remulcld 11149	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ∈ ℝ)
37		chtcl 27047	. . . 4 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) ∈ ℝ)
38	11, 37	syl 17	. . 3 ⊢ (𝜑 → (θ‘𝑁) ∈ ℝ)
39		1red 11120	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
40		1lt2 12298	. . . . . . . 8 ⊢ 1 < 2
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 2)
42	39, 17, 11, 41, 20	ltletrd 11280	. . . . . 6 ⊢ (𝜑 → 1 < 𝑁)
43	11, 42	rplogcld 26566	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
44	1, 43	rpmulcld 12952	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ⁺)
45	14, 31	resubcld 11552	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ∈ ℝ)
46		ppinncl 27112	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 2 ≤ 𝑁) → (π‘𝑁) ∈ ℕ)
47	10, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ)
48	31, 47	nndivred 12186	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) ∈ ℝ)
49		chtppilim.4	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))
50	48, 39, 2, 49	ltsub13d 11730	. . . . . . 7 ⊢ (𝜑 → 𝐴 < (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
51	31	recnd 11147	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℂ)
52	47	nnrpd 12934	. . . . . . . . . 10 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 12945	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0))
54		divsubdir 11822	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
56		divid 11814	. . . . . . . . . 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 7367	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
59	55, 58	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
60	50, 59	breqtrrd 5121	. . . . . 6 ⊢ (𝜑 → 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)))
61	2, 45, 52	ltmuldivd 12983	. . . . . 6 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ↔ 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁))))
62	60, 61	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (𝑁↑_𝑐𝐴)))
63		ppiltx 27115	. . . . . . 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 11737	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
66	27, 45, 35, 62, 65	lttrd 11281	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
67	27, 35, 44, 66	ltmul1dd 12991	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
68		fzfid 13882	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin)
69		inss1 4186	. . . . . 6 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))
70		ssfi 9089	. . . . . 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 586	. . . . 5 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
72		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ))
73	72	elin2d 4154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
74		prmnn 16587	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
75	74	nnrpd 12934	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ⁺)
76	73, 75	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
77	76	relogcld 26560	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
78	71, 77	fsumrecl 15643	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ∈ ℝ)
79	28	recnd 11147	. . . . . . 7 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℂ)
80		fsumconst 15699	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
82		ppifl 27098	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (π‘(⌊‘𝑁)) = (π‘𝑁))
83	11, 82	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘𝑁)) = (π‘𝑁))
84		ppifl 27098	. . . . . . . . . 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 7370	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
87	39, 11, 42	ltled 11268	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
88		chtppilim.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 1)
89		1re 11119	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
90		ltle 11208	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 → 𝐴 ≤ 1))
91	2, 89, 90	sylancl 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 1 → 𝐴 ≤ 1))
92	88, 91	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 1)
93	11, 87, 2, 39, 92	cxplead 26658	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ (𝑁↑_𝑐1))
94	11	recnd 11147	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
95	94	cxp1d 26643	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐1) = 𝑁)
96	93, 95	breqtrd 5119	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ 𝑁)
97		flword2 13719	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁↑_𝑐𝐴) ≤ 𝑁) → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
98	31, 11, 96, 97	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
99		ppidif 27101	. . . . . . . . 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 2770	. . . . . . 7 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
102	101	oveq1d 7367	. . . . . 6 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
103	81, 102	eqtr4d 2771	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
104	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ∈ ℝ)
105	31	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ∈ ℝ)
106		reflcl 13702	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
107		peano2re 11293	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
108	31, 106, 107	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
109	108	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
110	76	rpred 12936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
111		fllep1 13707	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
112	31, 111	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
113	112	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
114	72	elin1d 4153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)))
115		elfzle1 13429	. . . . . . . . . 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 11277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ 𝑝)
118	22	rpne0d 12941	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
119	94, 118, 3	cxpefd 26649	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑁))))
120	119	eqcomd 2739	. . . . . . . . 9 ⊢ (𝜑 → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
121	120	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
122	76	reeflogd 26561	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
123	117, 121, 122	3brtr4d 5125	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝)))
124		efle 16029	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((𝐴 · (log‘𝑁)) ≤ (log‘𝑝) ↔ (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝))))
126	123, 125	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ≤ (log‘𝑝))
127	71, 104, 77, 126	fsumle 15708	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
128	103, 127	eqbrtrrd 5117	. . . 4 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
129		fzfid 13882	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘𝑁)) ∈ Fin)
130		inss1 4186	. . . . . . 7 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))
131		ssfi 9089	. . . . . . 7 ⊢ (((1...(⌊‘𝑁)) ∈ Fin ∧ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))) → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
132	129, 130, 131	sylancl 586	. . . . . 6 ⊢ (𝜑 → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
133		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ))
134	133	elin2d 4154	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
135		prmuz2 16609	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
137		eluz2b2 12821	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
138	136, 137	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (𝑝 ∈ ℕ ∧ 1 < 𝑝))
139	138	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℕ)
140	139	nnred 12147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
141	138	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 1 < 𝑝)
142	140, 141	rplogcld 26566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
143	142	rpred 12936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
144	142	rpge0d 12940	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
145	30	rpge0d 12940	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑁↑_𝑐𝐴))
146		flge0nn0 13726	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 0 ≤ (𝑁↑_𝑐𝐴)) → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
147	31, 145, 146	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
148		nn0p1nn 12427	. . . . . . . . 9 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
149	147, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
150		nnuz 12777	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
151	149, 150	eleqtrdi 2843	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1))
152		fzss1 13465	. . . . . . 7 ⊢ (((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)))
153		ssrin 4191	. . . . . . 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 15705	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
156		chtval 27048	. . . . . . 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 12782	. . . . . . . 8 ⊢ 2 ∈ (ℤ_≥‘1)
159		ppisval2 27043	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ (ℤ_≥‘1)) → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
160	11, 158, 159	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
161	160	sumeq1d 15609	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
162	157, 161	eqtrd 2768	. . . . 5 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
163	155, 162	breqtrrd 5121	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ (θ‘𝑁))
164	36, 78, 38, 128, 163	letrd 11277	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ (θ‘𝑁))
165	29, 36, 38, 67, 164	ltletrd 11280	. 2 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (θ‘𝑁))
166	26, 165	eqbrtrd 5115	1 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∩ cin 3897 ⊆ wss 3898 class class class wbr 5093 ‘cfv 6486 (class class class)co 7352 Fincfn 8875 ℂcc 11011 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 +∞cpnf 11150 < clt 11153 ≤ cle 11154 − cmin 11351 / cdiv 11781 ℕcn 12132 2c2 12187 ℕ₀cn0 12388 ℤ_≥cuz 12738 ℝ⁺crp 12892 [,)cico 13249 [,]cicc 13250 ...cfz 13409 ⌊cfl 13696 ↑cexp 13970 ♯chash 14239 Σcsu 15595 expce 15970 ℙcprime 16584 logclog 26491 ↑_𝑐ccxp 26492 θccht 27029 πcppi 27032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-dju 9801 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-xnn0 12462 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ioc 13252 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-fl 13698 df-mod 13776 df-seq 13911 df-exp 13971 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14976 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-limsup 15380 df-clim 15397 df-rlim 15398 df-sum 15596 df-ef 15976 df-sin 15978 df-cos 15979 df-pi 15981 df-dvds 16166 df-prm 16585 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-fbas 21290 df-fg 21291 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-ntr 22936 df-cls 22937 df-nei 23014 df-lp 23052 df-perf 23053 df-cn 23143 df-cnp 23144 df-haus 23231 df-tx 23478 df-hmeo 23671 df-fil 23762 df-fm 23854 df-flim 23855 df-flf 23856 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-limc 25795 df-dv 25796 df-log 26493 df-cxp 26494 df-cht 27035 df-ppi 27038

This theorem is referenced by: chtppilimlem2 27413

W3C validator