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Theorem pntpbnd1a 25274

Description: Lemma for pntpbnd 25277. (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 11870	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
3		pntpbnd.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4	3	pntrf 25252	. . . . . . 7 ⊢ 𝑅:ℝ⁺⟶ℝ
5	4	ffvelrni 6358	. . . . . 6 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) ∈ ℝ)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℝ)
7	6, 2	rerpdivcld 11903	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℝ)
8	7	recnd 10068	. . 3 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℂ)
9	8	abscld 14175	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ∈ ℝ)
10	2	relogcld 24369	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
11	10, 2	rerpdivcld 11903	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ∈ ℝ)
12		ioossre 12235	. . 3 ⊢ (0(,)1) ⊆ ℝ
13		pntpbnd1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
14	12, 13	sseldi 3601	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ)
15	6	recnd 10068	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℂ)
16	1	nnred 11035	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	recnd 10068	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	1	nnne0d 11065	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
19	15, 17, 18	absdivd 14194	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)))
20	1	nnnn0d 11351	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 11354	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁)
22	16, 21	absidd 14161	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) = 𝑁)
23	22	oveq2d 6666	. . . 4 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
24	19, 23	eqtrd 2656	. . 3 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
25	15	abscld 14175	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ∈ ℝ)
26	1	peano2nnd 11037	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27		vmacl 24844	. . . . . . . . 9 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℝ)
29		peano2rem 10348	. . . . . . . 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 10068	. . . . . 6 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℂ)
32	31	abscld 14175	. . . . 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 11870	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
35	3	pntrval 25251	. . . . . . . . . 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 25251	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
38	2, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
39	36, 38	oveq12d 6668	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)))
40		peano2re 10209	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
41	16, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
42		chpcl 24850	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℝ → (ψ‘(𝑁 + 1)) ∈ ℝ)
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℝ)
44	43	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℂ)
45	41	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
46		chpcl 24850	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (ψ‘𝑁) ∈ ℝ)
47	16, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℝ)
48	47	recnd 10068	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℂ)
49	44, 45, 48, 17	sub4d 10441	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)) = (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)))
50		chpp1 24881	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
51	20, 50	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) = ((ψ‘𝑁) + (Λ‘(𝑁 + 1))))
52	51	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (((ψ‘𝑁) + (Λ‘(𝑁 + 1))) − (ψ‘𝑁)))
53	28	recnd 10068	. . . . . . . . . . 11 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℂ)
54	48, 53	pncan2d 10394	. . . . . . . . . 10 ⊢ (𝜑 → (((ψ‘𝑁) + (Λ‘(𝑁 + 1))) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
55	52, 54	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
56		ax-1cn 9994	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
57		pncan2 10288	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 𝑁) = 1)
58	17, 56, 57	sylancl 694	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) − 𝑁) = 1)
59	55, 58	oveq12d 6668	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
60	39, 49, 59	3eqtrd 2660	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
61	60	fveq2d 6195	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))) = (abs‘((Λ‘(𝑁 + 1)) − 1)))
62	33, 61	breqtrd 4679	. . . . 5 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((Λ‘(𝑁 + 1)) − 1)))
63		1red 10055	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
64	63, 10	resubcld 10458	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ∈ ℝ)
65		0red 10041	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
66		2re 11090	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
67		eliooord 12233	. . . . . . . . . . . . . 14 ⊢ (𝐸 ∈ (0(,)1) → (0 < 𝐸 ∧ 𝐸 < 1))
68	13, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 < 𝐸 ∧ 𝐸 < 1))
69	68	simpld 475	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐸)
70	14, 69	elrpd 11869	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
71		rerpdivcl 11861	. . . . . . . . . . 11 ⊢ ((2 ∈ ℝ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ)
72	66, 70, 71	sylancr 695	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ)
73	66	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
74		1lt2 11194	. . . . . . . . . . . 12 ⊢ 1 < 2
75	74	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
76		2cn 11091	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
77	76	div1i 10753	. . . . . . . . . . . 12 ⊢ (2 / 1) = 2
78	68	simprd 479	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 < 1)
79		0lt1 10550	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
81		2pos 11112	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
82	81	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
83		ltdiv2 10909	. . . . . . . . . . . . . 14 ⊢ (((𝐸 ∈ ℝ ∧ 0 < 𝐸) ∧ (1 ∈ ℝ ∧ 0 < 1) ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
84	14, 69, 63, 80, 73, 82, 83	syl222anc 1342	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 < 1 ↔ (2 / 1) < (2 / 𝐸)))
85	78, 84	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 1) < (2 / 𝐸))
86	77, 85	syl5eqbrr 4689	. . . . . . . . . . 11 ⊢ (𝜑 → 2 < (2 / 𝐸))
87	63, 73, 72, 75, 86	lttrd 10198	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (2 / 𝐸))
88		pntpbnd1.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = (exp‘(2 / 𝐸))
89	72	rpefcld 14835	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (exp‘(2 / 𝐸)) ∈ ℝ⁺)
90	88, 89	syl5eqel 2705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
91	90	rpred 11872	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
92		pntpbnd1.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
93	90	rpxrd 11873	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
94		elioopnf 12267	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℝ^* → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
95	93, 94	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑌 ∈ (𝑋(,)+∞) ↔ (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌)))
96	92, 95	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝑋 < 𝑌))
97	96	simpld 475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ℝ)
98	96	simprd 479	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 < 𝑌)
99		pntpbnd1a.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌 < 𝑁 ∧ 𝑁 ≤ (𝐾 · 𝑌)))
100	99	simpld 475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < 𝑁)
101	91, 97, 16, 98, 100	lttrd 10198	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < 𝑁)
102	88, 101	syl5eqbrr 4689	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < 𝑁)
103	2	reeflogd 24370	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
104	102, 103	breqtrrd 4681	. . . . . . . . . . 11 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁)))
105		eflt 14847	. . . . . . . . . . . 12 ⊢ (((2 / 𝐸) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
106	72, 10, 105	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 / 𝐸) < (log‘𝑁) ↔ (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁))))
107	104, 106	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → (2 / 𝐸) < (log‘𝑁))
108	63, 72, 10, 87, 107	lttrd 10198	. . . . . . . . 9 ⊢ (𝜑 → 1 < (log‘𝑁))
109	63, 10, 108	ltled 10185	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝑁))
110		1re 10039	. . . . . . . . 9 ⊢ 1 ∈ ℝ
111		suble0 10542	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
112	110, 10, 111	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → ((1 − (log‘𝑁)) ≤ 0 ↔ 1 ≤ (log‘𝑁)))
113	109, 112	mpbird 247	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ 0)
114		vmage0 24847	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → 0 ≤ (Λ‘(𝑁 + 1)))
115	26, 114	syl 17	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (Λ‘(𝑁 + 1)))
116	64, 65, 28, 113, 115	letrd 10194	. . . . . 6 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)))
117	34	relogcld 24369	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ∈ ℝ)
118		readdcl 10019	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → (1 + (log‘𝑁)) ∈ ℝ)
119	110, 10, 118	sylancr 695	. . . . . . 7 ⊢ (𝜑 → (1 + (log‘𝑁)) ∈ ℝ)
120		vmalelog 24930	. . . . . . . 8 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
121	26, 120	syl 17	. . . . . . 7 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (log‘(𝑁 + 1)))
122	73, 16	remulcld 10070	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
123		epr 14936	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
124		rpmulcl 11855	. . . . . . . . . . . 12 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e · 𝑁) ∈ ℝ⁺)
125	123, 2, 124	sylancr 695	. . . . . . . . . . 11 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ⁺)
126	125	rpred 11872	. . . . . . . . . 10 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ)
127	1	nnge1d 11063	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
128	63, 16, 16, 127	leadd2dd 10642	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ≤ (𝑁 + 𝑁))
129	17	2timesd 11275	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
130	128, 129	breqtrrd 4681	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ≤ (2 · 𝑁))
131		ere 14819	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
132		egt2lt3 14934	. . . . . . . . . . . . . 14 ⊢ (2 < e ∧ e < 3)
133	132	simpli 474	. . . . . . . . . . . . 13 ⊢ 2 < e
134	66, 131, 133	ltleii 10160	. . . . . . . . . . . 12 ⊢ 2 ≤ e
135	134	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ e)
136	131	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
137	1	nngt0d 11064	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑁)
138		lemul1 10875	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ e ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
139	73, 136, 16, 137, 138	syl112anc 1330	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ e ↔ (2 · 𝑁) ≤ (e · 𝑁)))
140	135, 139	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ≤ (e · 𝑁))
141	41, 122, 126, 130, 140	letrd 10194	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ≤ (e · 𝑁))
142	34, 125	logled 24373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + 1) ≤ (e · 𝑁) ↔ (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁))))
143	141, 142	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (log‘(e · 𝑁)))
144		relogmul 24338	. . . . . . . . . 10 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
145	123, 2, 144	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → (log‘(e · 𝑁)) = ((log‘e) + (log‘𝑁)))
146		loge 24333	. . . . . . . . . 10 ⊢ (log‘e) = 1
147	146	oveq1i 6660	. . . . . . . . 9 ⊢ ((log‘e) + (log‘𝑁)) = (1 + (log‘𝑁))
148	145, 147	syl6eq 2672	. . . . . . . 8 ⊢ (𝜑 → (log‘(e · 𝑁)) = (1 + (log‘𝑁)))
149	143, 148	breqtrd 4679	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
150	28, 117, 119, 121, 149	letrd 10194	. . . . . 6 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
151	28, 63, 10	absdifled 14173	. . . . . 6 ⊢ (𝜑 → ((abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁) ↔ ((1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)) ∧ (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))))
152	116, 150, 151	mpbir2and 957	. . . . 5 ⊢ (𝜑 → (abs‘((Λ‘(𝑁 + 1)) − 1)) ≤ (log‘𝑁))
153	25, 32, 10, 62, 152	letrd 10194	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (log‘𝑁))
154	25, 10, 2, 153	lediv1dd 11930	. . 3 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / 𝑁) ≤ ((log‘𝑁) / 𝑁))
155	24, 154	eqbrtrd 4675	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ ((log‘𝑁) / 𝑁))
156	90	relogcld 24369	. . . . 5 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ)
157	156, 90	rerpdivcld 11903	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) ∈ ℝ)
158	63, 72, 87	ltled 10185	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (2 / 𝐸))
159		efle 14848	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (2 / 𝐸) ∈ ℝ) → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
160	110, 72, 159	sylancr 695	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ (2 / 𝐸) ↔ (exp‘1) ≤ (exp‘(2 / 𝐸))))
161	158, 160	mpbid 222	. . . . . . 7 ⊢ (𝜑 → (exp‘1) ≤ (exp‘(2 / 𝐸)))
162		df-e 14799	. . . . . . 7 ⊢ e = (exp‘1)
163	161, 162, 88	3brtr4g 4687	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑋)
164	146, 109	syl5eqbr 4688	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑁))
165		logleb 24349	. . . . . . . 8 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
166	123, 2, 165	sylancr 695	. . . . . . 7 ⊢ (𝜑 → (e ≤ 𝑁 ↔ (log‘e) ≤ (log‘𝑁)))
167	164, 166	mpbird 247	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑁)
168		logdivlt 24367	. . . . . 6 ⊢ (((𝑋 ∈ ℝ ∧ e ≤ 𝑋) ∧ (𝑁 ∈ ℝ ∧ e ≤ 𝑁)) → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
169	91, 163, 16, 167, 168	syl22anc 1327	. . . . 5 ⊢ (𝜑 → (𝑋 < 𝑁 ↔ ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋)))
170	101, 169	mpbid 222	. . . 4 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < ((log‘𝑋) / 𝑋))
171	88	fveq2i 6194	. . . . . . 7 ⊢ (log‘𝑋) = (log‘(exp‘(2 / 𝐸)))
172	72	relogefd 24374	. . . . . . 7 ⊢ (𝜑 → (log‘(exp‘(2 / 𝐸))) = (2 / 𝐸))
173	171, 172	syl5eq 2668	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) = (2 / 𝐸))
174	173	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) = ((2 / 𝐸) / 𝑋))
175		2rp 11837	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
176		rpdivcl 11856	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ⁺)
177	175, 70, 176	sylancr 695	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ⁺)
178	177	rpcnd 11874	. . . . . . . . . . 11 ⊢ (𝜑 → (2 / 𝐸) ∈ ℂ)
179	178	sqvald 13005	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸)↑2) = ((2 / 𝐸) · (2 / 𝐸)))
180		2cnd 11093	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
181	70	rpcnne0d 11881	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
182		div12 10707	. . . . . . . . . . 11 ⊢ (((2 / 𝐸) ∈ ℂ ∧ 2 ∈ ℂ ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
183	178, 180, 181, 182	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) · (2 / 𝐸)) = (2 · ((2 / 𝐸) / 𝐸)))
184	179, 183	eqtrd 2656	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) = (2 · ((2 / 𝐸) / 𝐸)))
185	184	oveq1d 6665	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 · ((2 / 𝐸) / 𝐸)) / 2))
186	177, 70	rpdivcld 11889	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℝ⁺)
187	186	rpcnd 11874	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℂ)
188		2ne0 11113	. . . . . . . . . 10 ⊢ 2 ≠ 0
189	188	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
190	187, 180, 189	divcan3d 10806	. . . . . . . 8 ⊢ (𝜑 → ((2 · ((2 / 𝐸) / 𝐸)) / 2) = ((2 / 𝐸) / 𝐸))
191	185, 190	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 / 𝐸) / 𝐸))
192	72	resqcld 13035	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) ∈ ℝ)
193	192	rehalfcld 11279	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) ∈ ℝ)
194		1rp 11836	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
195		rpaddcl 11854	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (2 / 𝐸) ∈ ℝ⁺) → (1 + (2 / 𝐸)) ∈ ℝ⁺)
196	194, 177, 195	sylancr 695	. . . . . . . . . 10 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ⁺)
197	196	rpred 11872	. . . . . . . . 9 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ)
198	197, 193	readdcld 10069	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) ∈ ℝ)
199	193, 196	ltaddrp2d 11906	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)))
200		efgt1p2 14844	. . . . . . . . . 10 ⊢ ((2 / 𝐸) ∈ ℝ⁺ → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
201	177, 200	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < (exp‘(2 / 𝐸)))
202	201, 88	syl6breqr 4695	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < 𝑋)
203	193, 198, 91, 199, 202	lttrd 10198	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < 𝑋)
204	191, 203	eqbrtrrd 4677	. . . . . 6 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) < 𝑋)
205	72, 70, 90, 204	ltdiv23d 11937	. . . . 5 ⊢ (𝜑 → ((2 / 𝐸) / 𝑋) < 𝐸)
206	174, 205	eqbrtrd 4675	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) < 𝐸)
207	11, 157, 14, 170, 206	lttrd 10198	. . 3 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < 𝐸)
208	11, 14, 207	ltled 10185	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ≤ 𝐸)
209	9, 11, 14, 155, 208	letrd 10194	1 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 ℕ₀cn0 11292 ℝ⁺crp 11832 (,)cioo 12175 ↑cexp 12860 abscabs 13974 expce 14792 eceu 14793 logclog 24301 Λcvma 24818 ψcchp 24819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-vma 24824 df-chp 24825

This theorem is referenced by: pntpbnd1 25275

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