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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  cc 10529  cr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  +∞cpnf 10666   < clt 10669  cle 10670  cmin 10864   / cdiv 11291  cn 11632  2c2 11687  3c3 11688  4c4 11689  cdc 12093  +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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