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Theorem pntlemk 25647
Description: Lemma for pnt 25655. 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 11387 . . . . 5 2 ∈ ℝ
2 fzfid 13027 . . . . . 6 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
3 elfznn 12624 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
43adantl 474 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
54nnrpd 12115 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
65relogcld 24710 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
76, 4nndivred 11367 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℝ)
82, 7fsumrecl 14806 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ)
9 remulcl 10309 . . . . 5 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ∈ ℝ)
101, 8, 9sylancr 582 . . . 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 25638 . . . . . . . 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 1173 . . . . . . 7 (𝜑𝑍 ∈ ℝ+)
2827relogcld 24710 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ)
29 peano2re 10499 . . . . . 6 ((log‘𝑍) ∈ ℝ → ((log‘𝑍) + 1) ∈ ℝ)
3028, 29syl 17 . . . . 5 (𝜑 → ((log‘𝑍) + 1) ∈ ℝ)
3130resqcld 13291 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ∈ ℝ)
32 3re 11393 . . . . . 6 3 ∈ ℝ
33 readdcl 10307 . . . . . 6 (((log‘𝑍) ∈ ℝ ∧ 3 ∈ ℝ) → ((log‘𝑍) + 3) ∈ ℝ)
3428, 32, 33sylancl 581 . . . . 5 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3534, 28remulcld 10359 . . . 4 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) ∈ ℝ)
3627rpred 12117 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℝ)
3721simpld 489 . . . . . . . . . . 11 (𝜑𝑌 ∈ ℝ+)
3836, 37rerpdivcld 12148 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
39 1red 10329 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
4027rpsqrtcld 14491 . . . . . . . . . . . 12 (𝜑 → (√‘𝑍) ∈ ℝ+)
4140rpred 12117 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ∈ ℝ)
42 ere 15155 . . . . . . . . . . . . 13 e ∈ ℝ
4342a1i 11 . . . . . . . . . . . 12 (𝜑 → e ∈ ℝ)
44 1re 10328 . . . . . . . . . . . . . 14 1 ∈ ℝ
45 1lt2 11491 . . . . . . . . . . . . . . 15 1 < 2
46 egt2lt3 15270 . . . . . . . . . . . . . . . 16 (2 < e ∧ e < 3)
4746simpli 477 . . . . . . . . . . . . . . 15 2 < e
4844, 1, 42lttri 10453 . . . . . . . . . . . . . . 15 ((1 < 2 ∧ 2 < e) → 1 < e)
4945, 47, 48mp2an 684 . . . . . . . . . . . . . 14 1 < e
5044, 42, 49ltleii 10450 . . . . . . . . . . . . 13 1 ≤ e
5150a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ≤ e)
5226simp2d 1174 . . . . . . . . . . . . 13 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
5352simp2d 1174 . . . . . . . . . . . 12 (𝜑 → e ≤ (√‘𝑍))
5439, 43, 41, 51, 53letrd 10484 . . . . . . . . . . 11 (𝜑 → 1 ≤ (√‘𝑍))
5552simp3d 1175 . . . . . . . . . . 11 (𝜑 → (√‘𝑍) ≤ (𝑍 / 𝑌))
5639, 41, 38, 54, 55letrd 10484 . . . . . . . . . 10 (𝜑 → 1 ≤ (𝑍 / 𝑌))
57 flge1nn 12877 . . . . . . . . . 10 (((𝑍 / 𝑌) ∈ ℝ ∧ 1 ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5838, 56, 57syl2anc 580 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℕ)
5958nnrpd 12115 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ+)
6059relogcld 24710 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ∈ ℝ)
6160, 39readdcld 10358 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ∈ ℝ)
6261resqcld 13291 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ∈ ℝ)
63 logdivbnd 25597 . . . . . . 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 11423 . . . . . . . 8 0 < 2
6766a1i 11 . . . . . . 7 (𝜑 → 0 < 2)
68 lemuldiv2 11196 . . . . . . 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 1494 . . . . . 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 249 . . . . 5 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2))
71 reflcl 12852 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
7238, 71syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ∈ ℝ)
73 flle 12855 . . . . . . . . . 10 ((𝑍 / 𝑌) ∈ ℝ → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7438, 73syl 17 . . . . . . . . 9 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ (𝑍 / 𝑌))
7521simprd 490 . . . . . . . . . . 11 (𝜑 → 1 ≤ 𝑌)
76 1rp 12078 . . . . . . . . . . . . 13 1 ∈ ℝ+
7776a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
7877, 37, 27lediv2d 12141 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑌 ↔ (𝑍 / 𝑌) ≤ (𝑍 / 1)))
7975, 78mpbid 224 . . . . . . . . . 10 (𝜑 → (𝑍 / 𝑌) ≤ (𝑍 / 1))
8036recnd 10357 . . . . . . . . . . 11 (𝜑𝑍 ∈ ℂ)
8180div1d 11085 . . . . . . . . . 10 (𝜑 → (𝑍 / 1) = 𝑍)
8279, 81breqtrd 4869 . . . . . . . . 9 (𝜑 → (𝑍 / 𝑌) ≤ 𝑍)
8372, 38, 36, 74, 82letrd 10484 . . . . . . . 8 (𝜑 → (⌊‘(𝑍 / 𝑌)) ≤ 𝑍)
8459, 27logled 24714 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / 𝑌)) ≤ 𝑍 ↔ (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍)))
8583, 84mpbid 224 . . . . . . 7 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ (log‘𝑍))
8660, 28, 39, 85leadd1dd 10933 . . . . . 6 (𝜑 → ((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1))
87 0red 10332 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
88 log1 24673 . . . . . . . . 9 (log‘1) = 0
8958nnge1d 11361 . . . . . . . . . 10 (𝜑 → 1 ≤ (⌊‘(𝑍 / 𝑌)))
90 logleb 24690 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℝ+) → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9176, 59, 90sylancr 582 . . . . . . . . . 10 (𝜑 → (1 ≤ (⌊‘(𝑍 / 𝑌)) ↔ (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌)))))
9289, 91mpbid 224 . . . . . . . . 9 (𝜑 → (log‘1) ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9388, 92syl5eqbrr 4879 . . . . . . . 8 (𝜑 → 0 ≤ (log‘(⌊‘(𝑍 / 𝑌))))
9460lep1d 11247 . . . . . . . 8 (𝜑 → (log‘(⌊‘(𝑍 / 𝑌))) ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9587, 60, 61, 93, 94letrd 10484 . . . . . . 7 (𝜑 → 0 ≤ ((log‘(⌊‘(𝑍 / 𝑌))) + 1))
9687, 61, 30, 95, 86letrd 10484 . . . . . . 7 (𝜑 → 0 ≤ ((log‘𝑍) + 1))
9761, 30, 95, 96le2sqd 13300 . . . . . 6 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1) ≤ ((log‘𝑍) + 1) ↔ (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2)))
9886, 97mpbid 224 . . . . 5 (𝜑 → (((log‘(⌊‘(𝑍 / 𝑌))) + 1)↑2) ≤ (((log‘𝑍) + 1)↑2))
9910, 62, 31, 70, 98letrd 10484 . . . 4 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 1)↑2))
10028resqcld 13291 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
10165, 28remulcld 10359 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℝ)
102100, 101readdcld 10358 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + (2 · (log‘𝑍))) ∈ ℝ)
103 loge 24674 . . . . . . 7 (log‘e) = 1
10440rpge0d 12121 . . . . . . . . . . 11 (𝜑 → 0 ≤ (√‘𝑍))
10541, 41, 104, 54lemulge12d 11254 . . . . . . . . . 10 (𝜑 → (√‘𝑍) ≤ ((√‘𝑍) · (√‘𝑍)))
10627rprege0d 12124 . . . . . . . . . . 11 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
107 remsqsqrt 14338 . . . . . . . . . . 11 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
108106, 107syl 17 . . . . . . . . . 10 (𝜑 → ((√‘𝑍) · (√‘𝑍)) = 𝑍)
109105, 108breqtrd 4869 . . . . . . . . 9 (𝜑 → (√‘𝑍) ≤ 𝑍)
11043, 41, 36, 53, 109letrd 10484 . . . . . . . 8 (𝜑 → e ≤ 𝑍)
111 epr 15272 . . . . . . . . 9 e ∈ ℝ+
112 logleb 24690 . . . . . . . . 9 ((e ∈ ℝ+𝑍 ∈ ℝ+) → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
113111, 27, 112sylancr 582 . . . . . . . 8 (𝜑 → (e ≤ 𝑍 ↔ (log‘e) ≤ (log‘𝑍)))
114110, 113mpbid 224 . . . . . . 7 (𝜑 → (log‘e) ≤ (log‘𝑍))
115103, 114syl5eqbrr 4879 . . . . . 6 (𝜑 → 1 ≤ (log‘𝑍))
11639, 28, 102, 115leadd2dd 10934 . . . . 5 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + 1) ≤ ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
11728recnd 10357 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
118 binom21 13234 . . . . . 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 13259 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
121 df-3 11377 . . . . . . . . . 10 3 = (2 + 1)
122121oveq1i 6888 . . . . . . . . 9 (3 · (log‘𝑍)) = ((2 + 1) · (log‘𝑍))
123 2cnd 11391 . . . . . . . . . 10 (𝜑 → 2 ∈ ℂ)
124 1cnd 10323 . . . . . . . . . 10 (𝜑 → 1 ∈ ℂ)
125123, 124, 117adddird 10354 . . . . . . . . 9 (𝜑 → ((2 + 1) · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
126122, 125syl5eq 2845 . . . . . . . 8 (𝜑 → (3 · (log‘𝑍)) = ((2 · (log‘𝑍)) + (1 · (log‘𝑍))))
127117mulid2d 10347 . . . . . . . . 9 (𝜑 → (1 · (log‘𝑍)) = (log‘𝑍))
128127oveq2d 6894 . . . . . . . 8 (𝜑 → ((2 · (log‘𝑍)) + (1 · (log‘𝑍))) = ((2 · (log‘𝑍)) + (log‘𝑍)))
129126, 128eqtr2d 2834 . . . . . . 7 (𝜑 → ((2 · (log‘𝑍)) + (log‘𝑍)) = (3 · (log‘𝑍)))
130120, 129oveq12d 6896 . . . . . 6 (𝜑 → (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
131117sqcld 13260 . . . . . . 7 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
132 2cn 11388 . . . . . . . 8 2 ∈ ℂ
133 mulcl 10308 . . . . . . . 8 ((2 ∈ ℂ ∧ (log‘𝑍) ∈ ℂ) → (2 · (log‘𝑍)) ∈ ℂ)
134132, 117, 133sylancr 582 . . . . . . 7 (𝜑 → (2 · (log‘𝑍)) ∈ ℂ)
135131, 134, 117addassd 10351 . . . . . 6 (𝜑 → ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)) = (((log‘𝑍)↑2) + ((2 · (log‘𝑍)) + (log‘𝑍))))
136 3cn 11394 . . . . . . . 8 3 ∈ ℂ
137136a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℂ)
138117, 137, 117adddird 10354 . . . . . 6 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = (((log‘𝑍) · (log‘𝑍)) + (3 · (log‘𝑍))))
139130, 135, 1383eqtr4rd 2844 . . . . 5 (𝜑 → (((log‘𝑍) + 3) · (log‘𝑍)) = ((((log‘𝑍)↑2) + (2 · (log‘𝑍))) + (log‘𝑍)))
140116, 119, 1393brtr4d 4875 . . . 4 (𝜑 → (((log‘𝑍) + 1)↑2) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14110, 31, 35, 99, 140letrd 10484 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) ≤ (((log‘𝑍) + 3) · (log‘𝑍)))
14210, 35, 17lemul2d 12161 . . 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 224 . 2 (𝜑 → (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) ≤ (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
14417rpred 12117 . . . . . . . . 9 (𝜑𝑈 ∈ ℝ)
145144adantr 473 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
146145recnd 10357 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℂ)
1476recnd 10357 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
1485rpcnne0d 12126 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
149 div23 10996 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = ((𝑈 / 𝑛) · (log‘𝑛)))
150 divass 10995 . . . . . . . 8 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 · (log‘𝑛)) / 𝑛) = (𝑈 · ((log‘𝑛) / 𝑛)))
151149, 150eqtr3d 2835 . . . . . . 7 ((𝑈 ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
152146, 147, 148, 151syl3anc 1491 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · ((log‘𝑛) / 𝑛)))
153152sumeq2dv 14774 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
154144recnd 10357 . . . . . 6 (𝜑𝑈 ∈ ℂ)
1557recnd 10357 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((log‘𝑛) / 𝑛) ∈ ℂ)
1562, 154, 155fsummulc2 14854 . . . . 5 (𝜑 → (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(𝑈 · ((log‘𝑛) / 𝑛)))
157153, 156eqtr4d 2836 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) = (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛)))
158157oveq2d 6894 . . 3 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
1598recnd 10357 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛) ∈ ℂ)
160123, 154, 159mul12d 10535 . . 3 (𝜑 → (2 · (𝑈 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
161158, 160eqtrd 2833 . 2 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) = (𝑈 · (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((log‘𝑛) / 𝑛))))
16234recnd 10357 . . 3 (𝜑 → ((log‘𝑍) + 3) ∈ ℂ)
163154, 162, 117mulassd 10352 . 2 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) = (𝑈 · (((log‘𝑍) + 3) · (log‘𝑍))))
164143, 161, 1633brtr4d 4875 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090   class class class wbr 4843  cmpt 4922  cfv 6101  (class class class)co 6878  cc 10222  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   · cmul 10229  +∞cpnf 10360   < clt 10363  cle 10364  cmin 10556   / cdiv 10976  cn 11312  2c2 11368  3c3 11369  4c4 11370  cdc 11783  +crp 12074  (,)cioo 12424  [,)cico 12426  [,]cicc 12427  ...cfz 12580  cfl 12846  cexp 13114  csqrt 14314  abscabs 14315  Σcsu 14757  expce 15128  eceu 15129  logclog 24642  ψcchp 25171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302  ax-addf 10303  ax-mulf 10304
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-of 7131  df-om 7300  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-ixp 8149  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-fi 8559  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11581  df-z 11667  df-dec 11784  df-uz 11931  df-q 12034  df-rp 12075  df-xneg 12193  df-xadd 12194  df-xmul 12195  df-ioo 12428  df-ioc 12429  df-ico 12430  df-icc 12431  df-fz 12581  df-fzo 12721  df-fl 12848  df-mod 12924  df-seq 13056  df-exp 13115  df-fac 13314  df-bc 13343  df-hash 13371  df-shft 14148  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-abs 14317  df-limsup 14543  df-clim 14560  df-rlim 14561  df-sum 14758  df-ef 15134  df-e 15135  df-sin 15136  df-cos 15137  df-pi 15139  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-ress 16192  df-plusg 16280  df-mulr 16281  df-starv 16282  df-sca 16283  df-vsca 16284  df-ip 16285  df-tset 16286  df-ple 16287  df-ds 16289  df-unif 16290  df-hom 16291  df-cco 16292  df-rest 16398  df-topn 16399  df-0g 16417  df-gsum 16418  df-topgen 16419  df-pt 16420  df-prds 16423  df-xrs 16477  df-qtop 16482  df-imas 16483  df-xps 16485  df-mre 16561  df-mrc 16562  df-acs 16564  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-submnd 17651  df-mulg 17857  df-cntz 18062  df-cmn 18510  df-psmet 20060  df-xmet 20061  df-met 20062  df-bl 20063  df-mopn 20064  df-fbas 20065  df-fg 20066  df-cnfld 20069  df-top 21027  df-topon 21044  df-topsp 21066  df-bases 21079  df-cld 21152  df-ntr 21153  df-cls 21154  df-nei 21231  df-lp 21269  df-perf 21270  df-cn 21360  df-cnp 21361  df-haus 21448  df-tx 21694  df-hmeo 21887  df-fil 21978  df-fm 22070  df-flim 22071  df-flf 22072  df-xms 22453  df-ms 22454  df-tms 22455  df-cncf 23009  df-limc 23971  df-dv 23972  df-log 24644  df-em 25071
This theorem is referenced by:  pntlemo  25648
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