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Theorem pntpbnd1a 27536

Description: Lemma for pntpbnd 27539. (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 13046	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
3		pntpbnd.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4	3	pntrf 27514	. . . . . . 7 ⊢ 𝑅:ℝ⁺⟶ℝ
5	4	ffvelcdmi 7088	. . . . . 6 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) ∈ ℝ)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℝ)
7	6, 2	rerpdivcld 13079	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℝ)
8	7	recnd 11272	. . 3 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℂ)
9	8	abscld 15415	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ∈ ℝ)
10	2	relogcld 26575	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
11	10, 2	rerpdivcld 13079	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ∈ ℝ)
12		ioossre 13417	. . 3 ⊢ (0(,)1) ⊆ ℝ
13		pntpbnd1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
14	12, 13	sselid 3970	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ)
15	6	recnd 11272	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℂ)
16	1	nnred 12257	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	recnd 11272	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	1	nnne0d 12292	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
19	15, 17, 18	absdivd 15434	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)))
20	1	nnnn0d 12562	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 12565	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁)
22	16, 21	absidd 15401	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) = 𝑁)
23	22	oveq2d 7432	. . . 4 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
24	19, 23	eqtrd 2765	. . 3 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
25	15	abscld 15415	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ∈ ℝ)
26	1	peano2nnd 12259	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27		vmacl 27068	. . . . . . . . 9 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℝ)
29		peano2rem 11557	. . . . . . . 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 11272	. . . . . 6 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℂ)
32	31	abscld 15415	. . . . 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 13046	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
35	3	pntrval 27513	. . . . . . . . . 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 27513	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
38	2, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
39	36, 38	oveq12d 7434	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)))
40		peano2re 11417	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
41	16, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
42		chpcl 27074	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℝ → (ψ‘(𝑁 + 1)) ∈ ℝ)
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℝ)
44	43	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℂ)
45	41	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
46		chpcl 27074	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (ψ‘𝑁) ∈ ℝ)
47	16, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℝ)
48	47	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℂ)
49	44, 45, 48, 17	sub4d 11650	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)) = (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)))
50	28	recnd 11272	. . . . . . . . . 10 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℂ)
51		chpp1 27105	. . . . . . . . . . 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 11657	. . . . . . . . 9 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
54		ax-1cn 11196	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
55		pncan2 11497	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 𝑁) = 1)
56	17, 54, 55	sylancl 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − 𝑁) = 1)
57	53, 56	oveq12d 7434	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
58	39, 49, 57	3eqtrd 2769	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
59	58	fveq2d 6896	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))) = (abs‘((Λ‘(𝑁 + 1)) − 1)))
60	33, 59	breqtrd 5169	. . . . 5 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((Λ‘(𝑁 + 1)) − 1)))
61		1red 11245	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
62	61, 10	resubcld 11672	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ∈ ℝ)
63		0red 11247	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
64		2re 12316	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
65		eliooord 13415	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ (0(,)1) → (0 < 𝐸 ∧ 𝐸 < 1))
66	13, 65	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝐸 ∧ 𝐸 < 1))
67	66	simpld 493	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
68	14, 67	elrpd 13045	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
69		rerpdivcl 13036	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ)
70	64, 68, 69	sylancr 585	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ)
71	64	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
72		1lt2 12413	. . . . . . . . . . . 12 ⊢ 1 < 2
73	72	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
74		2cn 12317	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
75	74	div1i 11972	. . . . . . . . . . . 12 ⊢ (2 / 1) = 2
76	66	simprd 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 < 1)
77		0lt1 11766	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
78	77	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
79		2pos 12345	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
81		ltdiv2 12130	. . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
83	76, 82	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 1) < (2 / 𝐸))
84	75, 83	eqbrtrrid 5179	. . . . . . . . . . 11 ⊢ (𝜑 → 2 < (2 / 𝐸))
85	61, 71, 70, 73, 84	lttrd 11405	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (2 / 𝐸))
86		pntpbnd1.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = (exp‘(2 / 𝐸))
87	70	rpefcld 16081	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (exp‘(2 / 𝐸)) ∈ ℝ⁺)
88	86, 87	eqeltrid 2829	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
89	88	rpred 13048	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
90		pntpbnd1.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
91	88	rpxrd 13049	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
92		elioopnf 13452	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌))
95	94	simpld 493	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
96	94	simprd 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 < 𝑌)
97		pntpbnd1a.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌)))
98	97	simpld 493	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < 𝑁)
99	89, 95, 16, 96, 98	lttrd 11405	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < 𝑁)
100	86, 99	eqbrtrrid 5179	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < 𝑁)
101	2	reeflogd 26576	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
102	100, 101	breqtrrd 5171	. . . . . . . . . . 11 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁)))
103		eflt 16093	. . . . . . . . . . . 12 ⊢ (((2 / 𝐸) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
104	70, 10, 103	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
105	102, 104	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) < (log‘𝑁))
106	61, 70, 10, 85, 105	lttrd 11405	. . . . . . . . 9 ⊢ (𝜑 → 1 < (log‘𝑁))
107	61, 10, 106	ltled 11392	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝑁))
108		1re 11244	. . . . . . . . 9 ⊢ 1 ∈ ℝ
109		suble0 11758	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
110	108, 10, 109	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
111	107, 110	mpbird 256	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ 0)
112		vmage0 27071	. . . . . . . 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 11401	. . . . . 6 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)))
115	34	relogcld 26575	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ∈ ℝ)
116		readdcl 11221	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → (1 + (log‘𝑁)) ∈ ℝ)
117	108, 10, 116	sylancr 585	. . . . . . 7 ⊢ (𝜑 → (1 + (log‘𝑁)) ∈ ℝ)
118		vmalelog 27156	. . . . . . . 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 11274	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
121		epr 16184	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
122		rpmulcl 13029	. . . . . . . . . . . 12 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e · 𝑁) ∈ ℝ⁺)
123	121, 2, 122	sylancr 585	. . . . . . . . . . 11 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ⁺)
124	123	rpred 13048	. . . . . . . . . 10 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ)
125	1	nnge1d 12290	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
126	61, 16, 16, 125	leadd2dd 11859	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ≤ (𝑁 + 𝑁))
127	17	2timesd 12485	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
128	126, 127	breqtrrd 5171	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ≤ (2 · 𝑁))
129		ere 16065	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
130		egt2lt3 16182	. . . . . . . . . . . . . 14 ⊢ (2 < e ∧ e < 3)
131	130	simpli 482	. . . . . . . . . . . . 13 ⊢ 2 < e
132	64, 129, 131	ltleii 11367	. . . . . . . . . . . 12 ⊢ 2 ≤ e
133	132	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ e)
134	129	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
135	1	nngt0d 12291	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑁)
136		lemul1 12096	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ e ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
137	71, 134, 16, 135, 136	syl112anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
138	133, 137	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ≤ (e · 𝑁))
139	41, 120, 124, 128, 138	letrd 11401	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ≤ (e · 𝑁))
140	34, 123	logled 26579	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) ≤ (e · 𝑁) ↔ (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁))))
141	139, 140	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁)))
142		relogmul 26544	. . . . . . . . . 10 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
143	121, 2, 142	sylancr 585	. . . . . . . . 9 ⊢ (𝜑 → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
144		loge 26538	. . . . . . . . . 10 ⊢ (log‘e) = 1
145	144	oveq1i 7426	. . . . . . . . 9 ⊢ ((log‘e) + (log‘𝑁)) = (1 + (log‘𝑁))
146	143, 145	eqtrdi 2781	. . . . . . . 8 ⊢ (𝜑 → (log‘(e · 𝑁)) = (1 + (log‘𝑁)))
147	141, 146	breqtrd 5169	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
148	28, 115, 117, 119, 147	letrd 11401	. . . . . 6 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
149	28, 61, 10	absdifled 15413	. . . . . 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 711	. . . . 5 ⊢ (𝜑 → (abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁))
151	25, 32, 10, 60, 150	letrd 11401	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (log‘𝑁))
152	25, 10, 2, 151	lediv1dd 13106	. . 3 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / 𝑁) ≤ ((log‘𝑁) / 𝑁))
153	24, 152	eqbrtrd 5165	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ ((log‘𝑁) / 𝑁))
154	88	relogcld 26575	. . . . 5 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ)
155	154, 88	rerpdivcld 13079	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) ∈ ℝ)
156	61, 70, 85	ltled 11392	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (2 / 𝐸))
157		efle 16094	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (2 / 𝐸) ∈ ℝ) → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
158	108, 70, 157	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
159	156, 158	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (exp‘1) ≤ (exp‘(2 / 𝐸)))
160		df-e 16044	. . . . . . 7 ⊢ e = (exp‘1)
161	159, 160, 86	3brtr4g 5177	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑋)
162	144, 107	eqbrtrid 5178	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑁))
163		logleb 26555	. . . . . . . 8 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
164	121, 2, 163	sylancr 585	. . . . . . 7 ⊢ (𝜑 → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
165	162, 164	mpbird 256	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑁)
166		logdivlt 26573	. . . . . 6 ⊢ (((𝑋 ∈ ℝ ∧ e ≤ 𝑋) ∧ (𝑁 ∈ ℝ ∧ e ≤ 𝑁)) → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
167	89, 161, 16, 165, 166	syl22anc 837	. . . . 5 ⊢ (𝜑 → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
168	99, 167	mpbid 231	. . . 4 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋))
169	86	fveq2i 6895	. . . . . . 7 ⊢ (log‘𝑋) = (log‘(exp‘(2 / 𝐸)))
170	70	relogefd 26580	. . . . . . 7 ⊢ (𝜑 → (log‘(exp‘(2 / 𝐸))) = (2 / 𝐸))
171	169, 170	eqtrid 2777	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) = (2 / 𝐸))
172	171	oveq1d 7431	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) = ((2 / 𝐸) / 𝑋))
173		2rp 13011	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
174		rpdivcl 13031	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ⁺)
175	173, 68, 174	sylancr 585	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ⁺)
176	175	rpcnd 13050	. . . . . . . . . . 11 ⊢ (𝜑 → (2 / 𝐸) ∈ ℂ)
177	176	sqvald 14139	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸)↑2) = ((2 / 𝐸) · (2 / 𝐸)))
178		2cnd 12320	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
179	68	rpcnne0d 13057	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
180		div12 11924	. . . . . . . . . . 11 ⊢ (((2 / 𝐸) ∈ ℂ ∧ 2 ∈ ℂ ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
181	176, 178, 179, 180	syl3anc 1368	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
182	177, 181	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) = (2 · ((2 / 𝐸) / 𝐸)))
183	182	oveq1d 7431	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 · ((2 / 𝐸) / 𝐸)) / 2))
184	175, 68	rpdivcld 13065	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℝ⁺)
185	184	rpcnd 13050	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℂ)
186		2ne0 12346	. . . . . . . . . 10 ⊢ 2 ≠ 0
187	186	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
188	185, 178, 187	divcan3d 12025	. . . . . . . 8 ⊢ (𝜑 → ((2 · ((2 / 𝐸) / 𝐸)) / 2) = ((2 / 𝐸) / 𝐸))
189	183, 188	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 / 𝐸) / 𝐸))
190	70	resqcld 14121	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) ∈ ℝ)
191	190	rehalfcld 12489	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) ∈ ℝ)
192		1rp 13010	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
193		rpaddcl 13028	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (2 / 𝐸) ∈ ℝ⁺) → (1 + (2 / 𝐸)) ∈ ℝ⁺)
194	192, 175, 193	sylancr 585	. . . . . . . . . 10 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ⁺)
195	194	rpred 13048	. . . . . . . . 9 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ)
196	195, 191	readdcld 11273	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) ∈ ℝ)
197	191, 194	ltaddrp2d 13082	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)))
198		efgt1p2 16090	. . . . . . . . . 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 5185	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < 𝑋)
201	191, 196, 89, 197, 200	lttrd 11405	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < 𝑋)
202	189, 201	eqbrtrrd 5167	. . . . . 6 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) < 𝑋)
203	70, 68, 88, 202	ltdiv23d 13115	. . . . 5 ⊢ (𝜑 → ((2 / 𝐸) / 𝑋) < 𝐸)
204	172, 203	eqbrtrd 5165	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) < 𝐸)
205	11, 155, 14, 168, 204	lttrd 11405	. . 3 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < 𝐸)
206	11, 14, 205	ltled 11392	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ≤ 𝐸)
207	9, 11, 14, 153, 206	letrd 11401	1 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 +∞cpnf 11275 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 − cmin 11474 / cdiv 11901 ℕcn 12242 2c2 12297 3c3 12298 ℕ₀cn0 12502 ℝ⁺crp 13006 (,)cioo 13356 ↑cexp 14058 abscabs 15213 expce 16037 eceu 16038 logclog 26506 Λcvma 27042 ψcchp 27043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-e 16044 df-sin 16045 df-cos 16046 df-pi 16048 df-dvds 16231 df-gcd 16469 df-prm 16642 df-pc 16805 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-vma 27048 df-chp 27049

This theorem is referenced by: pntpbnd1 27537

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