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Theorem pntpbnd1a 27919

Description: Lemma for pntpbnd 27922. (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 13167	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
3		pntpbnd.r	. . . . . . . 8 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
4	3	pntrf 27897	. . . . . . 7 ⊢ 𝑅:ℝ⁺⟶ℝ
5	4	ffvelcdmi 7086	. . . . . 6 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) ∈ ℝ)
6	2, 5	syl 18	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℝ)
7	6, 2	rerpdivcld 13200	. . . 4 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℝ)
8	7	recnd 11342	. . 3 ⊢ (𝜑 → ((𝑅‘𝑁) / 𝑁) ∈ ℂ)
9	8	abscld 15613	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ∈ ℝ)
10	2	relogcld 26958	. . 3 ⊢ (𝜑 → (log‘𝑁) ∈ ℝ)
11	10, 2	rerpdivcld 13200	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ∈ ℝ)
12		ioossre 13543	. . 3 ⊢ (0(,)1) ⊆ ℝ
13		pntpbnd1.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1))
14	12, 13	sselid 3907	. 2 ⊢ (𝜑 → 𝐸 ∈ ℝ)
15	6	recnd 11342	. . . . 5 ⊢ (𝜑 → (𝑅‘𝑁) ∈ ℂ)
16	1	nnred 12355	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	recnd 11342	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℂ)
18	1	nnne0d 12393	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
19	15, 17, 18	absdivd 15632	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)))
20	1	nnnn0d 12672	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	nn0ge0d 12675	. . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑁)
22	16, 21	absidd 15597	. . . . 5 ⊢ (𝜑 → (abs‘𝑁) = 𝑁)
23	22	oveq2d 7438	. . . 4 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / (abs‘𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
24	19, 23	eqtrd 2795	. . 3 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) = ((abs‘(𝑅‘𝑁)) / 𝑁))
25	15	abscld 15613	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ∈ ℝ)
26	1	peano2nnd 12357	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
27		vmacl 27452	. . . . . . . . 9 ⊢ ((𝑁 + 1) ∈ ℕ → (Λ‘(𝑁 + 1)) ∈ ℝ)
28	26, 27	syl 18	. . . . . . . 8 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℝ)
29		peano2rem 11630	. . . . . . . 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 11342	. . . . . 6 ⊢ (𝜑 → ((Λ‘(𝑁 + 1)) − 1) ∈ ℂ)
32	31	abscld 15613	. . . . 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 13167	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ⁺)
35	3	pntrval 27896	. . . . . . . . . 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 27896	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ⁺ → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
38	2, 37	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝑁) = ((ψ‘𝑁) − 𝑁))
39	36, 38	oveq12d 7440	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)))
40		peano2re 11488	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
41	16, 40	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ∈ ℝ)
42		chpcl 27458	. . . . . . . . . . 11 ⊢ ((𝑁 + 1) ∈ ℝ → (ψ‘(𝑁 + 1)) ∈ ℝ)
43	41, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℝ)
44	43	recnd 11342	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘(𝑁 + 1)) ∈ ℂ)
45	41	recnd 11342	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ∈ ℂ)
46		chpcl 27458	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (ψ‘𝑁) ∈ ℝ)
47	16, 46	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℝ)
48	47	recnd 11342	. . . . . . . . 9 ⊢ (𝜑 → (ψ‘𝑁) ∈ ℂ)
49	44, 45, 48, 17	sub4d 11723	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (𝑁 + 1)) − ((ψ‘𝑁) − 𝑁)) = (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)))
50	28	recnd 11342	. . . . . . . . . 10 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ∈ ℂ)
51		chpp1 27489	. . . . . . . . . . 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 11733	. . . . . . . . 9 ⊢ (𝜑 → ((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) = (Λ‘(𝑁 + 1)))
54		ax-1cn 11263	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
55		pncan2 11569	. . . . . . . . . 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 7440	. . . . . . . 8 ⊢ (𝜑 → (((ψ‘(𝑁 + 1)) − (ψ‘𝑁)) − ((𝑁 + 1) − 𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
58	39, 49, 57	3eqtrd 2799	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁)) = ((Λ‘(𝑁 + 1)) − 1))
59	58	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → (abs‘((𝑅‘(𝑁 + 1)) − (𝑅‘𝑁))) = (abs‘((Λ‘(𝑁 + 1)) − 1)))
60	33, 59	breqtrd 5130	. . . . 5 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (abs‘((Λ‘(𝑁 + 1)) − 1)))
61		1red 11314	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
62	61, 10	resubcld 11749	. . . . . . 7 ⊢ (𝜑 → (1 − (log‘𝑁)) ∈ ℝ)
63		0red 11316	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℝ)
64		2re 12422	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
65		eliooord 13541	. . . . . . . . . . . . . 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 13166	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
69		rerpdivcl 13157	. . . . . . . . . . 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 12520	. . . . . . . . . . . 12 ⊢ 1 < 2
73	72	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 1 < 2)
74		2cn 12423	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
75	74	div1i 12050	. . . . . . . . . . . 12 ⊢ (2 / 1) = 2
76	66	simprd 501	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 < 1)
77		0lt1 11843	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
78	77	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 1)
79		2pos 12452	. . . . . . . . . . . . . . 15 ⊢ 0 < 2
80	79	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 2)
81		ltdiv2 12208	. . . . . . . . . . . . . 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 5140	. . . . . . . . . . 11 ⊢ (𝜑 → 2 < (2 / 𝐸))
85	61, 71, 70, 73, 84	lttrd 11476	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (2 / 𝐸))
86		pntpbnd1.x	. . . . . . . . . . . . 13 ⊢ 𝑋 = (exp‘(2 / 𝐸))
87	70	rpefcld 16280	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (exp‘(2 / 𝐸)) ∈ ℝ⁺)
88	86, 87	eqeltrid 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
89	88	rpred 13169	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ ℝ)
90		pntpbnd1.y	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑌 ∈ (𝑋(,)+∞))
91	88	rpxrd 13170	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ∈ ℝ^*)
92		elioopnf 13579	. . . . . . . . . . . . . . . . 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 11476	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 < 𝑁)
100	86, 99	eqbrtrrid 5140	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < 𝑁)
101	2	reeflogd 26959	. . . . . . . . . . . 12 ⊢ (𝜑 → (exp‘(log‘𝑁)) = 𝑁)
102	100, 101	breqtrrd 5132	. . . . . . . . . . 11 ⊢ (𝜑 → (exp‘(2 / 𝐸)) < (exp‘(log‘𝑁)))
103		eflt 16292	. . . . . . . . . . . 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 11476	. . . . . . . . 9 ⊢ (𝜑 → 1 < (log‘𝑁))
107	61, 10, 106	ltled 11463	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (log‘𝑁))
108		1re 11313	. . . . . . . . 9 ⊢ 1 ∈ ℝ
109		suble0 11835	. . . . . . . . 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 27455	. . . . . . . 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 11472	. . . . . 6 ⊢ (𝜑 → (1 − (log‘𝑁)) ≤ (Λ‘(𝑁 + 1)))
115	34	relogcld 26958	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ∈ ℝ)
116		readdcl 11288	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) → (1 + (log‘𝑁)) ∈ ℝ)
117	108, 10, 116	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (1 + (log‘𝑁)) ∈ ℝ)
118		vmalelog 27539	. . . . . . . 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 11344	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
121		epr 16383	. . . . . . . . . . . 12 ⊢ e ∈ ℝ⁺
122		rpmulcl 13150	. . . . . . . . . . . 12 ⊢ ((e ∈ ℝ⁺ ∧ 𝑁 ∈ ℝ⁺) → (e · 𝑁) ∈ ℝ⁺)
123	121, 2, 122	sylancr 599	. . . . . . . . . . 11 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ⁺)
124	123	rpred 13169	. . . . . . . . . 10 ⊢ (𝜑 → (e · 𝑁) ∈ ℝ)
125	1	nnge1d 12391	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
126	61, 16, 16, 125	leadd2dd 11936	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + 1) ≤ (𝑁 + 𝑁))
127	17	2timesd 12594	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
128	126, 127	breqtrrd 5132	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 + 1) ≤ (2 · 𝑁))
129		ere 16262	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
130		egt2lt3 16381	. . . . . . . . . . . . . 14 ⊢ (2 < e ∧ e < 3)
131	130	simpli 489	. . . . . . . . . . . . 13 ⊢ 2 < e
132	64, 129, 131	ltleii 11438	. . . . . . . . . . . 12 ⊢ 2 ≤ e
133	132	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ e)
134	129	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
135	1	nngt0d 12392	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑁)
136		lemul1 12174	. . . . . . . . . . . 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 11472	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 + 1) ≤ (e · 𝑁))
140	34, 123	logled 26962	. . . . . . . . 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 26927	. . . . . . . . . 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 26921	. . . . . . . . . 10 ⊢ (log‘e) = 1
145	144	oveq1i 7432	. . . . . . . . 9 ⊢ ((log‘e) + (log‘𝑁)) = (1 + (log‘𝑁))
146	143, 145	eqtrdi 2811	. . . . . . . 8 ⊢ (𝜑 → (log‘(e · 𝑁)) = (1 + (log‘𝑁)))
147	141, 146	breqtrd 5130	. . . . . . 7 ⊢ (𝜑 → (log‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
148	28, 115, 117, 119, 147	letrd 11472	. . . . . 6 ⊢ (𝜑 → (Λ‘(𝑁 + 1)) ≤ (1 + (log‘𝑁)))
149	28, 61, 10	absdifled 15611	. . . . . 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 11472	. . . 4 ⊢ (𝜑 → (abs‘(𝑅‘𝑁)) ≤ (log‘𝑁))
152	25, 10, 2, 151	lediv1dd 13227	. . 3 ⊢ (𝜑 → ((abs‘(𝑅‘𝑁)) / 𝑁) ≤ ((log‘𝑁) / 𝑁))
153	24, 152	eqbrtrd 5126	. 2 ⊢ (𝜑 → (abs‘((𝑅‘𝑁) / 𝑁)) ≤ ((log‘𝑁) / 𝑁))
154	88	relogcld 26958	. . . . 5 ⊢ (𝜑 → (log‘𝑋) ∈ ℝ)
155	154, 88	rerpdivcld 13200	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) ∈ ℝ)
156	61, 70, 85	ltled 11463	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ (2 / 𝐸))
157		efle 16293	. . . . . . . . 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 16241	. . . . . . 7 ⊢ e = (exp‘1)
161	159, 160, 86	3brtr4g 5138	. . . . . 6 ⊢ (𝜑 → e ≤ 𝑋)
162	144, 107	eqbrtrid 5139	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑁))
163		logleb 26938	. . . . . . . 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 26956	. . . . . 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 6891	. . . . . . 7 ⊢ (log‘𝑋) = (log‘(exp‘(2 / 𝐸)))
170	70	relogefd 26963	. . . . . . 7 ⊢ (𝜑 → (log‘(exp‘(2 / 𝐸))) = (2 / 𝐸))
171	169, 170	eqtrid 2807	. . . . . 6 ⊢ (𝜑 → (log‘𝑋) = (2 / 𝐸))
172	171	oveq1d 7437	. . . . 5 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) = ((2 / 𝐸) / 𝑋))
173		2rp 13130	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ⁺
174		rpdivcl 13152	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ⁺ ∧ 𝐸 ∈ ℝ⁺) → (2 / 𝐸) ∈ ℝ⁺)
175	173, 68, 174	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 / 𝐸) ∈ ℝ⁺)
176	175	rpcnd 13171	. . . . . . . . . . 11 ⊢ (𝜑 → (2 / 𝐸) ∈ ℂ)
177	176	sqvald 14291	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸)↑2) = ((2 / 𝐸) · (2 / 𝐸)))
178		2cnd 12426	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℂ)
179	68	rpcnne0d 13178	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
180		div12 12001	. . . . . . . . . . 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 2795	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) = (2 · ((2 / 𝐸) / 𝐸)))
183	182	oveq1d 7437	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 · ((2 / 𝐸) / 𝐸)) / 2))
184	175, 68	rpdivcld 13186	. . . . . . . . . 10 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℝ⁺)
185	184	rpcnd 13171	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) ∈ ℂ)
186		2ne0 12454	. . . . . . . . . 10 ⊢ 2 ≠ 0
187	186	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ≠ 0)
188	185, 178, 187	divcan3d 12103	. . . . . . . 8 ⊢ (𝜑 → ((2 · ((2 / 𝐸) / 𝐸)) / 2) = ((2 / 𝐸) / 𝐸))
189	183, 188	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) = ((2 / 𝐸) / 𝐸))
190	70	resqcld 14273	. . . . . . . . 9 ⊢ (𝜑 → ((2 / 𝐸)↑2) ∈ ℝ)
191	190	rehalfcld 12598	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) ∈ ℝ)
192		1rp 13129	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ⁺
193		rpaddcl 13149	. . . . . . . . . . 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 13169	. . . . . . . . 9 ⊢ (𝜑 → (1 + (2 / 𝐸)) ∈ ℝ)
196	195, 191	readdcld 11343	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) ∈ ℝ)
197	191, 194	ltaddrp2d 13203	. . . . . . . 8 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)))
198		efgt1p2 16289	. . . . . . . . . 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 5146	. . . . . . . 8 ⊢ (𝜑 → ((1 + (2 / 𝐸)) + (((2 / 𝐸)↑2) / 2)) < 𝑋)
201	191, 196, 89, 197, 200	lttrd 11476	. . . . . . 7 ⊢ (𝜑 → (((2 / 𝐸)↑2) / 2) < 𝑋)
202	189, 201	eqbrtrrd 5128	. . . . . 6 ⊢ (𝜑 → ((2 / 𝐸) / 𝐸) < 𝑋)
203	70, 68, 88, 202	ltdiv23d 13236	. . . . 5 ⊢ (𝜑 → ((2 / 𝐸) / 𝑋) < 𝐸)
204	172, 203	eqbrtrd 5126	. . . 4 ⊢ (𝜑 → ((log‘𝑋) / 𝑋) < 𝐸)
205	11, 155, 14, 168, 204	lttrd 11476	. . 3 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) < 𝐸)
206	11, 14, 205	ltled 11463	. 2 ⊢ (𝜑 → ((log‘𝑁) / 𝑁) ≤ 𝐸)
207	9, 11, 14, 153, 206	letrd 11472	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 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6541 (class class class)co 7422 ℂcc 11203 ℝcr 11204 0cc0 11205 1c1 11206 + caddc 11208 · cmul 11210 +∞cpnf 11345 ℝ^*cxr 11347 < clt 11348 ≤ cle 11349 − cmin 11546 / cdiv 11978 ℕcn 12340 2c2 12402 3c3 12403 ℕ₀cn0 12611 ℝ⁺crp 13125 (,)cioo 13481 ↑cexp 14209 abscabs 15408 expce 16234 eceu 16235 logclog 26889 Λcvma 27426 ψcchp 27427

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-inf2 9647 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283 ax-addf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-of 7695 df-om 7880 df-1st 8003 df-2nd 8004 df-supp 8182 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-oadd 8485 df-er 8722 df-map 8854 df-pm 8855 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-fsupp 9359 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9509 df-dju 9987 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-q 13081 df-rp 13126 df-xneg 13246 df-xadd 13247 df-xmul 13248 df-ioo 13485 df-ioc 13486 df-ico 13487 df-icc 13488 df-fz 13645 df-fzo 13794 df-fl 13937 df-mod 14015 df-seq 14150 df-exp 14210 df-fac 14423 df-bc 14452 df-hash 14480 df-shft 15227 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-limsup 15645 df-clim 15662 df-rlim 15663 df-sum 15861 df-ef 16240 df-e 16241 df-sin 16242 df-cos 16243 df-pi 16245 df-dvds 16430 df-gcd 16672 df-prm 16854 df-pc 17022 df-struct 17332 df-sets 17349 df-slot 17367 df-ndx 17379 df-base 17395 df-ress 17416 df-plusg 17448 df-mulr 17449 df-starv 17450 df-sca 17451 df-vsca 17452 df-ip 17453 df-tset 17454 df-ple 17455 df-ds 17457 df-unif 17458 df-hom 17459 df-cco 17460 df-rest 17600 df-topn 17601 df-0g 17619 df-gsum 17620 df-topgen 17621 df-pt 17622 df-prds 17625 df-xrs 17681 df-qtop 17686 df-imas 17687 df-xps 17689 df-mre 17763 df-mrc 17764 df-acs 17766 df-mgm 18823 df-sgrp 18915 df-mnd 18931 df-submnd 18986 df-mulg 19285 df-cntz 19538 df-cmn 20003 df-psmet 21677 df-xmet 21678 df-met 21679 df-bl 21680 df-mopn 21681 df-fbas 21682 df-fg 21683 df-cnfld 21686 df-top 23219 df-topon 23236 df-topsp 23258 df-bases 23271 df-cld 23344 df-ntr 23345 df-cls 23346 df-nei 23423 df-lp 23461 df-perf 23462 df-cn 23552 df-cnp 23553 df-haus 23640 df-tx 23888 df-hmeo 24081 df-fil 24172 df-fm 24264 df-flim 24265 df-flf 24266 df-xms 24646 df-ms 24647 df-tms 24648 df-cncf 25206 df-limc 26193 df-dv 26194 df-log 26891 df-vma 27432 df-chp 27433

This theorem is used by: pntpbnd1 27920

W3C validator