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Theorem pntpbnd1a 27905

Description: Lemma for pntpbnd 27908. (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 13155	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
3		pntpbnd.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4	3	pntrf 27883	. . . . . . 7 ⊢ 𝑅:ℝ⁺⟶ℝ
5	4	ffvelcdmi 7081	. . . . . 6 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) ∈ ℝ)
6	2, 5	syl 18	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℝ)
7	6, 2	rerpdivcld 13188	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℝ)
8	7	recnd 11330	. . 3 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℂ)
9	8	abscld 15599	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ∈ ℝ)
10	2	relogcld 26944	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
11	10, 2	rerpdivcld 13188	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ∈ ℝ)
12		ioossre 13531	. . 3 ⊢ (0(,)1) ⊆ ℝ
13		pntpbnd1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
14	12, 13	sselid 3929	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ)
15	6	recnd 11330	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℂ)
16	1	nnred 12343	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	recnd 11330	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	1	nnne0d 12381	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
19	15, 17, 18	absdivd 15618	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)))
20	1	nnnn0d 12660	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 12663	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁)
22	16, 21	absidd 15583	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) = 𝑁)
23	22	oveq2d 7434	. . . 4 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
24	19, 23	eqtrd 2796	. . 3 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
25	15	abscld 15599	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ∈ ℝ)
26	1	peano2nnd 12345	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27		vmacl 27438	. . . . . . . . 9 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ∈ ℝ)
28	26, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℝ)
29		peano2rem 11618	. . . . . . . 8 ⊢ ((Λ‘(𝑁 + 1)) ∈ ℝ → ((Λ‘(𝑁 + 1)) − 1) ∈ ℝ)
30	28, 29	syl 18	. . . . . . 7 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℝ)
31	30	recnd 11330	. . . . . 6 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℂ)
32	31	abscld 15599	. . . . 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 13155	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
35	3	pntrval 27882	. . . . . . . . . 10 ⊢ ((𝑁 + 1) ∈ ℝ⁺ → (𝑅‘(𝑁 + 1)) = ((ψ‘(𝑁 + 1)) − (𝑁 + 1)))
36	34, 35	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘(𝑁 + 1)) = ((ψ‘(𝑁 + 1)) − (𝑁 + 1)))
37	3	pntrval 27882	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
38	2, 37	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
39	36, 38	oveq12d 7436	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)))
40		peano2re 11476	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
41	16, 40	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
42		chpcl 27444	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℝ → (ψ‘(𝑁 + 1)) ∈ ℝ)
43	41, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℝ)
44	43	recnd 11330	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℂ)
45	41	recnd 11330	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
46		chpcl 27444	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (ψ‘𝑁) ∈ ℝ)
47	16, 46	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℝ)
48	47	recnd 11330	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℂ)
49	44, 45, 48, 17	sub4d 11711	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)) = (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)))
50	28	recnd 11330	. . . . . . . . . 10 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℂ)
51		chpp1 27475	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
52	20, 51	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
53	48, 50, 52	mvrladdd 11721	. . . . . . . . 9 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
54		ax-1cn 11251	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
55		pncan2 11557	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 𝑁) = 1)
56	17, 54, 55	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − 𝑁) = 1)
57	53, 56	oveq12d 7436	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
58	39, 49, 57	3eqtrd 2800	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
59	58	fveq2d 6887	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))) = (abs‘((Λ‘(𝑁 + 1)) − 1)))
60	33, 59	breqtrd 5131	. . . . 5 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((Λ‘(𝑁 + 1)) − 1)))
61		1red 11302	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
62	61, 10	resubcld 11737	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ∈ ℝ)
63		0red 11304	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
64		2re 12410	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
65		eliooord 13529	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ (0(,)1) → (0 < 𝐸 ∧ 𝐸 < 1))
66	13, 65	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝐸 ∧ 𝐸 < 1))
67	66	simpld 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
68	14, 67	elrpd 13154	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
69		rerpdivcl 13145	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ)
70	64, 68, 69	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ)
71	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
72		1lt2 12508	. . . . . . . . . . . 12 ⊢ 1 < 2
73	72	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
74		2cn 12411	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
75	74	div1i 12038	. . . . . . . . . . . 12 ⊢ (2 / 1) = 2
76	66	simprd 501	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 < 1)
77		0lt1 11831	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
78	77	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
79		2pos 12440	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
81		ltdiv2 12196	. . . . . . . . . . . . . 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 1413	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
83	76, 82	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 1) < (2 / 𝐸))
84	75, 83	eqbrtrrid 5141	. . . . . . . . . . 11 ⊢ (𝜑 → 2 < (2 / 𝐸))
85	61, 71, 70, 73, 84	lttrd 11464	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (2 / 𝐸))
86		pntpbnd1.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = (exp‘(2 / 𝐸))
87	70	rpefcld 16266	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (exp‘(2 / 𝐸)) ∈ ℝ⁺)
88	86, 87	eqeltrid 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
89	88	rpred 13157	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
90		pntpbnd1.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
91	88	rpxrd 13158	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
92		elioopnf 13567	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℝ^* → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
93	91, 92	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
94	90, 93	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌))
95	94	simpld 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
96	94	simprd 501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 < 𝑌)
97		pntpbnd1a.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌)))
98	97	simpld 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < 𝑁)
99	89, 95, 16, 96, 98	lttrd 11464	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < 𝑁)
100	86, 99	eqbrtrrid 5141	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < 𝑁)
101	2	reeflogd 26945	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
102	100, 101	breqtrrd 5133	. . . . . . . . . . 11 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁)))
103		eflt 16278	. . . . . . . . . . . 12 ⊢ (((2 / 𝐸) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
104	70, 10, 103	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
105	102, 104	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) < (log‘𝑁))
106	61, 70, 10, 85, 105	lttrd 11464	. . . . . . . . 9 ⊢ (𝜑 → 1 < (log‘𝑁))
107	61, 10, 106	ltled 11451	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝑁))
108		1re 11301	. . . . . . . . 9 ⊢ 1 ∈ ℝ
109		suble0 11823	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
110	108, 10, 109	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
111	107, 110	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ 0)
112		vmage0 27441	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → 0 ≤ (Λ‘(𝑁 + 1)))
113	26, 112	syl 18	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (Λ‘(𝑁 + 1)))
114	62, 63, 28, 111, 113	letrd 11460	. . . . . 6 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)))
115	34	relogcld 26944	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ∈ ℝ)
116		readdcl 11276	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → (1 + (log‘𝑁)) ∈ ℝ)
117	108, 10, 116	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1 + (log‘𝑁)) ∈ ℝ)
118		vmalelog 27525	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
119	26, 118	syl 18	. . . . . . 7 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
120	71, 16	remulcld 11332	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
121		epr 16369	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
122		rpmulcl 13138	. . . . . . . . . . . 12 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e · 𝑁) ∈ ℝ⁺)
123	121, 2, 122	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ⁺)
124	123	rpred 13157	. . . . . . . . . 10 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ)
125	1	nnge1d 12379	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
126	61, 16, 16, 125	leadd2dd 11924	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ≤ (𝑁 + 𝑁))
127	17	2timesd 12582	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
128	126, 127	breqtrrd 5133	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ≤ (2 · 𝑁))
129		ere 16248	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
130		egt2lt3 16367	. . . . . . . . . . . . . 14 ⊢ (2 < e ∧ e < 3)
131	130	simpli 489	. . . . . . . . . . . . 13 ⊢ 2 < e
132	64, 129, 131	ltleii 11426	. . . . . . . . . . . 12 ⊢ 2 ≤ e
133	132	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ e)
134	129	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
135	1	nngt0d 12380	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑁)
136		lemul1 12162	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ e ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
137	71, 134, 16, 135, 136	syl112anc 1401	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
138	133, 137	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ≤ (e · 𝑁))
139	41, 120, 124, 128, 138	letrd 11460	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ≤ (e · 𝑁))
140	34, 123	logled 26948	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) ≤ (e · 𝑁) ↔ (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁))))
141	139, 140	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁)))
142		relogmul 26913	. . . . . . . . . 10 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
143	121, 2, 142	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
144		loge 26907	. . . . . . . . . 10 ⊢ (log‘e) = 1
145	144	oveq1i 7428	. . . . . . . . 9 ⊢ ((log‘e) + (log‘𝑁)) = (1 + (log‘𝑁))
146	143, 145	eqtrdi 2812	. . . . . . . 8 ⊢ (𝜑 → (log‘(e · 𝑁)) = (1 + (log‘𝑁)))
147	141, 146	breqtrd 5131	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
148	28, 115, 117, 119, 147	letrd 11460	. . . . . 6 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
149	28, 61, 10	absdifled 15597	. . . . . 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 726	. . . . 5 ⊢ (𝜑 → (abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁))
151	25, 32, 10, 60, 150	letrd 11460	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (log‘𝑁))
152	25, 10, 2, 151	lediv1dd 13215	. . 3 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / 𝑁) ≤ ((log‘𝑁) / 𝑁))
153	24, 152	eqbrtrd 5127	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ ((log‘𝑁) / 𝑁))
154	88	relogcld 26944	. . . . 5 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ)
155	154, 88	rerpdivcld 13188	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) ∈ ℝ)
156	61, 70, 85	ltled 11451	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (2 / 𝐸))
157		efle 16279	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (2 / 𝐸) ∈ ℝ) → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
158	108, 70, 157	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
159	156, 158	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (exp‘1) ≤ (exp‘(2 / 𝐸)))
160		df-e 16227	. . . . . . 7 ⊢ e = (exp‘1)
161	159, 160, 86	3brtr4g 5139	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑋)
162	144, 107	eqbrtrid 5140	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑁))
163		logleb 26924	. . . . . . . 8 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
164	121, 2, 163	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
165	162, 164	mpbird 260	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑁)
166		logdivlt 26942	. . . . . 6 ⊢ (((𝑋 ∈ ℝ ∧ e ≤ 𝑋) ∧ (𝑁 ∈ ℝ ∧ e ≤ 𝑁)) → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
167	89, 161, 16, 165, 166	syl22anc 852	. . . . 5 ⊢ (𝜑 → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
168	99, 167	mpbid 235	. . . 4 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋))
169	86	fveq2i 6886	. . . . . . 7 ⊢ (log‘𝑋) = (log‘(exp‘(2 / 𝐸)))
170	70	relogefd 26949	. . . . . . 7 ⊢ (𝜑 → (log‘(exp‘(2 / 𝐸))) = (2 / 𝐸))
171	169, 170	eqtrid 2808	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) = (2 / 𝐸))
172	171	oveq1d 7433	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) = ((2 / 𝐸) / 𝑋))
173		2rp 13118	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
174		rpdivcl 13140	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ⁺)
175	173, 68, 174	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ⁺)
176	175	rpcnd 13159	. . . . . . . . . . 11 ⊢ (𝜑 → (2 / 𝐸) ∈ ℂ)
177	176	sqvald 14279	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸)↑2) = ((2 / 𝐸) · (2 / 𝐸)))
178		2cnd 12414	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
179	68	rpcnne0d 13166	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
180		div12 11989	. . . . . . . . . . 11 ⊢ (((2 / 𝐸) ∈ ℂ ∧ 2 ∈ ℂ ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
181	176, 178, 179, 180	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
182	177, 181	eqtrd 2796	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) = (2 · ((2 / 𝐸) / 𝐸)))
183	182	oveq1d 7433	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 · ((2 / 𝐸) / 𝐸)) / 2))
184	175, 68	rpdivcld 13174	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℝ⁺)
185	184	rpcnd 13159	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℂ)
186		2ne0 12442	. . . . . . . . . 10 ⊢ 2 ≠ 0
187	186	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
188	185, 178, 187	divcan3d 12091	. . . . . . . 8 ⊢ (𝜑 → ((2 · ((2 / 𝐸) / 𝐸)) / 2) = ((2 / 𝐸) / 𝐸))
189	183, 188	eqtrd 2796	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 / 𝐸) / 𝐸))
190	70	resqcld 14261	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) ∈ ℝ)
191	190	rehalfcld 12586	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) ∈ ℝ)
192		1rp 13117	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
193		rpaddcl 13137	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (2 / 𝐸) ∈ ℝ⁺) → (1 + (2 / 𝐸)) ∈ ℝ⁺)
194	192, 175, 193	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ⁺)
195	194	rpred 13157	. . . . . . . . 9 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ)
196	195, 191	readdcld 11331	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) ∈ ℝ)
197	191, 194	ltaddrp2d 13191	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)))
198		efgt1p2 16275	. . . . . . . . . 10 ⊢ ((2 / 𝐸) ∈ ℝ⁺ → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
199	175, 198	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
200	199, 86	breqtrrdi 5147	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < 𝑋)
201	191, 196, 89, 197, 200	lttrd 11464	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < 𝑋)
202	189, 201	eqbrtrrd 5129	. . . . . 6 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) < 𝑋)
203	70, 68, 88, 202	ltdiv23d 13224	. . . . 5 ⊢ (𝜑 → ((2 / 𝐸) / 𝑋) < 𝐸)
204	172, 203	eqbrtrd 5127	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) < 𝐸)
205	11, 155, 14, 168, 204	lttrd 11464	. . 3 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < 𝐸)
206	11, 14, 205	ltled 11451	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ≤ 𝐸)
207	9, 11, 14, 153, 206	letrd 11460	1 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 +∞cpnf 11333 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 / cdiv 11966 ℕcn 12328 2c2 12390 3c3 12391 ℕ₀cn0 12599 ℝ⁺crp 13113 (,)cioo 13469 ↑cexp 14197 abscabs 15394 expce 16220 eceu 16221 logclog 26875 Λcvma 27412 ψcchp 27413

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-mod 14003 df-seq 14138 df-exp 14198 df-fac 14411 df-bc 14440 df-hash 14468 df-shft 15213 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-limsup 15631 df-clim 15648 df-rlim 15649 df-sum 15847 df-ef 16226 df-e 16227 df-sin 16228 df-cos 16229 df-pi 16231 df-dvds 16416 df-gcd 16658 df-prm 16840 df-pc 17008 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180 df-log 26877 df-vma 27418 df-chp 27419

This theorem is used by: pntpbnd1 27906

W3C validator