MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemk Structured version   Visualization version   GIF version

Theorem pntlemk 25340
Description: Lemma for pnt 25348. 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 11128 . . . . 5 2 ∈ ℝ
2 fzfid 12812 . . . . . 6 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3 elfznn 12408 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
43adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
54nnrpd 11908 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
65relogcld 24414 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
76, 4nndivred 11107 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
82, 7fsumrecl 14509 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9 remulcl 10059 . . . . 5 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
101, 8, 9sylancr 696 . . . 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 25331 . . . . . . . 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 1093 . . . . . . 7 (𝜑𝑍 ∈ ℝ+)
2827relogcld 24414 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ)
29 peano2re 10247 . . . . . 6 ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
3028, 29syl 17 . . . . 5 (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
3130resqcld 13075 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32 3re 11132 . . . . . 6 3 ∈ ℝ
33 readdcl 10057 . . . . . 6 (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
3428, 32, 33sylancl 695 . . . . 5 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3534, 28remulcld 10108 . . . 4 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
3627rpred 11910 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℝ)
3721simpld 474 . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
3836, 37rerpdivcld 11941 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39 1red 10093 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
4027rpsqrtcld 14194 . . . . . . . . . . . 12 (𝜑 → (√‘𝑍) ∈ ℝ+)
4140rpred 11910 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ∈ ℝ)
42 ere 14863 . . . . . . . . . . . . 13 e ∈ ℝ
4342a1i 11 . . . . . . . . . . . 12 (𝜑 → e ∈ ℝ)
44 1re 10077 . . . . . . . . . . . . . 14 1 ∈ ℝ
45 1lt2 11232 . . . . . . . . . . . . . . 15 1 < 2
46 egt2lt3 14978 . . . . . . . . . . . . . . . 16 (2 < e ∧ e < 3)
4746simpli 473 . . . . . . . . . . . . . . 15 2 < e
4844, 1, 42lttri 10201 . . . . . . . . . . . . . . 15 ((1 < 2 ∧ 2 < e) → 1 < e)
4945, 47, 48mp2an 708 . . . . . . . . . . . . . 14 1 < e
5044, 42, 49ltleii 10198 . . . . . . . . . . . . 13 1 ≤ e
5150a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ e)
5226simp2d 1094 . . . . . . . . . . . . 13 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5352simp2d 1094 . . . . . . . . . . . 12 (𝜑 → e ≤ (√‘𝑍))
5439, 43, 41, 51, 53letrd 10232 . . . . . . . . . . 11 (𝜑 → 1 ≤ (√‘𝑍))
5552simp3d 1095 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
5639, 41, 38, 54, 55letrd 10232 . . . . . . . . . 10 (𝜑 → 1 ≤ (𝑍 / 𝑌))
57 flge1nn 12662 . . . . . . . . . 10 (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5838, 56, 57syl2anc 694 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5958nnrpd 11908 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ+)
6059relogcld 24414 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
6160, 39readdcld 10107 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
6261resqcld 13075 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63 logdivbnd 25290 . . . . . . 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 11150 . . . . . . . 8 0 < 2
6766a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
68 lemuldiv2 10942 . . . . . . 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 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)))
7064, 69mpbird 247 . . . . 5 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71 reflcl 12637 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
7238, 71syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73 flle 12640 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7438, 73syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7521simprd 478 . . . . . . . . . . 11 (𝜑 → 1 ≤ 𝑌)
76 1rp 11874 . . . . . . . . . . . . 13 1 ∈ ℝ+
7776a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
7877, 37, 27lediv2d 11934 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
7975, 78mpbid 222 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
8036recnd 10106 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℂ)
8180div1d 10831 . . . . . . . . . 10 (𝜑 → (𝑍 / 1) = 𝑍)
8279, 81breqtrd 4711 . . . . . . . . 9 (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
8372, 38, 36, 74, 82letrd 10232 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
8459, 27logled 24418 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
8583, 84mpbid 222 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
8660, 28, 39, 85leadd1dd 10679 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87 0red 10079 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
88 log1 24377 . . . . . . . . 9 (log‘1) = 0
8958nnge1d 11101 . . . . . . . . . 10 (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90 logleb 24394 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ+) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9176, 59, 90sylancr 696 . . . . . . . . . 10 (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9289, 91mpbid 222 . . . . . . . . 9 (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9388, 92syl5eqbrr 4721 . . . . . . . 8 (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9460lep1d 10993 . . . . . . . 8 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9587, 60, 61, 93, 94letrd 10232 . . . . . . 7 (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9687, 61, 30, 95, 86letrd 10232 . . . . . . 7 (𝜑 → 0 ≤ ((log‘𝑍) + 1))
9761, 30, 95, 96le2sqd 13084 . . . . . 6 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
9886, 97mpbid 222 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
9910, 62, 31, 70, 98letrd 10232 . . . 4 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
10028resqcld 13075 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
10165, 28remulcld 10108 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102100, 101readdcld 10107 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103 loge 24378 . . . . . . 7 (log‘e) = 1
10440rpge0d 11914 . . . . . . . . . . 11 (𝜑 → 0 ≤ (√‘𝑍))
10541, 41, 104, 54lemulge12d 11000 . . . . . . . . . 10 (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
10627rprege0d 11917 . . . . . . . . . . 11 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107 remsqsqrt 14041 . . . . . . . . . . 11 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108106, 107syl 17 . . . . . . . . . 10 (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109105, 108breqtrd 4711 . . . . . . . . 9 (𝜑 → (√‘𝑍) ≤ 𝑍)
11043, 41, 36, 53, 109letrd 10232 . . . . . . . 8 (𝜑 → e ≤ 𝑍)
111 epr 14980 . . . . . . . . 9 e ∈ ℝ+
112 logleb 24394 . . . . . . . . 9 ((e ∈ ℝ+𝑍 ∈ ℝ+) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113111, 27, 112sylancr 696 . . . . . . . 8 (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114110, 113mpbid 222 . . . . . . 7 (𝜑 → (log‘e) ≤ (log‘𝑍))
115103, 114syl5eqbrr 4721 . . . . . 6 (𝜑 → 1 ≤ (log‘𝑍))
11639, 28, 102, 115leadd2dd 10680 . . . . 5 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
11728recnd 10106 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
118 binom21 13020 . . . . . 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 13045 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121 df-3 11118 . . . . . . . . . 10 3 = (2 + 1)
122121oveq1i 6700 . . . . . . . . 9 (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123 2cnd 11131 . . . . . . . . . 10 (𝜑 → 2 ∈ ℂ)
124 1cnd 10094 . . . . . . . . . 10 (𝜑 → 1 ∈ ℂ)
125123, 124, 117adddird 10103 . . . . . . . . 9 (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126122, 125syl5eq 2697 . . . . . . . 8 (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127117mulid2d 10096 . . . . . . . . 9 (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128127oveq2d 6706 . . . . . . . 8 (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129126, 128eqtr2d 2686 . . . . . . 7 (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130120, 129oveq12d 6708 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131117sqcld 13046 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132 2cn 11129 . . . . . . . 8 2 ∈ ℂ
133 mulcl 10058 . . . . . . . 8 ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134132, 117, 133sylancr 696 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135131, 134, 117addassd 10100 . . . . . 6 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136 3cn 11133 . . . . . . . 8 3 ∈ ℂ
137136a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℂ)
138117, 137, 117adddird 10103 . . . . . 6 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139130, 135, 1383eqtr4rd 2696 . . . . 5 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140116, 119, 1393brtr4d 4717 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14110, 31, 35, 99, 140letrd 10232 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14210, 35, 17lemul2d 11954 . . 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 222 . 2 (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
14417rpred 11910 . . . . . . . . 9 (𝜑𝑈 ∈ ℝ)
145144adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146145recnd 10106 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
1476recnd 10106 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
1485rpcnne0d 11919 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149 div23 10742 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150 divass 10741 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151149, 150eqtr3d 2687 . . . . . . 7 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152146, 147, 148, 151syl3anc 1366 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153152sumeq2dv 14477 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154144recnd 10106 . . . . . 6 (𝜑𝑈 ∈ ℂ)
1557recnd 10106 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
1562, 154, 155fsummulc2 14560 . . . . 5 (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157153, 156eqtr4d 2688 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158157oveq2d 6706 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
1598recnd 10106 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160123, 154, 159mul12d 10283 . . 3 (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161158, 160eqtrd 2685 . 2 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
16234recnd 10106 . . 3 (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163154, 162, 117mulassd 10101 . 2 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164143, 161, 1633brtr4d 4717 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  3c3 11109  4c4 11110  cdc 11531  +crp 11870  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  ...cfz 12364  cfl 12631  cexp 12900  csqrt 14017  abscabs 14018  Σcsu 14460  expce 14836  eceu 14837  logclog 24346  ψcchp 24864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-em 24764
This theorem is referenced by:  pntlemo  25341
  Copyright terms: Public domain W3C validator