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Theorem pntlemk 26182

Description: Lemma for pnt 26190. 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 11712	. . . . 5 ⊢ 2 ∈ ℝ
2		fzfid 13342	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3		elfznn 12937	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
4	3	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
5	4	nnrpd 12430	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
6	5	relogcld 25206	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
7	6, 4	nndivred 11692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
8	2, 7	fsumrecl 15091	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9		remulcl 10622	. . . . 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 589	. . . 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 26173	. . . . . . . 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 1138	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
28	27	relogcld 25206	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
29		peano2re 10813	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
31	30	resqcld 13612	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32		3re 11718	. . . . . 6 ⊢ 3 ∈ ℝ
33		readdcl 10620	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
34	28, 32, 33	sylancl 588	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
35	34, 28	remulcld 10671	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
36	27	rpred 12432	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	21	simpld 497	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
38	36, 37	rerpdivcld 12463	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39		1red 10642	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
40	27	rpsqrtcld 14771	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ⁺)
41	40	rpred 12432	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ)
42		ere 15442	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
44		1re 10641	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45		1lt2 11809	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
46		egt2lt3 15559	. . . . . . . . . . . . . . . 16 ⊢ (2 < e ∧ e < 3)
47	46	simpli 486	. . . . . . . . . . . . . . 15 ⊢ 2 < e
48	44, 1, 42	lttri 10766	. . . . . . . . . . . . . . 15 ⊢ ((1 < 2 ∧ 2 < e) → 1 < e)
49	45, 47, 48	mp2an 690	. . . . . . . . . . . . . 14 ⊢ 1 < e
50	44, 42, 49	ltleii 10763	. . . . . . . . . . . . 13 ⊢ 1 ≤ e
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ e)
52	26	simp2d 1139	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
53	52	simp2d 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → e ≤ (√‘𝑍))
54	39, 43, 41, 51, 53	letrd 10797	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ (√‘𝑍))
55	52	simp3d 1140	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
56	39, 41, 38, 54, 55	letrd 10797	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (𝑍 / 𝑌))
57		flge1nn 13192	. . . . . . . . . 10 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
58	38, 56, 57	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
59	58	nnrpd 12430	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺)
60	59	relogcld 25206	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
61	60, 39	readdcld 10670	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
62	61	resqcld 13612	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63		logdivbnd 26132	. . . . . . 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 11741	. . . . . . . 8 ⊢ 0 < 2
67	66	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
68		lemuldiv2 11521	. . . . . . 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 1370	. . . . . 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 259	. . . . 5 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71		reflcl 13167	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
72	38, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73		flle 13170	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
74	38, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
75	21	simprd 498	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑌)
76		1rp 12394	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
78	77, 37, 27	lediv2d 12456	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
79	75, 78	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
80	36	recnd 10669	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℂ)
81	80	div1d 11408	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 1) = 𝑍)
82	79, 81	breqtrd 5092	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
83	72, 38, 36, 74, 82	letrd 10797	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
84	59, 27	logled 25210	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
85	83, 84	mpbid 234	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
86	60, 28, 39, 85	leadd1dd 11254	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87		0red 10644	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
88		log1 25169	. . . . . . . . 9 ⊢ (log‘1) = 0
89	58	nnge1d 11686	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90		logleb 25186	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
91	76, 59, 90	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
92	89, 91	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
93	88, 92	eqbrtrrid 5102	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
94	60	lep1d 11571	. . . . . . . 8 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
95	87, 60, 61, 93, 94	letrd 10797	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
96	87, 61, 30, 95, 86	letrd 10797	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘𝑍) + 1))
97	61, 30, 95, 96	le2sqd 13621	. . . . . 6 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
98	86, 97	mpbid 234	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
99	10, 62, 31, 70, 98	letrd 10797	. . . 4 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
100	28	resqcld 13612	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
101	65, 28	remulcld 10671	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102	100, 101	readdcld 10670	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103		loge 25170	. . . . . . 7 ⊢ (log‘e) = 1
104	40	rpge0d 12436	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (√‘𝑍))
105	41, 41, 104, 54	lemulge12d 11578	. . . . . . . . . 10 ⊢ (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
106	27	rprege0d 12439	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107		remsqsqrt 14616	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109	105, 108	breqtrd 5092	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝑍) ≤ 𝑍)
110	43, 41, 36, 53, 109	letrd 10797	. . . . . . . 8 ⊢ (𝜑 → e ≤ 𝑍)
111		epr 15561	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
112		logleb 25186	. . . . . . . . 9 ⊢ ((e ∈ ℝ⁺ ∧ 𝑍 ∈ ℝ⁺) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113	111, 27, 112	sylancr 589	. . . . . . . 8 ⊢ (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114	110, 113	mpbid 234	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑍))
115	103, 114	eqbrtrrid 5102	. . . . . 6 ⊢ (𝜑 → 1 ≤ (log‘𝑍))
116	39, 28, 102, 115	leadd2dd 11255	. . . . 5 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
117	28	recnd 10669	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
118		binom21 13581	. . . . . 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 13508	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121		df-3 11702	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
122	121	oveq1i 7166	. . . . . . . . 9 ⊢ (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123		2cnd 11716	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
124		1cnd 10636	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
125	123, 124, 117	adddird 10666	. . . . . . . . 9 ⊢ (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126	122, 125	syl5eq 2868	. . . . . . . 8 ⊢ (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127	117	mulid2d 10659	. . . . . . . . 9 ⊢ (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128	127	oveq2d 7172	. . . . . . . 8 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129	126, 128	eqtr2d 2857	. . . . . . 7 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130	120, 129	oveq12d 7174	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131	117	sqcld 13509	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132		2cn 11713	. . . . . . . 8 ⊢ 2 ∈ ℂ
133		mulcl 10621	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134	132, 117, 133	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135	131, 134, 117	addassd 10663	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136		3cn 11719	. . . . . . . 8 ⊢ 3 ∈ ℂ
137	136	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℂ)
138	117, 137, 117	adddird 10666	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139	130, 135, 138	3eqtr4rd 2867	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140	116, 119, 139	3brtr4d 5098	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
141	10, 31, 35, 99, 140	letrd 10797	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
142	10, 35, 17	lemul2d 12476	. . 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 234	. 2 ⊢ (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
144	17	rpred 12432	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
145	144	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146	145	recnd 10669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
147	6	recnd 10669	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
148	5	rpcnne0d 12441	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149		div23 11317	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150		divass 11316	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151	149, 150	eqtr3d 2858	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152	146, 147, 148, 151	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153	152	sumeq2dv 15060	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154	144	recnd 10669	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℂ)
155	7	recnd 10669	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
156	2, 154, 155	fsummulc2 15139	. . . . 5 ⊢ (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157	153, 156	eqtr4d 2859	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158	157	oveq2d 7172	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
159	8	recnd 10669	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160	123, 154, 159	mul12d 10849	. . 3 ⊢ (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161	158, 160	eqtrd 2856	. 2 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
162	34	recnd 10669	. . 3 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163	154, 162, 117	mulassd 10664	. 2 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164	143, 161, 163	3brtr4d 5098	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 +∞cpnf 10672 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 ℕcn 11638 2c2 11693 3c3 11694 4c4 11695 cdc 12099 ℝ⁺crp 12390 (,)cioo 12739 [,)cico 12741 [,]cicc 12742 ...cfz 12893 ⌊cfl 13161 ↑cexp 13430 √csqrt 14592 abscabs 14593 Σcsu 15042 expce 15415 eceu 15416 logclog 25138 ψcchp 25670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-tan 15425 df-pi 15426 df-dvds 15608 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-ulm 24965 df-log 25140 df-atan 25445 df-em 25570

This theorem is referenced by: pntlemo 26183

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