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Theorem pntpbnd1a 27518

Description: Lemma for pntpbnd 27521. (Contributed by Mario Carneiro, 11-Apr-2016.) Replace reference to OLD theorem. (Revised by Wolf Lammen, 8-Sep-2020.)

**Hypotheses**
Ref	Expression
pntpbnd.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntpbnd1.e	⊢ (𝜑 → 𝐸 ∈ (0(,)1))
pntpbnd1.x	⊢ 𝑋 = (exp‘(2 / 𝐸))
pntpbnd1.y	⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
pntpbnd1a.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
pntpbnd1a.2	⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌)))
pntpbnd1a.3	⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))))

**Assertion**
Ref	Expression
pntpbnd1a	⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸)

Distinct variable group: 𝑁,𝑎 Allowed substitution hints: 𝜑(𝑎) 𝑅(𝑎) 𝐸(𝑎) 𝐾(𝑎) 𝑋(𝑎) 𝑌(𝑎)

**Proof of Theorem pntpbnd1a**
Step	Hyp	Ref	Expression
1		pntpbnd1a.1	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnrpd 12927	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
3		pntpbnd.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4	3	pntrf 27496	. . . . . . 7 ⊢ 𝑅:ℝ⁺⟶ℝ
5	4	ffvelcdmi 7011	. . . . . 6 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) ∈ ℝ)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℝ)
7	6, 2	rerpdivcld 12960	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℝ)
8	7	recnd 11135	. . 3 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℂ)
9	8	abscld 15341	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ∈ ℝ)
10	2	relogcld 26554	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
11	10, 2	rerpdivcld 12960	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ∈ ℝ)
12		ioossre 13302	. . 3 ⊢ (0(,)1) ⊆ ℝ
13		pntpbnd1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
14	12, 13	sselid 3927	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ)
15	6	recnd 11135	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℂ)
16	1	nnred 12135	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	recnd 11135	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	1	nnne0d 12170	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
19	15, 17, 18	absdivd 15360	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)))
20	1	nnnn0d 12437	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 12440	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁)
22	16, 21	absidd 15325	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) = 𝑁)
23	22	oveq2d 7357	. . . 4 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
24	19, 23	eqtrd 2766	. . 3 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
25	15	abscld 15341	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ∈ ℝ)
26	1	peano2nnd 12137	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27		vmacl 27050	. . . . . . . . 9 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℝ)
29		peano2rem 11423	. . . . . . . 8 ⊢ ((Λ‘(𝑁 + 1)) ∈ ℝ → ((Λ‘(𝑁 + 1)) − 1) ∈ ℝ)
30	28, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℝ)
31	30	recnd 11135	. . . . . 6 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℂ)
32	31	abscld 15341	. . . . 5 ⊢ (𝜑 → (abs‘((Λ‘(𝑁 + 1)) − 1)) ∈ ℝ)
33		pntpbnd1a.3	. . . . . 6 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))))
34	26	nnrpd 12927	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
35	3	pntrval 27495	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℝ⁺ → (𝑅‘(𝑁 + 1)) = ((ψ‘(𝑁 + 1)) − (𝑁 + 1)))
36	34, 35	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘(𝑁 + 1)) = ((ψ‘(𝑁 + 1)) − (𝑁 + 1)))
37	3	pntrval 27495	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
38	2, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
39	36, 38	oveq12d 7359	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)))
40		peano2re 11281	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
41	16, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
42		chpcl 27056	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℝ → (ψ‘(𝑁 + 1)) ∈ ℝ)
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℝ)
44	43	recnd 11135	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℂ)
45	41	recnd 11135	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
46		chpcl 27056	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (ψ‘𝑁) ∈ ℝ)
47	16, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℝ)
48	47	recnd 11135	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℂ)
49	44, 45, 48, 17	sub4d 11516	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)) = (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)))
50	28	recnd 11135	. . . . . . . . . 10 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℂ)
51		chpp1 27087	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
52	20, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
53	48, 50, 52	mvrladdd 11525	. . . . . . . . 9 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
54		ax-1cn 11059	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
55		pncan2 11362	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 𝑁) = 1)
56	17, 54, 55	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − 𝑁) = 1)
57	53, 56	oveq12d 7359	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
58	39, 49, 57	3eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
59	58	fveq2d 6821	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))) = (abs‘((Λ‘(𝑁 + 1)) − 1)))
60	33, 59	breqtrd 5112	. . . . 5 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((Λ‘(𝑁 + 1)) − 1)))
61		1red 11108	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
62	61, 10	resubcld 11540	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ∈ ℝ)
63		0red 11110	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
64		2re 12194	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
65		eliooord 13300	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ (0(,)1) → (0 < 𝐸 ∧ 𝐸 < 1))
66	13, 65	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝐸 ∧ 𝐸 < 1))
67	66	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
68	14, 67	elrpd 12926	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
69		rerpdivcl 12917	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ)
70	64, 68, 69	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ)
71	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
72		1lt2 12286	. . . . . . . . . . . 12 ⊢ 1 < 2
73	72	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
74		2cn 12195	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
75	74	div1i 11844	. . . . . . . . . . . 12 ⊢ (2 / 1) = 2
76	66	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 < 1)
77		0lt1 11634	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
78	77	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
79		2pos 12223	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
81		ltdiv2 12003	. . . . . . . . . . . . . 14 ⊢ (((𝐸 ∈ ℝ ∧ 0 < 𝐸) ∧ (1 ∈ ℝ ∧ 0 < 1) ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
82	14, 67, 61, 78, 71, 80, 81	syl222anc 1388	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
83	76, 82	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 1) < (2 / 𝐸))
84	75, 83	eqbrtrrid 5122	. . . . . . . . . . 11 ⊢ (𝜑 → 2 < (2 / 𝐸))
85	61, 71, 70, 73, 84	lttrd 11269	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (2 / 𝐸))
86		pntpbnd1.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = (exp‘(2 / 𝐸))
87	70	rpefcld 16009	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (exp‘(2 / 𝐸)) ∈ ℝ⁺)
88	86, 87	eqeltrid 2835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
89	88	rpred 12929	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
90		pntpbnd1.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
91	88	rpxrd 12930	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
92		elioopnf 13338	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℝ^* → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
93	91, 92	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
94	90, 93	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌))
95	94	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
96	94	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 < 𝑌)
97		pntpbnd1a.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌)))
98	97	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < 𝑁)
99	89, 95, 16, 96, 98	lttrd 11269	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < 𝑁)
100	86, 99	eqbrtrrid 5122	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < 𝑁)
101	2	reeflogd 26555	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
102	100, 101	breqtrrd 5114	. . . . . . . . . . 11 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁)))
103		eflt 16021	. . . . . . . . . . . 12 ⊢ (((2 / 𝐸) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
104	70, 10, 103	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
105	102, 104	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) < (log‘𝑁))
106	61, 70, 10, 85, 105	lttrd 11269	. . . . . . . . 9 ⊢ (𝜑 → 1 < (log‘𝑁))
107	61, 10, 106	ltled 11256	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝑁))
108		1re 11107	. . . . . . . . 9 ⊢ 1 ∈ ℝ
109		suble0 11626	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
110	108, 10, 109	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
111	107, 110	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ 0)
112		vmage0 27053	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → 0 ≤ (Λ‘(𝑁 + 1)))
113	26, 112	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (Λ‘(𝑁 + 1)))
114	62, 63, 28, 111, 113	letrd 11265	. . . . . 6 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)))
115	34	relogcld 26554	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ∈ ℝ)
116		readdcl 11084	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → (1 + (log‘𝑁)) ∈ ℝ)
117	108, 10, 116	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1 + (log‘𝑁)) ∈ ℝ)
118		vmalelog 27138	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
119	26, 118	syl 17	. . . . . . 7 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
120	71, 16	remulcld 11137	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
121		epr 16112	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
122		rpmulcl 12910	. . . . . . . . . . . 12 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e · 𝑁) ∈ ℝ⁺)
123	121, 2, 122	sylancr 587	. . . . . . . . . . 11 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ⁺)
124	123	rpred 12929	. . . . . . . . . 10 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ)
125	1	nnge1d 12168	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
126	61, 16, 16, 125	leadd2dd 11727	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ≤ (𝑁 + 𝑁))
127	17	2timesd 12359	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
128	126, 127	breqtrrd 5114	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ≤ (2 · 𝑁))
129		ere 15991	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
130		egt2lt3 16110	. . . . . . . . . . . . . 14 ⊢ (2 < e ∧ e < 3)
131	130	simpli 483	. . . . . . . . . . . . 13 ⊢ 2 < e
132	64, 129, 131	ltleii 11231	. . . . . . . . . . . 12 ⊢ 2 ≤ e
133	132	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ e)
134	129	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
135	1	nngt0d 12169	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑁)
136		lemul1 11968	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ e ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
137	71, 134, 16, 135, 136	syl112anc 1376	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
138	133, 137	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ≤ (e · 𝑁))
139	41, 120, 124, 128, 138	letrd 11265	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ≤ (e · 𝑁))
140	34, 123	logled 26558	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) ≤ (e · 𝑁) ↔ (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁))))
141	139, 140	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁)))
142		relogmul 26523	. . . . . . . . . 10 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
143	121, 2, 142	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
144		loge 26517	. . . . . . . . . 10 ⊢ (log‘e) = 1
145	144	oveq1i 7351	. . . . . . . . 9 ⊢ ((log‘e) + (log‘𝑁)) = (1 + (log‘𝑁))
146	143, 145	eqtrdi 2782	. . . . . . . 8 ⊢ (𝜑 → (log‘(e · 𝑁)) = (1 + (log‘𝑁)))
147	141, 146	breqtrd 5112	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
148	28, 115, 117, 119, 147	letrd 11265	. . . . . 6 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
149	28, 61, 10	absdifled 15339	. . . . . 6 ⊢ (𝜑 → ((abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁) ↔ ((1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)) ∧ (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))))
150	114, 148, 149	mpbir2and 713	. . . . 5 ⊢ (𝜑 → (abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁))
151	25, 32, 10, 60, 150	letrd 11265	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (log‘𝑁))
152	25, 10, 2, 151	lediv1dd 12987	. . 3 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / 𝑁) ≤ ((log‘𝑁) / 𝑁))
153	24, 152	eqbrtrd 5108	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ ((log‘𝑁) / 𝑁))
154	88	relogcld 26554	. . . . 5 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ)
155	154, 88	rerpdivcld 12960	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) ∈ ℝ)
156	61, 70, 85	ltled 11256	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (2 / 𝐸))
157		efle 16022	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (2 / 𝐸) ∈ ℝ) → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
158	108, 70, 157	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
159	156, 158	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (exp‘1) ≤ (exp‘(2 / 𝐸)))
160		df-e 15970	. . . . . . 7 ⊢ e = (exp‘1)
161	159, 160, 86	3brtr4g 5120	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑋)
162	144, 107	eqbrtrid 5121	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑁))
163		logleb 26534	. . . . . . . 8 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
164	121, 2, 163	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
165	162, 164	mpbird 257	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑁)
166		logdivlt 26552	. . . . . 6 ⊢ (((𝑋 ∈ ℝ ∧ e ≤ 𝑋) ∧ (𝑁 ∈ ℝ ∧ e ≤ 𝑁)) → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
167	89, 161, 16, 165, 166	syl22anc 838	. . . . 5 ⊢ (𝜑 → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
168	99, 167	mpbid 232	. . . 4 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋))
169	86	fveq2i 6820	. . . . . . 7 ⊢ (log‘𝑋) = (log‘(exp‘(2 / 𝐸)))
170	70	relogefd 26559	. . . . . . 7 ⊢ (𝜑 → (log‘(exp‘(2 / 𝐸))) = (2 / 𝐸))
171	169, 170	eqtrid 2778	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) = (2 / 𝐸))
172	171	oveq1d 7356	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) = ((2 / 𝐸) / 𝑋))
173		2rp 12890	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
174		rpdivcl 12912	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ⁺)
175	173, 68, 174	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ⁺)
176	175	rpcnd 12931	. . . . . . . . . . 11 ⊢ (𝜑 → (2 / 𝐸) ∈ ℂ)
177	176	sqvald 14045	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸)↑2) = ((2 / 𝐸) · (2 / 𝐸)))
178		2cnd 12198	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
179	68	rpcnne0d 12938	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
180		div12 11793	. . . . . . . . . . 11 ⊢ (((2 / 𝐸) ∈ ℂ ∧ 2 ∈ ℂ ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
181	176, 178, 179, 180	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
182	177, 181	eqtrd 2766	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) = (2 · ((2 / 𝐸) / 𝐸)))
183	182	oveq1d 7356	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 · ((2 / 𝐸) / 𝐸)) / 2))
184	175, 68	rpdivcld 12946	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℝ⁺)
185	184	rpcnd 12931	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℂ)
186		2ne0 12224	. . . . . . . . . 10 ⊢ 2 ≠ 0
187	186	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
188	185, 178, 187	divcan3d 11897	. . . . . . . 8 ⊢ (𝜑 → ((2 · ((2 / 𝐸) / 𝐸)) / 2) = ((2 / 𝐸) / 𝐸))
189	183, 188	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 / 𝐸) / 𝐸))
190	70	resqcld 14027	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) ∈ ℝ)
191	190	rehalfcld 12363	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) ∈ ℝ)
192		1rp 12889	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
193		rpaddcl 12909	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (2 / 𝐸) ∈ ℝ⁺) → (1 + (2 / 𝐸)) ∈ ℝ⁺)
194	192, 175, 193	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ⁺)
195	194	rpred 12929	. . . . . . . . 9 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ)
196	195, 191	readdcld 11136	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) ∈ ℝ)
197	191, 194	ltaddrp2d 12963	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)))
198		efgt1p2 16018	. . . . . . . . . 10 ⊢ ((2 / 𝐸) ∈ ℝ⁺ → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
199	175, 198	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
200	199, 86	breqtrrdi 5128	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < 𝑋)
201	191, 196, 89, 197, 200	lttrd 11269	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < 𝑋)
202	189, 201	eqbrtrrd 5110	. . . . . 6 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) < 𝑋)
203	70, 68, 88, 202	ltdiv23d 12996	. . . . 5 ⊢ (𝜑 → ((2 / 𝐸) / 𝑋) < 𝐸)
204	172, 203	eqbrtrd 5108	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) < 𝐸)
205	11, 155, 14, 168, 204	lttrd 11269	. . 3 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < 𝐸)
206	11, 14, 205	ltled 11256	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ≤ 𝐸)
207	9, 11, 14, 153, 206	letrd 11265	1 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 +∞cpnf 11138 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 ℕcn 12120 2c2 12175 3c3 12176 ℕ₀cn0 12376 ℝ⁺crp 12885 (,)cioo 13240 ↑cexp 13963 abscabs 15136 expce 15963 eceu 15964 logclog 26485 Λcvma 27024 ψcchp 27025

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-e 15970 df-sin 15971 df-cos 15972 df-pi 15974 df-dvds 16159 df-gcd 16401 df-prm 16578 df-pc 16744 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19224 df-cmn 19689 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790 df-log 26487 df-vma 27030 df-chp 27031

This theorem is referenced by: pntpbnd1 27519

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