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Theorem chtppilimlem1 26976

Description: Lemma for chtppilim 26978. (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 13016	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	recnd 11242	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
4	3	sqvald 14108	. . . 4 ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴))
5	4	oveq1d 7424	. . 3 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))))
6		chtppilim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (2[,)+∞))
7		2re 12286	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8		elicopnf 13422	. . . . . . . . . 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 217	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ∈ ℝ ∧ 2 ≤ 𝑁))
11	10	simpld 496	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
12		ppicl 26635	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (π‘𝑁) ∈ ℕ₀)
13	11, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ₀)
14	13	nn0red 12533	. . . . 5 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ)
15	14	recnd 11242	. . . 4 ⊢ (𝜑 → (π‘𝑁) ∈ ℂ)
16		0red 11217	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
17	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℝ)
18		2pos 12315	. . . . . . . . 9 ⊢ 0 < 2
19	18	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 2)
20	10	simprd 497	. . . . . . . 8 ⊢ (𝜑 → 2 ≤ 𝑁)
21	16, 17, 11, 19, 20	ltletrd 11374	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	11, 21	elrpd 13013	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
23	22	relogcld 26131	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
24	23	recnd 11242	. . . 4 ⊢ (𝜑 → (log‘𝑁) ∈ ℂ)
25	3, 3, 15, 24	mul4d 11426	. . 3 ⊢ (𝜑 → ((𝐴 · 𝐴) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
26	5, 25	eqtrd 2773	. 2 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) = ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))))
27	2, 14	remulcld 11244	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) ∈ ℝ)
28	2, 23	remulcld 11244	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ)
29	27, 28	remulcld 11244	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) ∈ ℝ)
30	22, 2	rpcxpcld 26241	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ⁺)
31	30	rpred 13016	. . . . . . 7 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℝ)
32		ppicl 26635	. . . . . . 7 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
33	31, 32	syl 17	. . . . . 6 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
34	33	nn0red 12533	. . . . 5 ⊢ (𝜑 → (π‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
35	14, 34	resubcld 11642	. . . 4 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) ∈ ℝ)
36	35, 28	remulcld 11244	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ∈ ℝ)
37		chtcl 26613	. . . 4 ⊢ (𝑁 ∈ ℝ → (θ‘𝑁) ∈ ℝ)
38	11, 37	syl 17	. . 3 ⊢ (𝜑 → (θ‘𝑁) ∈ ℝ)
39		1red 11215	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ)
40		1lt2 12383	. . . . . . . 8 ⊢ 1 < 2
41	40	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 < 2)
42	39, 17, 11, 41, 20	ltletrd 11374	. . . . . 6 ⊢ (𝜑 → 1 < 𝑁)
43	11, 42	rplogcld 26137	. . . . 5 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ⁺)
44	1, 43	rpmulcld 13032	. . . 4 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℝ⁺)
45	14, 31	resubcld 11642	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) ∈ ℝ)
46		ppinncl 26678	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 2 ≤ 𝑁) → (π‘𝑁) ∈ ℕ)
47	10, 46	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘𝑁) ∈ ℕ)
48	31, 47	nndivred 12266	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) ∈ ℝ)
49		chtppilim.4	. . . . . . . 8 ⊢ (𝜑 → ((𝑁↑_𝑐𝐴) / (π‘𝑁)) < (1 − 𝐴))
50	48, 39, 2, 49	ltsub13d 11820	. . . . . . 7 ⊢ (𝜑 → 𝐴 < (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
51	31	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ∈ ℂ)
52	47	nnrpd 13014	. . . . . . . . . 10 ⊢ (𝜑 → (π‘𝑁) ∈ ℝ⁺)
53	52	rpcnne0d 13025	. . . . . . . . 9 ⊢ (𝜑 → ((π‘𝑁) ∈ ℂ ∧ (π‘𝑁) ≠ 0))
54		divsubdir 11908	. . . . . . . . 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 1372	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
56		divid 11901	. . . . . . . . . 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 7424	. . . . . . . 8 ⊢ (𝜑 → (((π‘𝑁) / (π‘𝑁)) − ((𝑁↑_𝑐𝐴) / (π‘𝑁))) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
59	55, 58	eqtrd 2773	. . . . . . 7 ⊢ (𝜑 → (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)) = (1 − ((𝑁↑_𝑐𝐴) / (π‘𝑁))))
60	50, 59	breqtrrd 5177	. . . . . 6 ⊢ (𝜑 → 𝐴 < (((π‘𝑁) − (𝑁↑_𝑐𝐴)) / (π‘𝑁)))
61	2, 45, 52	ltmuldivd 13063	. . . . . 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 26681	. . . . . . 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 11827	. . . . 5 ⊢ (𝜑 → ((π‘𝑁) − (𝑁↑_𝑐𝐴)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
66	27, 45, 35, 62, 65	lttrd 11375	. . . 4 ⊢ (𝜑 → (𝐴 · (π‘𝑁)) < ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
67	27, 35, 44, 66	ltmul1dd 13071	. . 3 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
68		fzfid 13938	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∈ Fin)
69		inss1 4229	. . . . . 6 ⊢ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ⊆ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁))
70		ssfi 9173	. . . . . 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 587	. . . . 5 ⊢ (𝜑 → ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
72		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ))
73	72	elin2d 4200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
74		prmnn 16611	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
75	74	nnrpd 13014	. . . . . . 7 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℝ⁺)
76	73, 75	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
77	76	relogcld 26131	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
78	71, 77	fsumrecl 15680	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ∈ ℝ)
79	28	recnd 11242	. . . . . . 7 ⊢ (𝜑 → (𝐴 · (log‘𝑁)) ∈ ℂ)
80		fsumconst 15736	. . . . . . 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 585	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
82		ppifl 26664	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → (π‘(⌊‘𝑁)) = (π‘𝑁))
83	11, 82	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (π‘(⌊‘𝑁)) = (π‘𝑁))
84		ppifl 26664	. . . . . . . . . 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 7427	. . . . . . . 8 ⊢ (𝜑 → ((π‘(⌊‘𝑁)) − (π‘(⌊‘(𝑁↑_𝑐𝐴)))) = ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))))
87	39, 11, 42	ltled 11362	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
88		chtppilim.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 1)
89		1re 11214	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
90		ltle 11302	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 → 𝐴 ≤ 1))
91	2, 89, 90	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 < 1 → 𝐴 ≤ 1))
92	88, 91	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≤ 1)
93	11, 87, 2, 39, 92	cxplead 26229	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ (𝑁↑_𝑐1))
94	11	recnd 11242	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
95	94	cxp1d 26214	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁↑_𝑐1) = 𝑁)
96	93, 95	breqtrd 5175	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ 𝑁)
97		flword2 13778	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁↑_𝑐𝐴) ≤ 𝑁) → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
98	31, 11, 96, 97	syl3anc 1372	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘𝑁) ∈ (ℤ_≥‘(⌊‘(𝑁↑_𝑐𝐴))))
99		ppidif 26667	. . . . . . . . 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 2775	. . . . . . 7 ⊢ (𝜑 → ((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) = (♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)))
102	101	oveq1d 7424	. . . . . 6 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) = ((♯‘((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) · (𝐴 · (log‘𝑁))))
103	81, 102	eqtr4d 2776	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) = (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))))
104	28	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝐴 · (log‘𝑁)) ∈ ℝ)
105	31	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ∈ ℝ)
106		reflcl 13761	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ)
107		peano2re 11387	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℝ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
108	31, 106, 107	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
109	108	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℝ)
110	76	rpred 13016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
111		fllep1 13766	. . . . . . . . . . 11 ⊢ ((𝑁↑_𝑐𝐴) ∈ ℝ → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
112	31, 111	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
113	112	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ ((⌊‘(𝑁↑_𝑐𝐴)) + 1))
114	72	elin1d 4199	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)))
115		elfzle1 13504	. . . . . . . . . 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 11371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (𝑁↑_𝑐𝐴) ≤ 𝑝)
118	22	rpne0d 13021	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
119	94, 118, 3	cxpefd 26220	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑁))))
120	119	eqcomd 2739	. . . . . . . . 9 ⊢ (𝜑 → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
121	120	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) = (𝑁↑_𝑐𝐴))
122	76	reeflogd 26132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(log‘𝑝)) = 𝑝)
123	117, 121, 122	3brtr4d 5181	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)) → (exp‘(𝐴 · (log‘𝑁))) ≤ (exp‘(log‘𝑝)))
124		efle 16061	. . . . . . . 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 585	. . . . . . 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 15745	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(𝐴 · (log‘𝑁)) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
128	103, 127	eqbrtrrd 5173	. . . 4 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
129		fzfid 13938	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘𝑁)) ∈ Fin)
130		inss1 4229	. . . . . . 7 ⊢ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))
131		ssfi 9173	. . . . . . 7 ⊢ (((1...(⌊‘𝑁)) ∈ Fin ∧ ((1...(⌊‘𝑁)) ∩ ℙ) ⊆ (1...(⌊‘𝑁))) → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
132	129, 130, 131	sylancl 587	. . . . . 6 ⊢ (𝜑 → ((1...(⌊‘𝑁)) ∩ ℙ) ∈ Fin)
133		simpr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ))
134	133	elin2d 4200	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℙ)
135		prmuz2 16633	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
137		eluz2b2 12905	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
138	136, 137	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (𝑝 ∈ ℕ ∧ 1 < 𝑝))
139	138	simpld 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℕ)
140	139	nnred 12227	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 𝑝 ∈ ℝ)
141	138	simprd 497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 1 < 𝑝)
142	140, 141	rplogcld 26137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
143	142	rpred 13016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
144	142	rpge0d 13020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)) → 0 ≤ (log‘𝑝))
145	30	rpge0d 13020	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (𝑁↑_𝑐𝐴))
146		flge0nn0 13785	. . . . . . . . . 10 ⊢ (((𝑁↑_𝑐𝐴) ∈ ℝ ∧ 0 ≤ (𝑁↑_𝑐𝐴)) → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
147	31, 145, 146	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀)
148		nn0p1nn 12511	. . . . . . . . 9 ⊢ ((⌊‘(𝑁↑_𝑐𝐴)) ∈ ℕ₀ → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
149	147, 148	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ ℕ)
150		nnuz 12865	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
151	149, 150	eleqtrdi 2844	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1))
152		fzss1 13540	. . . . . . 7 ⊢ (((⌊‘(𝑁↑_𝑐𝐴)) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ⊆ (1...(⌊‘𝑁)))
153		ssrin 4234	. . . . . . 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 15742	. . . . 5 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
156		chtval 26614	. . . . . . 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 12878	. . . . . . . 8 ⊢ 2 ∈ (ℤ_≥‘1)
159		ppisval2 26609	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 2 ∈ (ℤ_≥‘1)) → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
160	11, 158, 159	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ((0[,]𝑁) ∩ ℙ) = ((1...(⌊‘𝑁)) ∩ ℙ))
161	160	sumeq1d 15647	. . . . . 6 ⊢ (𝜑 → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
162	157, 161	eqtrd 2773	. . . . 5 ⊢ (𝜑 → (θ‘𝑁) = Σ𝑝 ∈ ((1...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝))
163	155, 162	breqtrrd 5177	. . . 4 ⊢ (𝜑 → Σ𝑝 ∈ ((((⌊‘(𝑁↑_𝑐𝐴)) + 1)...(⌊‘𝑁)) ∩ ℙ)(log‘𝑝) ≤ (θ‘𝑁))
164	36, 78, 38, 128, 163	letrd 11371	. . 3 ⊢ (𝜑 → (((π‘𝑁) − (π‘(𝑁↑_𝑐𝐴))) · (𝐴 · (log‘𝑁))) ≤ (θ‘𝑁))
165	29, 36, 38, 67, 164	ltletrd 11374	. 2 ⊢ (𝜑 → ((𝐴 · (π‘𝑁)) · (𝐴 · (log‘𝑁))) < (θ‘𝑁))
166	26, 165	eqbrtrd 5171	1 ⊢ (𝜑 → ((𝐴↑2) · ((π‘𝑁) · (log‘𝑁))) < (θ‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∩ cin 3948 ⊆ wss 3949 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤ_≥cuz 12822 ℝ⁺crp 12974 [,)cico 13326 [,]cicc 13327 ...cfz 13484 ⌊cfl 13755 ↑cexp 14027 ♯chash 14290 Σcsu 15632 expce 16005 ℙcprime 16608 logclog 26063 ↑_𝑐ccxp 26064 θccht 26595 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-dju 9896 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-prm 16609 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-cxp 26066 df-cht 26601 df-ppi 26604

This theorem is referenced by: chtppilimlem2 26977

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