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

Theorem pntlemk 25095
 Description: Lemma for pnt 25103. 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 10967 . . . . 5 2 ∈ ℝ
2 fzfid 12634 . . . . . 6 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3 elfznn 12241 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
43adantl 481 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
54nnrpd 11746 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
65relogcld 24173 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
76, 4nndivred 10946 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
82, 7fsumrecl 14312 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9 remulcl 9900 . . . . 5 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
101, 8, 9sylancr 694 . . . 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 25086 . . . . . . . 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 1066 . . . . . . 7 (𝜑𝑍 ∈ ℝ+)
2827relogcld 24173 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ)
29 peano2re 10088 . . . . . 6 ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
3028, 29syl 17 . . . . 5 (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
3130resqcld 12897 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32 3re 10971 . . . . . 6 3 ∈ ℝ
33 readdcl 9898 . . . . . 6 (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
3428, 32, 33sylancl 693 . . . . 5 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3534, 28remulcld 9949 . . . 4 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
3627rpred 11748 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℝ)
3721simpld 474 . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
3836, 37rerpdivcld 11779 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39 1red 9934 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
4027rpsqrtcld 13998 . . . . . . . . . . . 12 (𝜑 → (√‘𝑍) ∈ ℝ+)
4140rpred 11748 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ∈ ℝ)
42 ere 14658 . . . . . . . . . . . . 13 e ∈ ℝ
4342a1i 11 . . . . . . . . . . . 12 (𝜑 → e ∈ ℝ)
44 1re 9918 . . . . . . . . . . . . . 14 1 ∈ ℝ
45 1lt2 11071 . . . . . . . . . . . . . . 15 1 < 2
46 egt2lt3 14773 . . . . . . . . . . . . . . . 16 (2 < e ∧ e < 3)
4746simpli 473 . . . . . . . . . . . . . . 15 2 < e
4844, 1, 42lttri 10042 . . . . . . . . . . . . . . 15 ((1 < 2 ∧ 2 < e) → 1 < e)
4945, 47, 48mp2an 704 . . . . . . . . . . . . . 14 1 < e
5044, 42, 49ltleii 10039 . . . . . . . . . . . . 13 1 ≤ e
5150a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ e)
5226simp2d 1067 . . . . . . . . . . . . 13 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5352simp2d 1067 . . . . . . . . . . . 12 (𝜑 → e ≤ (√‘𝑍))
5439, 43, 41, 51, 53letrd 10073 . . . . . . . . . . 11 (𝜑 → 1 ≤ (√‘𝑍))
5552simp3d 1068 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
5639, 41, 38, 54, 55letrd 10073 . . . . . . . . . 10 (𝜑 → 1 ≤ (𝑍 / 𝑌))
57 flge1nn 12484 . . . . . . . . . 10 (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5838, 56, 57syl2anc 691 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5958nnrpd 11746 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ+)
6059relogcld 24173 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
6160, 39readdcld 9948 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
6261resqcld 12897 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63 logdivbnd 25045 . . . . . . 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 10989 . . . . . . . 8 0 < 2
6766a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
68 lemuldiv2 10783 . . . . . . 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 1322 . . . . . 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 246 . . . . 5 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71 reflcl 12459 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
7238, 71syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73 flle 12462 . . . . . . . . . 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 11712 . . . . . . . . . . . . 13 1 ∈ ℝ+
7776a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
7877, 37, 27lediv2d 11772 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
7975, 78mpbid 221 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
8036recnd 9947 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℂ)
8180div1d 10672 . . . . . . . . . 10 (𝜑 → (𝑍 / 1) = 𝑍)
8279, 81breqtrd 4609 . . . . . . . . 9 (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
8372, 38, 36, 74, 82letrd 10073 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
8459, 27logled 24177 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
8583, 84mpbid 221 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
8660, 28, 39, 85leadd1dd 10520 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87 0red 9920 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
88 log1 24136 . . . . . . . . 9 (log‘1) = 0
8958nnge1d 10940 . . . . . . . . . 10 (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90 logleb 24153 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ+) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9176, 59, 90sylancr 694 . . . . . . . . . 10 (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9289, 91mpbid 221 . . . . . . . . 9 (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9388, 92syl5eqbrr 4619 . . . . . . . 8 (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9460lep1d 10834 . . . . . . . 8 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9587, 60, 61, 93, 94letrd 10073 . . . . . . 7 (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9687, 61, 30, 95, 86letrd 10073 . . . . . . 7 (𝜑 → 0 ≤ ((log‘𝑍) + 1))
9761, 30, 95, 96le2sqd 12906 . . . . . 6 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
9886, 97mpbid 221 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
9910, 62, 31, 70, 98letrd 10073 . . . 4 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
10028resqcld 12897 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
10165, 28remulcld 9949 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102100, 101readdcld 9948 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103 loge 24137 . . . . . . 7 (log‘e) = 1
10440rpge0d 11752 . . . . . . . . . . 11 (𝜑 → 0 ≤ (√‘𝑍))
10541, 41, 104, 54lemulge12d 10841 . . . . . . . . . 10 (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
10627rprege0d 11755 . . . . . . . . . . 11 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107 remsqsqrt 13845 . . . . . . . . . . 11 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108106, 107syl 17 . . . . . . . . . 10 (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109105, 108breqtrd 4609 . . . . . . . . 9 (𝜑 → (√‘𝑍) ≤ 𝑍)
11043, 41, 36, 53, 109letrd 10073 . . . . . . . 8 (𝜑 → e ≤ 𝑍)
111 epr 14775 . . . . . . . . 9 e ∈ ℝ+
112 logleb 24153 . . . . . . . . 9 ((e ∈ ℝ+𝑍 ∈ ℝ+) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113111, 27, 112sylancr 694 . . . . . . . 8 (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114110, 113mpbid 221 . . . . . . 7 (𝜑 → (log‘e) ≤ (log‘𝑍))
115103, 114syl5eqbrr 4619 . . . . . 6 (𝜑 → 1 ≤ (log‘𝑍))
11639, 28, 102, 115leadd2dd 10521 . . . . 5 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
11728recnd 9947 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
118 binom21 12842 . . . . . 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 12867 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121 df-3 10957 . . . . . . . . . 10 3 = (2 + 1)
122121oveq1i 6559 . . . . . . . . 9 (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123 2cnd 10970 . . . . . . . . . 10 (𝜑 → 2 ∈ ℂ)
124 1cnd 9935 . . . . . . . . . 10 (𝜑 → 1 ∈ ℂ)
125123, 124, 117adddird 9944 . . . . . . . . 9 (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126122, 125syl5eq 2656 . . . . . . . 8 (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127117mulid2d 9937 . . . . . . . . 9 (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128127oveq2d 6565 . . . . . . . 8 (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129126, 128eqtr2d 2645 . . . . . . 7 (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130120, 129oveq12d 6567 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131117sqcld 12868 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132 2cn 10968 . . . . . . . 8 2 ∈ ℂ
133 mulcl 9899 . . . . . . . 8 ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134132, 117, 133sylancr 694 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135131, 134, 117addassd 9941 . . . . . 6 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136 3cn 10972 . . . . . . . 8 3 ∈ ℂ
137136a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℂ)
138117, 137, 117adddird 9944 . . . . . 6 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139130, 135, 1383eqtr4rd 2655 . . . . 5 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140116, 119, 1393brtr4d 4615 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14110, 31, 35, 99, 140letrd 10073 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14210, 35, 17lemul2d 11792 . . 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 221 . 2 (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
14417rpred 11748 . . . . . . . . 9 (𝜑𝑈 ∈ ℝ)
145144adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146145recnd 9947 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
1476recnd 9947 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
1485rpcnne0d 11757 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149 div23 10583 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150 divass 10582 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151149, 150eqtr3d 2646 . . . . . . 7 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152146, 147, 148, 151syl3anc 1318 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153152sumeq2dv 14281 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154144recnd 9947 . . . . . 6 (𝜑𝑈 ∈ ℂ)
1557recnd 9947 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
1562, 154, 155fsummulc2 14358 . . . . 5 (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157153, 156eqtr4d 2647 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158157oveq2d 6565 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
1598recnd 9947 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160123, 154, 159mul12d 10124 . . 3 (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161158, 160eqtrd 2644 . 2 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
16234recnd 9947 . . 3 (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163154, 162, 117mulassd 9942 . 2 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164143, 161, 1633brtr4d 4615 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  3c3 10948  4c4 10949  ;cdc 11369  ℝ+crp 11708  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  ...cfz 12197  ⌊cfl 12453  ↑cexp 12722  √csqrt 13821  abscabs 13822  Σcsu 14264  expce 14631  eceu 14632  logclog 24105  ψcchp 24619 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-e 14638  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-em 24519 This theorem is referenced by:  pntlemo  25096
 Copyright terms: Public domain W3C validator