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Theorem pntlemk 27587

Description: Lemma for pnt 27595. 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 12246	. . . . 5 ⊢ 2 ∈ ℝ
2		fzfid 13926	. . . . . 6 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3		elfznn 13498	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
5	4	nnrpd 12975	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
6	5	relogcld 26605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
7	6, 4	nndivred 12222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
8	2, 7	fsumrecl 15687	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9		remulcl 11114	. . . . 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 593	. . . 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 27578	. . . . . . . 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 1148	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
28	27	relogcld 26605	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
29		peano2re 11310	. . . . . 6 ⊢ ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
31	30	resqcld 14078	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32		3re 12252	. . . . . 6 ⊢ 3 ∈ ℝ
33		readdcl 11112	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
34	28, 32, 33	sylancl 592	. . . . 5 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
35	34, 28	remulcld 11166	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
36	27	rpred 12977	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	21	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
38	36, 37	rerpdivcld 13008	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39		1red 11136	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
40	27	rpsqrtcld 15365	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ⁺)
41	40	rpred 12977	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ∈ ℝ)
42		ere 16045	. . . . . . . . . . . . 13 ⊢ e ∈ ℝ
43	42	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → e ∈ ℝ)
44		1re 11135	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
45		1lt2 12338	. . . . . . . . . . . . . . 15 ⊢ 1 < 2
46		egt2lt3 16164	. . . . . . . . . . . . . . . 16 ⊢ (2 < e ∧ e < 3)
47	46	simpli 484	. . . . . . . . . . . . . . 15 ⊢ 2 < e
48	44, 1, 42	lttri 11263	. . . . . . . . . . . . . . 15 ⊢ ((1 < 2 ∧ 2 < e) → 1 < e)
49	45, 47, 48	mp2an 698	. . . . . . . . . . . . . 14 ⊢ 1 < e
50	44, 42, 49	ltleii 11260	. . . . . . . . . . . . 13 ⊢ 1 ≤ e
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ e)
52	26	simp2d 1149	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
53	52	simp2d 1149	. . . . . . . . . . . 12 ⊢ (𝜑 → e ≤ (√‘𝑍))
54	39, 43, 41, 51, 53	letrd 11294	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ (√‘𝑍))
55	52	simp3d 1150	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
56	39, 41, 38, 54, 55	letrd 11294	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (𝑍 / 𝑌))
57		flge1nn 13771	. . . . . . . . . 10 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
58	38, 56, 57	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
59	58	nnrpd 12975	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺)
60	59	relogcld 26605	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
61	60, 39	readdcld 11165	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
62	61	resqcld 14078	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63		logdivbnd 27537	. . . . . . 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 12275	. . . . . . . 8 ⊢ 0 < 2
67	66	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
68		lemuldiv2 12028	. . . . . . 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 1382	. . . . . 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 258	. . . . 5 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71		reflcl 13746	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
72	38, 71	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73		flle 13749	. . . . . . . . . 10 ⊢ ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
74	38, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
75	21	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ≤ 𝑌)
76		1rp 12937	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ⁺)
78	77, 37, 27	lediv2d 13001	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
79	75, 78	mpbid 233	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
80	36	recnd 11164	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ ℂ)
81	80	div1d 11914	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / 1) = 𝑍)
82	79, 81	breqtrd 5098	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
83	72, 38, 36, 74, 82	letrd 11294	. . . . . . . 8 ⊢ (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
84	59, 27	logled 26609	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
85	83, 84	mpbid 233	. . . . . . 7 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
86	60, 28, 39, 85	leadd1dd 11755	. . . . . 6 ⊢ (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87		0red 11138	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
88		log1 26567	. . . . . . . . 9 ⊢ (log‘1) = 0
89	58	nnge1d 12216	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90		logleb 26585	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ⁺ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ⁺) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
91	76, 59, 90	sylancr 593	. . . . . . . . . 10 ⊢ (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
92	89, 91	mpbid 233	. . . . . . . . 9 ⊢ (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
93	88, 92	eqbrtrrid 5108	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
94	60	lep1d 12078	. . . . . . . 8 ⊢ (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
95	87, 60, 61, 93, 94	letrd 11294	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
96	87, 61, 30, 95, 86	letrd 11294	. . . . . . 7 ⊢ (𝜑 → 0 ≤ ((log‘𝑍) + 1))
97	61, 30, 95, 96	le2sqd 14210	. . . . . 6 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
98	86, 97	mpbid 233	. . . . 5 ⊢ (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
99	10, 62, 31, 70, 98	letrd 11294	. . . 4 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
100	28	resqcld 14078	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
101	65, 28	remulcld 11166	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102	100, 101	readdcld 11165	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103		loge 26568	. . . . . . 7 ⊢ (log‘e) = 1
104	40	rpge0d 12981	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (√‘𝑍))
105	41, 41, 104, 54	lemulge12d 12085	. . . . . . . . . 10 ⊢ (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
106	27	rprege0d 12984	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107		remsqsqrt 15209	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109	105, 108	breqtrd 5098	. . . . . . . . 9 ⊢ (𝜑 → (√‘𝑍) ≤ 𝑍)
110	43, 41, 36, 53, 109	letrd 11294	. . . . . . . 8 ⊢ (𝜑 → e ≤ 𝑍)
111		epr 16166	. . . . . . . . 9 ⊢ e ∈ ℝ⁺
112		logleb 26585	. . . . . . . . 9 ⊢ ((e ∈ ℝ⁺ ∧ 𝑍 ∈ ℝ⁺) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113	111, 27, 112	sylancr 593	. . . . . . . 8 ⊢ (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114	110, 113	mpbid 233	. . . . . . 7 ⊢ (𝜑 → (log‘e) ≤ (log‘𝑍))
115	103, 114	eqbrtrrid 5108	. . . . . 6 ⊢ (𝜑 → 1 ≤ (log‘𝑍))
116	39, 28, 102, 115	leadd2dd 11756	. . . . 5 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
117	28	recnd 11164	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
118		binom21 14172	. . . . . 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 14096	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121		df-3 12236	. . . . . . . . . 10 ⊢ 3 = (2 + 1)
122	121	oveq1i 7366	. . . . . . . . 9 ⊢ (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123		2cnd 12250	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
124		1cnd 11130	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
125	123, 124, 117	adddird 11161	. . . . . . . . 9 ⊢ (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126	122, 125	eqtrid 2786	. . . . . . . 8 ⊢ (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127	117	mullidd 11154	. . . . . . . . 9 ⊢ (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128	127	oveq2d 7372	. . . . . . . 8 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129	126, 128	eqtr2d 2775	. . . . . . 7 ⊢ (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130	120, 129	oveq12d 7374	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131	117	sqcld 14097	. . . . . . 7 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132		2cn 12247	. . . . . . . 8 ⊢ 2 ∈ ℂ
133		mulcl 11113	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134	132, 117, 133	sylancr 593	. . . . . . 7 ⊢ (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135	131, 134, 117	addassd 11158	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136		3cn 12253	. . . . . . . 8 ⊢ 3 ∈ ℂ
137	136	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℂ)
138	117, 137, 117	adddird 11161	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139	130, 135, 138	3eqtr4rd 2785	. . . . 5 ⊢ (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140	116, 119, 139	3brtr4d 5104	. . . 4 ⊢ (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
141	10, 31, 35, 99, 140	letrd 11294	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
142	10, 35, 17	lemul2d 13021	. . 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 233	. 2 ⊢ (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
144	17	rpred 12977	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ)
145	144	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146	145	recnd 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
147	6	recnd 11164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
148	5	rpcnne0d 12986	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149		div23 11819	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150		divass 11818	. . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151	149, 150	eqtr3d 2776	. . . . . . 7 ⊢ ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152	146, 147, 148, 151	syl3anc 1379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153	152	sumeq2dv 15655	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154	144	recnd 11164	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℂ)
155	7	recnd 11164	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
156	2, 154, 155	fsummulc2 15737	. . . . 5 ⊢ (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157	153, 156	eqtr4d 2777	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158	157	oveq2d 7372	. . 3 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
159	8	recnd 11164	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160	123, 154, 159	mul12d 11346	. . 3 ⊢ (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161	158, 160	eqtrd 2774	. 2 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
162	34	recnd 11164	. . 3 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163	154, 162, 117	mulassd 11159	. 2 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164	143, 161, 163	3brtr4d 5104	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 class class class wbr 5072 ↦ cmpt 5153 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 +∞cpnf 11167 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 2c2 12227 3c3 12228 4c4 12229 cdc 12635 ℝ⁺crp 12933 (,)cioo 13289 [,)cico 13291 [,]cicc 13292 ...cfz 13452 ⌊cfl 13740 ↑cexp 14014 √csqrt 15186 abscabs 15187 Σcsu 15639 expce 16017 eceu 16018 logclog 26536 ψcchp 27074

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-e 16024 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-dvds 16213 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-limc 25851 df-dv 25852 df-ulm 26360 df-log 26538 df-atan 26849 df-em 26974

This theorem is referenced by: pntlemo 27588

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