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Theorem pntlemk 27650

Description: Lemma for pnt 27658. 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 12340	. . . . 5 ⊢ 2 ∈ ℝ
2		fzfid 14014	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3		elfznn 13593	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
5	4	nnrpd 13075	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
6	5	relogcld 26665	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
7	6, 4	nndivred 12320	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
8	2, 7	fsumrecl 15770	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9		remulcl 11240	. . . . 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 587	. . . 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 27641	. . . . . . . 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 1143	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
28	27	relogcld 26665	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
29		peano2re 11434	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
31	30	resqcld 14165	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32		3re 12346	. . . . . 6 ⊢ 3 ∈ ℝ
33		readdcl 11238	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
34	28, 32, 33	sylancl 586	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
35	34, 28	remulcld 11291	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
36	27	rpred 13077	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	21	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
38	36, 37	rerpdivcld 13108	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39		1red 11262	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
40	27	rpsqrtcld 15450	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ⁺)
41	40	rpred 13077	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ)
42		ere 16125	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
44		1re 11261	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45		1lt2 12437	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
46		egt2lt3 16242	. . . . . . . . . . . . . . . 16 ⊢ (2 < e ∧ e < 3)
47	46	simpli 483	. . . . . . . . . . . . . . 15 ⊢ 2 < e
48	44, 1, 42	lttri 11387	. . . . . . . . . . . . . . 15 ⊢ ((1 < 2 ∧ 2 < e) → 1 < e)
49	45, 47, 48	mp2an 692	. . . . . . . . . . . . . 14 ⊢ 1 < e
50	44, 42, 49	ltleii 11384	. . . . . . . . . . . . 13 ⊢ 1 ≤ e
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ e)
52	26	simp2d 1144	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
53	52	simp2d 1144	. . . . . . . . . . . 12 ⊢ (𝜑 → e ≤ (√‘𝑍))
54	39, 43, 41, 51, 53	letrd 11418	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ (√‘𝑍))
55	52	simp3d 1145	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
56	39, 41, 38, 54, 55	letrd 11418	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (𝑍 / 𝑌))
57		flge1nn 13861	. . . . . . . . . 10 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
58	38, 56, 57	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
59	58	nnrpd 13075	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺)
60	59	relogcld 26665	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
61	60, 39	readdcld 11290	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
62	61	resqcld 14165	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63		logdivbnd 27600	. . . . . . 7 ⊢ ((⌊‘(𝑍 / 𝑌)) ∈ ℕ → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2))
64	58, 63	syl 17	. . . . . 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 12369	. . . . . . . 8 ⊢ 0 < 2
67	66	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
68		lemuldiv2 12149	. . . . . . 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 1376	. . . . . 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 257	. . . . 5 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71		reflcl 13836	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
72	38, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73		flle 13839	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
74	38, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
75	21	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑌)
76		1rp 13038	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
78	77, 37, 27	lediv2d 13101	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
79	75, 78	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
80	36	recnd 11289	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℂ)
81	80	div1d 12035	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 1) = 𝑍)
82	79, 81	breqtrd 5169	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
83	72, 38, 36, 74, 82	letrd 11418	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
84	59, 27	logled 26669	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
85	83, 84	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
86	60, 28, 39, 85	leadd1dd 11877	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87		0red 11264	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
88		log1 26627	. . . . . . . . 9 ⊢ (log‘1) = 0
89	58	nnge1d 12314	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90		logleb 26645	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
91	76, 59, 90	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
92	89, 91	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
93	88, 92	eqbrtrrid 5179	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
94	60	lep1d 12199	. . . . . . . 8 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
95	87, 60, 61, 93, 94	letrd 11418	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
96	87, 61, 30, 95, 86	letrd 11418	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘𝑍) + 1))
97	61, 30, 95, 96	le2sqd 14296	. . . . . 6 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
98	86, 97	mpbid 232	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
99	10, 62, 31, 70, 98	letrd 11418	. . . 4 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
100	28	resqcld 14165	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
101	65, 28	remulcld 11291	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102	100, 101	readdcld 11290	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103		loge 26628	. . . . . . 7 ⊢ (log‘e) = 1
104	40	rpge0d 13081	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (√‘𝑍))
105	41, 41, 104, 54	lemulge12d 12206	. . . . . . . . . 10 ⊢ (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
106	27	rprege0d 13084	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107		remsqsqrt 15295	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109	105, 108	breqtrd 5169	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝑍) ≤ 𝑍)
110	43, 41, 36, 53, 109	letrd 11418	. . . . . . . 8 ⊢ (𝜑 → e ≤ 𝑍)
111		epr 16244	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
112		logleb 26645	. . . . . . . . 9 ⊢ ((e ∈ ℝ⁺ ∧ 𝑍 ∈ ℝ⁺) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113	111, 27, 112	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114	110, 113	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑍))
115	103, 114	eqbrtrrid 5179	. . . . . 6 ⊢ (𝜑 → 1 ≤ (log‘𝑍))
116	39, 28, 102, 115	leadd2dd 11878	. . . . 5 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
117	28	recnd 11289	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
118		binom21 14258	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℂ → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
119	117, 118	syl 17	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
120	117	sqvald 14183	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121		df-3 12330	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
122	121	oveq1i 7441	. . . . . . . . 9 ⊢ (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123		2cnd 12344	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
124		1cnd 11256	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
125	123, 124, 117	adddird 11286	. . . . . . . . 9 ⊢ (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126	122, 125	eqtrid 2789	. . . . . . . 8 ⊢ (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127	117	mullidd 11279	. . . . . . . . 9 ⊢ (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128	127	oveq2d 7447	. . . . . . . 8 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129	126, 128	eqtr2d 2778	. . . . . . 7 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130	120, 129	oveq12d 7449	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131	117	sqcld 14184	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132		2cn 12341	. . . . . . . 8 ⊢ 2 ∈ ℂ
133		mulcl 11239	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134	132, 117, 133	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135	131, 134, 117	addassd 11283	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136		3cn 12347	. . . . . . . 8 ⊢ 3 ∈ ℂ
137	136	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℂ)
138	117, 137, 117	adddird 11286	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139	130, 135, 138	3eqtr4rd 2788	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140	116, 119, 139	3brtr4d 5175	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
141	10, 31, 35, 99, 140	letrd 11418	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
142	10, 35, 17	lemul2d 13121	. . 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 232	. 2 ⊢ (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
144	17	rpred 13077	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
145	144	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146	145	recnd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
147	6	recnd 11289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
148	5	rpcnne0d 13086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149		div23 11941	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150		divass 11940	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151	149, 150	eqtr3d 2779	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152	146, 147, 148, 151	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153	152	sumeq2dv 15738	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154	144	recnd 11289	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℂ)
155	7	recnd 11289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
156	2, 154, 155	fsummulc2 15820	. . . . 5 ⊢ (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157	153, 156	eqtr4d 2780	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158	157	oveq2d 7447	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
159	8	recnd 11289	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160	123, 154, 159	mul12d 11470	. . 3 ⊢ (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161	158, 160	eqtrd 2777	. 2 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
162	34	recnd 11289	. . 3 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163	154, 162, 117	mulassd 11284	. 2 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164	143, 161, 163	3brtr4d 5175	1 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 +∞cpnf 11292 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 cdc 12733 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 [,]cicc 13390 ...cfz 13547 ⌊cfl 13830 ↑cexp 14102 √csqrt 15272 abscabs 15273 Σcsu 15722 expce 16097 eceu 16098 logclog 26596 ψcchp 27136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-tan 16107 df-pi 16108 df-dvds 16291 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-ulm 26420 df-log 26598 df-atan 26910 df-em 27036

This theorem is referenced by: pntlemo 27651

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