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Theorem pntlemk 26188
 Description: Lemma for pnt 26196. 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 (𝜑𝐴 ∈ ℝ+)
pntlem1.b (𝜑𝐵 ∈ ℝ+)
pntlem1.l (𝜑𝐿 ∈ (0(,)1))
pntlem1.d 𝐷 = (𝐴 + 1)
pntlem1.f 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))
pntlem1.u (𝜑𝑈 ∈ ℝ+)
pntlem1.u2 (𝜑𝑈𝐴)
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 (𝜑𝐶 ∈ ℝ+)
pntlem1.w 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z (𝜑𝑍 ∈ (𝑊[,)+∞))
pntlem1.m 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
pntlem1.U (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)
pntlem1.K (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 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
StepHypRef Expression
1 2re 11706 . . . . 5 2 ∈ ℝ
2 fzfid 13343 . . . . . 6 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3 elfznn 12938 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
43adantl 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
54nnrpd 12424 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
65relogcld 25212 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
76, 4nndivred 11686 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
82, 7fsumrecl 15089 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9 remulcl 10616 . . . . 5 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
101, 8, 9sylancr 590 . . . 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 (𝜑𝐴 ∈ ℝ+)
13 pntlem1.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ+)
14 pntlem1.l . . . . . . . . 9 (𝜑𝐿 ∈ (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 (𝜑𝑈 ∈ ℝ+)
18 pntlem1.u2 . . . . . . . . 9 (𝜑𝑈𝐴)
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 (𝜑𝐶 ∈ ℝ+)
24 pntlem1.w . . . . . . . . 9 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
25 pntlem1.z . . . . . . . . 9 (𝜑𝑍 ∈ (𝑊[,)+∞))
2611, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25pntlemb 26179 . . . . . . . 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) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
2726simp1d 1139 . . . . . . 7 (𝜑𝑍 ∈ ℝ+)
2827relogcld 25212 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ)
29 peano2re 10807 . . . . . 6 ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
3028, 29syl 17 . . . . 5 (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
3130resqcld 13614 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32 3re 11712 . . . . . 6 3 ∈ ℝ
33 readdcl 10614 . . . . . 6 (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
3428, 32, 33sylancl 589 . . . . 5 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3534, 28remulcld 10665 . . . 4 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
3627rpred 12426 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℝ)
3721simpld 498 . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
3836, 37rerpdivcld 12457 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39 1red 10636 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
4027rpsqrtcld 14769 . . . . . . . . . . . 12 (𝜑 → (√‘𝑍) ∈ ℝ+)
4140rpred 12426 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ∈ ℝ)
42 ere 15440 . . . . . . . . . . . . 13 e ∈ ℝ
4342a1i 11 . . . . . . . . . . . 12 (𝜑 → e ∈ ℝ)
44 1re 10635 . . . . . . . . . . . . . 14 1 ∈ ℝ
45 1lt2 11803 . . . . . . . . . . . . . . 15 1 < 2
46 egt2lt3 15557 . . . . . . . . . . . . . . . 16 (2 < e ∧ e < 3)
4746simpli 487 . . . . . . . . . . . . . . 15 2 < e
4844, 1, 42lttri 10760 . . . . . . . . . . . . . . 15 ((1 < 2 ∧ 2 < e) → 1 < e)
4945, 47, 48mp2an 691 . . . . . . . . . . . . . 14 1 < e
5044, 42, 49ltleii 10757 . . . . . . . . . . . . 13 1 ≤ e
5150a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ e)
5226simp2d 1140 . . . . . . . . . . . . 13 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5352simp2d 1140 . . . . . . . . . . . 12 (𝜑 → e ≤ (√‘𝑍))
5439, 43, 41, 51, 53letrd 10791 . . . . . . . . . . 11 (𝜑 → 1 ≤ (√‘𝑍))
5552simp3d 1141 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
5639, 41, 38, 54, 55letrd 10791 . . . . . . . . . 10 (𝜑 → 1 ≤ (𝑍 / 𝑌))
57 flge1nn 13193 . . . . . . . . . 10 (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5838, 56, 57syl2anc 587 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5958nnrpd 12424 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ+)
6059relogcld 25212 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
6160, 39readdcld 10664 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
6261resqcld 13614 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63 logdivbnd 26138 . . . . . . 7 ((⌊‘(𝑍 / 𝑌)) ∈ ℕ → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2))
6458, 63syl 17 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ≤ ((((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) / 2))
651a1i 11 . . . . . . 7 (𝜑 → 2 ∈ ℝ)
66 2pos 11735 . . . . . . . 8 0 < 2
6766a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
68 lemuldiv2 11515 . . . . . . 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)))
698, 62, 65, 67, 68syl112anc 1371 . . . . . 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)))
7064, 69mpbird 260 . . . . 5 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71 reflcl 13168 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
7238, 71syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73 flle 13171 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7438, 73syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7521simprd 499 . . . . . . . . . . 11 (𝜑 → 1 ≤ 𝑌)
76 1rp 12388 . . . . . . . . . . . . 13 1 ∈ ℝ+
7776a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
7877, 37, 27lediv2d 12450 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
7975, 78mpbid 235 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
8036recnd 10663 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℂ)
8180div1d 11402 . . . . . . . . . 10 (𝜑 → (𝑍 / 1) = 𝑍)
8279, 81breqtrd 5079 . . . . . . . . 9 (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
8372, 38, 36, 74, 82letrd 10791 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
8459, 27logled 25216 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
8583, 84mpbid 235 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
8660, 28, 39, 85leadd1dd 11248 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87 0red 10638 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
88 log1 25175 . . . . . . . . 9 (log‘1) = 0
8958nnge1d 11680 . . . . . . . . . 10 (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90 logleb 25192 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ+) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9176, 59, 90sylancr 590 . . . . . . . . . 10 (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9289, 91mpbid 235 . . . . . . . . 9 (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9388, 92eqbrtrrid 5089 . . . . . . . 8 (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9460lep1d 11565 . . . . . . . 8 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9587, 60, 61, 93, 94letrd 10791 . . . . . . 7 (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9687, 61, 30, 95, 86letrd 10791 . . . . . . 7 (𝜑 → 0 ≤ ((log‘𝑍) + 1))
9761, 30, 95, 96le2sqd 13623 . . . . . 6 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
9886, 97mpbid 235 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
9910, 62, 31, 70, 98letrd 10791 . . . 4 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
10028resqcld 13614 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
10165, 28remulcld 10665 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102100, 101readdcld 10664 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103 loge 25176 . . . . . . 7 (log‘e) = 1
10440rpge0d 12430 . . . . . . . . . . 11 (𝜑 → 0 ≤ (√‘𝑍))
10541, 41, 104, 54lemulge12d 11572 . . . . . . . . . 10 (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
10627rprege0d 12433 . . . . . . . . . . 11 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107 remsqsqrt 14614 . . . . . . . . . . 11 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108106, 107syl 17 . . . . . . . . . 10 (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109105, 108breqtrd 5079 . . . . . . . . 9 (𝜑 → (√‘𝑍) ≤ 𝑍)
11043, 41, 36, 53, 109letrd 10791 . . . . . . . 8 (𝜑 → e ≤ 𝑍)
111 epr 15559 . . . . . . . . 9 e ∈ ℝ+
112 logleb 25192 . . . . . . . . 9 ((e ∈ ℝ+𝑍 ∈ ℝ+) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113111, 27, 112sylancr 590 . . . . . . . 8 (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114110, 113mpbid 235 . . . . . . 7 (𝜑 → (log‘e) ≤ (log‘𝑍))
115103, 114eqbrtrrid 5089 . . . . . 6 (𝜑 → 1 ≤ (log‘𝑍))
11639, 28, 102, 115leadd2dd 11249 . . . . 5 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
11728recnd 10663 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
118 binom21 13583 . . . . . 6 ((log‘𝑍) ∈ ℂ → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
119117, 118syl 17 . . . . 5 (𝜑 → (((log‘𝑍) + 1)↑2) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1))
120117sqvald 13510 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121 df-3 11696 . . . . . . . . . 10 3 = (2 + 1)
122121oveq1i 7156 . . . . . . . . 9 (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123 2cnd 11710 . . . . . . . . . 10 (𝜑 → 2 ∈ ℂ)
124 1cnd 10630 . . . . . . . . . 10 (𝜑 → 1 ∈ ℂ)
125123, 124, 117adddird 10660 . . . . . . . . 9 (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126122, 125syl5eq 2871 . . . . . . . 8 (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127117mulid2d 10653 . . . . . . . . 9 (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128127oveq2d 7162 . . . . . . . 8 (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129126, 128eqtr2d 2860 . . . . . . 7 (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130120, 129oveq12d 7164 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131117sqcld 13511 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132 2cn 11707 . . . . . . . 8 2 ∈ ℂ
133 mulcl 10615 . . . . . . . 8 ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134132, 117, 133sylancr 590 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135131, 134, 117addassd 10657 . . . . . 6 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136 3cn 11713 . . . . . . . 8 3 ∈ ℂ
137136a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℂ)
138117, 137, 117adddird 10660 . . . . . 6 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139130, 135, 1383eqtr4rd 2870 . . . . 5 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140116, 119, 1393brtr4d 5085 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14110, 31, 35, 99, 140letrd 10791 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14210, 35, 17lemul2d 12470 . . 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)))))
143141, 142mpbid 235 . 2 (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
14417rpred 12426 . . . . . . . . 9 (𝜑𝑈 ∈ ℝ)
145144adantr 484 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146145recnd 10663 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
1476recnd 10663 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
1485rpcnne0d 12435 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149 div23 11311 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150 divass 11310 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151149, 150eqtr3d 2861 . . . . . . 7 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152146, 147, 148, 151syl3anc 1368 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153152sumeq2dv 15058 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154144recnd 10663 . . . . . 6 (𝜑𝑈 ∈ ℂ)
1557recnd 10663 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
1562, 154, 155fsummulc2 15137 . . . . 5 (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157153, 156eqtr4d 2862 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158157oveq2d 7162 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
1598recnd 10663 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160123, 154, 159mul12d 10843 . . 3 (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161158, 160eqtrd 2859 . 2 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
16234recnd 10663 . . 3 (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163154, 162, 117mulassd 10658 . 2 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164143, 161, 1633brtr4d 5085 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134   class class class wbr 5053   ↦ cmpt 5133  ‘cfv 6344  (class class class)co 7146  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  +∞cpnf 10666   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  2c2 11687  3c3 11688  4c4 11689  ;cdc 12093  ℝ+crp 12384  (,)cioo 12733  [,)cico 12735  [,]cicc 12736  ...cfz 12892  ⌊cfl 13162  ↑cexp 13432  √csqrt 14590  abscabs 14591  Σcsu 15040  expce 15413  eceu 15414  logclog 25144  ψcchp 25676 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-inf2 9097  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-iin 4909  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-isom 6353  df-riota 7104  df-ov 7149  df-oprab 7150  df-mpo 7151  df-of 7400  df-om 7572  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8827  df-fi 8868  df-sup 8899  df-inf 8900  df-oi 8967  df-card 9361  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11695  df-3 11696  df-4 11697  df-5 11698  df-6 11699  df-7 11700  df-8 11701  df-9 11702  df-n0 11893  df-xnn0 11963  df-z 11977  df-dec 12094  df-uz 12239  df-q 12344  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ioo 12737  df-ioc 12738  df-ico 12739  df-icc 12740  df-fz 12893  df-fzo 13036  df-fl 13164  df-mod 13240  df-seq 13372  df-exp 13433  df-fac 13637  df-bc 13666  df-hash 13694  df-shft 14424  df-cj 14456  df-re 14457  df-im 14458  df-sqrt 14592  df-abs 14593  df-limsup 14826  df-clim 14843  df-rlim 14844  df-sum 15041  df-ef 15419  df-e 15420  df-sin 15421  df-cos 15422  df-tan 15423  df-pi 15424  df-dvds 15606  df-struct 16483  df-ndx 16484  df-slot 16485  df-base 16487  df-sets 16488  df-ress 16489  df-plusg 16576  df-mulr 16577  df-starv 16578  df-sca 16579  df-vsca 16580  df-ip 16581  df-tset 16582  df-ple 16583  df-ds 16585  df-unif 16586  df-hom 16587  df-cco 16588  df-rest 16694  df-topn 16695  df-0g 16713  df-gsum 16714  df-topgen 16715  df-pt 16716  df-prds 16719  df-xrs 16773  df-qtop 16778  df-imas 16779  df-xps 16781  df-mre 16855  df-mrc 16856  df-acs 16858  df-mgm 17850  df-sgrp 17899  df-mnd 17910  df-submnd 17955  df-mulg 18223  df-cntz 18445  df-cmn 18906  df-psmet 20532  df-xmet 20533  df-met 20534  df-bl 20535  df-mopn 20536  df-fbas 20537  df-fg 20538  df-cnfld 20541  df-top 21497  df-topon 21514  df-topsp 21536  df-bases 21549  df-cld 21622  df-ntr 21623  df-cls 21624  df-nei 21701  df-lp 21739  df-perf 21740  df-cn 21830  df-cnp 21831  df-haus 21918  df-cmp 21990  df-tx 22165  df-hmeo 22358  df-fil 22449  df-fm 22541  df-flim 22542  df-flf 22543  df-xms 22925  df-ms 22926  df-tms 22927  df-cncf 23481  df-limc 24467  df-dv 24468  df-ulm 24970  df-log 25146  df-atan 25451  df-em 25576 This theorem is referenced by:  pntlemo  26189
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