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Theorem pntlemk 27801

Description: Lemma for pnt 27809. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
pntlem1.e	⊢ 𝐸 = (𝑈 / 𝐷)
pntlem1.k	⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
pntlem1.y	⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
pntlem1.x	⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
pntlem1.c	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z	⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
pntlem1.m	⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n	⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
pntlem1.U	⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
pntlem1.K	⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))

**Assertion**
Ref	Expression
pntlemk	⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))

Distinct variable groups: 𝑧,𝐶 𝑦,𝑛,𝑧,𝑢,𝐿 𝑛,𝐾,𝑦,𝑧 𝑛,𝑀,𝑧 𝜑,𝑛 𝑛,𝑁,𝑧 𝑅,𝑛,𝑢,𝑦,𝑧 𝑈,𝑛,𝑧 𝑛,𝑊,𝑧 𝑛,𝑋,𝑦,𝑧 𝑛,𝑌,𝑧 𝑛,𝑎,𝑢,𝑦,𝑧,𝐸 𝑛,𝑍,𝑢,𝑧 Allowed substitution hints: 𝜑(𝑦, 𝑧, 𝑢, 𝑎) 𝐴(𝑦, 𝑧, 𝑢, 𝑛, 𝑎) 𝐵(𝑦, 𝑧, 𝑢, 𝑛, 𝑎) 𝐶(𝑦, 𝑢, 𝑛, 𝑎) 𝐷(𝑦, 𝑧, 𝑢, 𝑛, 𝑎) 𝑅(𝑎) 𝑈(𝑦, 𝑢, 𝑎) 𝐹(𝑦, 𝑧, 𝑢, 𝑛, 𝑎) 𝐾(𝑢, 𝑎) 𝐿(𝑎) 𝑀(𝑦, 𝑢, 𝑎) 𝑁(𝑦, 𝑢, 𝑎) 𝑊(𝑦, 𝑢, 𝑎) 𝑋(𝑢, 𝑎) 𝑌(𝑦, 𝑢, 𝑎) 𝑍(𝑦, 𝑎)

**Proof of Theorem pntlemk**
Step	Hyp	Ref	Expression
1		2re 12326	. . . . 5 ⊢ 2 ∈ ℝ
2		fzfid 14023	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3		elfznn 13594	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
4	3	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
5	4	nnrpd 13070	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
6	5	relogcld 26819	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
7	6, 4	nndivred 12301	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
8	2, 7	fsumrecl 15804	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9		remulcl 11196	. . . . 5 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
10	1, 8, 9	sylancr 599	. . . 4 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
11		pntlem1.r	. . . . . . . . 9 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12		pntlem1.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
13		pntlem1.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
14		pntlem1.l	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
15		pntlem1.d	. . . . . . . . 9 ⊢ 𝐷 = (𝐴 + 1)
16		pntlem1.f	. . . . . . . . 9 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
17		pntlem1.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
18		pntlem1.u2	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
19		pntlem1.e	. . . . . . . . 9 ⊢ 𝐸 = (𝑈 / 𝐷)
20		pntlem1.k	. . . . . . . . 9 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
21		pntlem1.y	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
22		pntlem1.x	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
23		pntlem1.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
24		pntlem1.w	. . . . . . . . 9 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
25		pntlem1.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
26	11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	pntlemb 27792	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ ℝ⁺ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
27	26	simp1d 1160	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
28	27	relogcld 26819	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
29		peano2re 11394	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
30	28, 29	syl 18	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
31	30	resqcld 14175	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32		3re 12332	. . . . . 6 ⊢ 3 ∈ ℝ
33		readdcl 11194	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
34	28, 32, 33	sylancl 598	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
35	34, 28	remulcld 11250	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
36	27	rpred 13072	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	21	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
38	36, 37	rerpdivcld 13103	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39		1red 11220	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
40	27	rpsqrtcld 15483	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ⁺)
41	40	rpred 13072	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ)
42		ere 16161	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
44		1re 11219	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45		1lt2 12424	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
46		egt2lt3 16280	. . . . . . . . . . . . . . . 16 ⊢ (2 < e ∧ e < 3)
47	46	simpli 489	. . . . . . . . . . . . . . 15 ⊢ 2 < e
48	44, 1, 42	lttri 11347	. . . . . . . . . . . . . . 15 ⊢ ((1 < 2 ∧ 2 < e) → 1 < e)
49	45, 47, 48	mp2an 705	. . . . . . . . . . . . . 14 ⊢ 1 < e
50	44, 42, 49	ltleii 11344	. . . . . . . . . . . . 13 ⊢ 1 ≤ e
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ e)
52	26	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
53	52	simp2d 1161	. . . . . . . . . . . 12 ⊢ (𝜑 → e ≤ (√‘𝑍))
54	39, 43, 41, 51, 53	letrd 11378	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ (√‘𝑍))
55	52	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
56	39, 41, 38, 54, 55	letrd 11378	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (𝑍 / 𝑌))
57		flge1nn 13868	. . . . . . . . . 10 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
58	38, 56, 57	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
59	58	nnrpd 13070	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺)
60	59	relogcld 26819	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
61	60, 39	readdcld 11249	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
62	61	resqcld 14175	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63		logdivbnd 27751	. . . . . . 7 ⊢ ((⌊‘(𝑍 / 𝑌)) ∈ ℕ → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2))
64	58, 63	syl 18	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2))
65	1	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
66		2pos 12356	. . . . . . . 8 ⊢ 0 < 2
67	66	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
68		lemuldiv2 12107	. . . . . . 7 ⊢ ((Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ ∧ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ↔ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2)))
69	8, 62, 65, 67, 68	syl112anc 1401	. . . . . 6 ⊢ (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ↔ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2)))
70	64, 69	mpbird 260	. . . . 5 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71		reflcl 13843	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
72	38, 71	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73		flle 13846	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
74	38, 73	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
75	21	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑌)
76		1rp 13032	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
78	77, 37, 27	lediv2d 13096	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
79	75, 78	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
80	36	recnd 11248	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℂ)
81	80	div1d 11994	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 1) = 𝑍)
82	79, 81	breqtrd 5139	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
83	72, 38, 36, 74, 82	letrd 11378	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
84	59, 27	logled 26823	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
85	83, 84	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
86	60, 28, 39, 85	leadd1dd 11839	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87		0red 11222	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
88		log1 26781	. . . . . . . . 9 ⊢ (log‘1) = 0
89	58	nnge1d 12295	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90		logleb 26799	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
91	76, 59, 90	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
92	89, 91	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
93	88, 92	eqbrtrrid 5149	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
94	60	lep1d 12157	. . . . . . . 8 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
95	87, 60, 61, 93, 94	letrd 11378	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
96	87, 61, 30, 95, 86	letrd 11378	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘𝑍) + 1))
97	61, 30, 95, 96	le2sqd 14307	. . . . . 6 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
98	86, 97	mpbid 235	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
99	10, 62, 31, 70, 98	letrd 11378	. . . 4 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
100	28	resqcld 14175	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
101	65, 28	remulcld 11250	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102	100, 101	readdcld 11249	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103		loge 26782	. . . . . . 7 ⊢ (log‘e) = 1
104	40	rpge0d 13076	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (√‘𝑍))
105	41, 41, 104, 54	lemulge12d 12164	. . . . . . . . . 10 ⊢ (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
106	27	rprege0d 13079	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107		remsqsqrt 15327	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108	106, 107	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109	105, 108	breqtrd 5139	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝑍) ≤ 𝑍)
110	43, 41, 36, 53, 109	letrd 11378	. . . . . . . 8 ⊢ (𝜑 → e ≤ 𝑍)
111		epr 16282	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
112		logleb 26799	. . . . . . . . 9 ⊢ ((e ∈ ℝ⁺ ∧ 𝑍 ∈ ℝ⁺) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113	111, 27, 112	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114	110, 113	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑍))
115	103, 114	eqbrtrrid 5149	. . . . . 6 ⊢ (𝜑 → 1 ≤ (log‘𝑍))
116	39, 28, 102, 115	leadd2dd 11840	. . . . 5 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
117	28	recnd 11248	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
118		binom21 14269	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℂ → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
119	117, 118	syl 18	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
120	117	sqvald 14193	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121		df-3 12315	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
122	121	oveq1i 7426	. . . . . . . . 9 ⊢ (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123		2cnd 12330	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
124		1cnd 11213	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
125	123, 124, 117	adddird 11245	. . . . . . . . 9 ⊢ (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126	122, 125	eqtrid 2812	. . . . . . . 8 ⊢ (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127	117	mullidd 11238	. . . . . . . . 9 ⊢ (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128	127	oveq2d 7432	. . . . . . . 8 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129	126, 128	eqtr2d 2801	. . . . . . 7 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130	120, 129	oveq12d 7434	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131	117	sqcld 14194	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132		2cn 12327	. . . . . . . 8 ⊢ 2 ∈ ℂ
133		mulcl 11195	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134	132, 117, 133	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135	131, 134, 117	addassd 11242	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136		3cn 12333	. . . . . . . 8 ⊢ 3 ∈ ℂ
137	136	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℂ)
138	117, 137, 117	adddird 11245	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139	130, 135, 138	3eqtr4rd 2811	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140	116, 119, 139	3brtr4d 5145	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
141	10, 31, 35, 99, 140	letrd 11378	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
142	10, 35, 17	lemul2d 13116	. . 3 ⊢ (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)) ↔ (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍)))))
143	141, 142	mpbid 235	. 2 ⊢ (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
144	17	rpred 13072	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
145	144	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146	145	recnd 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
147	6	recnd 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
148	5	rpcnne0d 13081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149		div23 11902	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150		divass 11901	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151	149, 150	eqtr3d 2802	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152	146, 147, 148, 151	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153	152	sumeq2dv 15773	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154	144	recnd 11248	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℂ)
155	7	recnd 11248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
156	2, 154, 155	fsummulc2 15854	. . . . 5 ⊢ (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157	153, 156	eqtr4d 2803	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158	157	oveq2d 7432	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
159	8	recnd 11248	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160	123, 154, 159	mul12d 11430	. . 3 ⊢ (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161	158, 160	eqtrd 2800	. 2 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
162	34	recnd 11248	. . 3 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163	154, 162, 117	mulassd 11243	. 2 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164	143, 161, 163	3brtr4d 5145	1 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 ℝcr 11110 0cc0 11111 1c1 11112 + caddc 11114 · cmul 11116 +∞cpnf 11251 < clt 11254 ≤ cle 11255 − cmin 11452 / cdiv 11882 ℕcn 12244 2c2 12306 3c3 12307 4c4 12308 cdc 12723 ℝ⁺crp 13028 (,)cioo 13384 [,)cico 13386 [,]cicc 13387 ...cfz 13547 ⌊cfl 13837 ↑cexp 14111 √csqrt 15304 abscabs 15305 Σcsu 15757 expce 16133 eceu 16134 logclog 26750 ψcchp 27288

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-xnn0 12589 df-z 12603 df-dec 12724 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13149 df-xadd 13150 df-xmul 13151 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13548 df-fzo 13696 df-fl 13839 df-mod 13917 df-seq 14052 df-exp 14112 df-fac 14324 df-bc 14353 df-hash 14381 df-shft 15124 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-limsup 15542 df-clim 15559 df-rlim 15560 df-sum 15758 df-ef 16139 df-e 16140 df-sin 16141 df-cos 16142 df-tan 16143 df-pi 16144 df-dvds 16329 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-starv 17343 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-unif 17351 df-hom 17352 df-cco 17353 df-rest 17493 df-topn 17494 df-0g 17512 df-gsum 17513 df-topgen 17514 df-pt 17515 df-prds 17518 df-xrs 17574 df-qtop 17579 df-imas 17580 df-xps 17582 df-mre 17656 df-mrc 17657 df-acs 17659 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-mulg 19158 df-cntz 19411 df-cmn 19876 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-fbas 21549 df-fg 21550 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cld 23206 df-ntr 23207 df-cls 23208 df-nei 23285 df-lp 23323 df-perf 23324 df-cn 23414 df-cnp 23415 df-haus 23502 df-cmp 23574 df-tx 23750 df-hmeo 23943 df-fil 24034 df-fm 24126 df-flim 24127 df-flf 24128 df-xms 24508 df-ms 24509 df-tms 24510 df-cncf 25068 df-limc 26056 df-dv 26057 df-ulm 26571 df-log 26752 df-atan 27063 df-em 27188

This theorem is used by: pntlemo 27802

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