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Theorem pntlemf 27664
Description: Lemma for pnt 27673. Add up the pieces in pntlemi 27663 to get an estimate slightly better than the naive lower bound 0. (Contributed by Mario Carneiro, 13-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
pntlemf (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
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 pntlemf
Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pntlem1.r . . . . . . 7 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
2 pntlem1.a . . . . . . 7 (𝜑𝐴 ∈ ℝ+)
3 pntlem1.b . . . . . . 7 (𝜑𝐵 ∈ ℝ+)
4 pntlem1.l . . . . . . 7 (𝜑𝐿 ∈ (0(,)1))
5 pntlem1.d . . . . . . 7 𝐷 = (𝐴 + 1)
6 pntlem1.f . . . . . . 7 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))
7 pntlem1.u . . . . . . 7 (𝜑𝑈 ∈ ℝ+)
8 pntlem1.u2 . . . . . . 7 (𝜑𝑈𝐴)
9 pntlem1.e . . . . . . 7 𝐸 = (𝑈 / 𝐷)
10 pntlem1.k . . . . . . 7 𝐾 = (exp‘(𝐵 / 𝐸))
111, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 27654 . . . . . 6 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
1211simp3d 1143 . . . . 5 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
1312simp3d 1143 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℝ+)
141, 2, 3, 4, 5, 6pntlemd 27653 . . . . . . . 8 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
1514simp1d 1141 . . . . . . 7 (𝜑𝐿 ∈ ℝ+)
1611simp1d 1141 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
17 2z 12647 . . . . . . . 8 2 ∈ ℤ
18 rpexpcl 14118 . . . . . . . 8 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
1916, 17, 18sylancl 586 . . . . . . 7 (𝜑 → (𝐸↑2) ∈ ℝ+)
2015, 19rpmulcld 13091 . . . . . 6 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
21 3nn0 12542 . . . . . . . . 9 3 ∈ ℕ0
22 2nn 12337 . . . . . . . . 9 2 ∈ ℕ
2321, 22decnncl 12751 . . . . . . . 8 32 ∈ ℕ
24 nnrp 13044 . . . . . . . 8 (32 ∈ ℕ → 32 ∈ ℝ+)
2523, 24ax-mp 5 . . . . . . 7 32 ∈ ℝ+
26 rpmulcl 13056 . . . . . . 7 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
2725, 3, 26sylancr 587 . . . . . 6 (𝜑 → (32 · 𝐵) ∈ ℝ+)
2820, 27rpdivcld 13092 . . . . 5 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ+)
29 pntlem1.y . . . . . . . . . 10 (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
30 pntlem1.x . . . . . . . . . 10 (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))
31 pntlem1.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ+)
32 pntlem1.w . . . . . . . . . 10 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
33 pntlem1.z . . . . . . . . . 10 (𝜑𝑍 ∈ (𝑊[,)+∞))
341, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33pntlemb 27656 . . . . . . . . 9 (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)))))
3534simp1d 1141 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
3635rpred 13075 . . . . . . 7 (𝜑𝑍 ∈ ℝ)
3734simp2d 1142 . . . . . . . 8 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
3837simp1d 1141 . . . . . . 7 (𝜑 → 1 < 𝑍)
3936, 38rplogcld 26686 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ+)
40 rpexpcl 14118 . . . . . 6 (((log‘𝑍) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ+)
4139, 17, 40sylancl 586 . . . . 5 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ+)
4228, 41rpmulcld 13091 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ+)
4313, 42rpmulcld 13091 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ+)
4443rpred 13075 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
4515, 16rpmulcld 13091 . . . . . . 7 (𝜑 → (𝐿 · 𝐸) ∈ ℝ+)
46 8re 12360 . . . . . . . 8 8 ∈ ℝ
47 8pos 12376 . . . . . . . 8 0 < 8
4846, 47elrpii 13035 . . . . . . 7 8 ∈ ℝ+
49 rpdivcl 13058 . . . . . . 7 (((𝐿 · 𝐸) ∈ ℝ+ ∧ 8 ∈ ℝ+) → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5045, 48, 49sylancl 586 . . . . . 6 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5150, 39rpmulcld 13091 . . . . 5 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ+)
5213, 51rpmulcld 13091 . . . 4 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
5352rpred 13075 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
54 pntlem1.m . . . . . . . 8 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
55 pntlem1.n . . . . . . . 8 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
561, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemg 27657 . . . . . . 7 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
5756simp1d 1141 . . . . . 6 (𝜑𝑀 ∈ ℕ)
5856simp2d 1142 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
59 eluznn 12958 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑁 ∈ ℕ)
6057, 58, 59syl2anc 584 . . . . 5 (𝜑𝑁 ∈ ℕ)
6160nnred 12279 . . . 4 (𝜑𝑁 ∈ ℝ)
6257nnred 12279 . . . 4 (𝜑𝑀 ∈ ℝ)
6361, 62resubcld 11689 . . 3 (𝜑 → (𝑁𝑀) ∈ ℝ)
6453, 63remulcld 11289 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ∈ ℝ)
65 fzfid 14011 . . 3 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
667rpred 13075 . . . . . 6 (𝜑𝑈 ∈ ℝ)
67 elfznn 13590 . . . . . 6 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68 nndivre 12305 . . . . . 6 ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
6966, 67, 68syl2an 596 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
7035adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
7167adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
7271nnrpd 13073 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
7370, 72rpdivcld 13092 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
741pntrf 27622 . . . . . . . . . 10 𝑅:ℝ+⟶ℝ
7574ffvelcdmi 7103 . . . . . . . . 9 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7673, 75syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7776, 70rerpdivcld 13106 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
7877recnd 11287 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
7978abscld 15472 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
8069, 79resubcld 11689 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
8172relogcld 26680 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
8280, 81remulcld 11289 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8365, 82fsumrecl 15767 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8445rpcnd 13077 . . . . . . . . 9 (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
8511simp2d 1142 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ+)
8685rpred 13075 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℝ)
8712simp2d 1142 . . . . . . . . . . . 12 (𝜑 → 1 < 𝐾)
8886, 87rplogcld 26686 . . . . . . . . . . 11 (𝜑 → (log‘𝐾) ∈ ℝ+)
8939, 88rpdivcld 13092 . . . . . . . . . 10 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
9089rpcnd 13077 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91 rpcnne0 13051 . . . . . . . . . 10 (8 ∈ ℝ+ → (8 ∈ ℂ ∧ 8 ≠ 0))
9248, 91mp1i 13 . . . . . . . . 9 (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93 4re 12348 . . . . . . . . . . 11 4 ∈ ℝ
94 4pos 12371 . . . . . . . . . . 11 0 < 4
9593, 94elrpii 13035 . . . . . . . . . 10 4 ∈ ℝ+
96 rpcnne0 13051 . . . . . . . . . 10 (4 ∈ ℝ+ → (4 ∈ ℂ ∧ 4 ≠ 0))
9795, 96mp1i 13 . . . . . . . . 9 (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98 divmuldiv 11965 . . . . . . . . 9 ((((𝐿 · 𝐸) ∈ ℂ ∧ ((log‘𝑍) / (log‘𝐾)) ∈ ℂ) ∧ ((8 ∈ ℂ ∧ 8 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0))) → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
9984, 90, 92, 97, 98syl22anc 839 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
10010fveq2i 6910 . . . . . . . . . . . . . 14 (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
1013, 16rpdivcld 13092 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 / 𝐸) ∈ ℝ+)
102101rpred 13075 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103102relogefd 26685 . . . . . . . . . . . . . 14 (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104100, 103eqtrid 2787 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105104oveq2d 7447 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
10639rpcnd 13077 . . . . . . . . . . . . 13 (𝜑 → (log‘𝑍) ∈ ℂ)
1073rpcnne0d 13084 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
10816rpcnne0d 13084 . . . . . . . . . . . . 13 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109 divdiv2 11977 . . . . . . . . . . . . 13 (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110106, 107, 108, 109syl3anc 1370 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111105, 110eqtrd 2775 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112111oveq2d 7447 . . . . . . . . . 10 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11316rpcnd 13077 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℂ)
114106, 113mulcld 11279 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115 divass 11938 . . . . . . . . . . 11 (((𝐿 · 𝐸) ∈ ℂ ∧ ((log‘𝑍) · 𝐸) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11684, 114, 107, 115syl3anc 1370 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11715rpcnd 13077 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ℂ)
118117, 113, 106, 113mul4d 11471 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119113sqvald 14180 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120119oveq2d 7447 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121113sqcld 14181 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) ∈ ℂ)
122117, 106, 121mul32d 11469 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123118, 120, 1223eqtr2d 2781 . . . . . . . . . . 11 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124123oveq1d 7446 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125112, 116, 1243eqtr2d 2781 . . . . . . . . 9 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126 8t4e32 12848 . . . . . . . . . 10 (8 · 4) = 32
127126a1i 11 . . . . . . . . 9 (𝜑 → (8 · 4) = 32)
128125, 127oveq12d 7449 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32))
12920rpcnd 13077 . . . . . . . . . . 11 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130129, 106mulcld 11279 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131 rpcnne0 13051 . . . . . . . . . . 11 (32 ∈ ℝ+ → (32 ∈ ℂ ∧ 32 ≠ 0))
13225, 131mp1i 13 . . . . . . . . . 10 (𝜑 → (32 ∈ ℂ ∧ 32 ≠ 0))
133 divdiv1 11976 . . . . . . . . . 10 ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (32 ∈ ℂ ∧ 32 ≠ 0)) → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
134130, 107, 132, 133syl3anc 1370 . . . . . . . . 9 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
13523nncni 12274 . . . . . . . . . . 11 32 ∈ ℂ
1363rpcnd 13077 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℂ)
137 mulcom 11239 . . . . . . . . . . 11 ((32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (32 · 𝐵) = (𝐵 · 32))
138135, 136, 137sylancr 587 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) = (𝐵 · 32))
139138oveq2d 7447 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
14027rpcnne0d 13084 . . . . . . . . . 10 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
141 div23 11939 . . . . . . . . . 10 (((𝐿 · (𝐸↑2)) ∈ ℂ ∧ (log‘𝑍) ∈ ℂ ∧ ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0)) → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
142129, 106, 140, 141syl3anc 1370 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
143134, 139, 1423eqtr2d 2781 . . . . . . . 8 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
14499, 128, 1433eqtrd 2779 . . . . . . 7 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
145144oveq1d 7446 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
14650rpcnd 13077 . . . . . . 7 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
14789rpred 13075 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148 4nn 12347 . . . . . . . . 9 4 ∈ ℕ
149 nndivre 12305 . . . . . . . . 9 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150147, 148, 149sylancl 586 . . . . . . . 8 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151150recnd 11287 . . . . . . 7 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152146, 106, 151mul32d 11469 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153106sqvald 14180 . . . . . . . 8 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154153oveq2d 7447 . . . . . . 7 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
15528rpcnd 13077 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
156155, 106, 106mulassd 11282 . . . . . . 7 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157154, 156eqtr4d 2778 . . . . . 6 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158145, 152, 1573eqtr4d 2785 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))
15956simp3d 1143 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
160150, 63, 51lemul2d 13119 . . . . . 6 (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀) ↔ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
161159, 160mpbid 232 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
162158, 161eqbrtrrd 5172 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
16342rpred 13075 . . . . 5 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
16451rpred 13075 . . . . . 6 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165164, 63remulcld 11289 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ∈ ℝ)
166163, 165, 13lemul2d 13119 . . . 4 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ↔ ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))))
167162, 166mpbid 232 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
16813rpcnd 13077 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℂ)
16951rpcnd 13077 . . . 4 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
17063recnd 11287 . . . 4 (𝜑 → (𝑁𝑀) ∈ ℂ)
171168, 169, 170mulassd 11282 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) = ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
172167, 171breqtrrd 5176 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
173 fzfid 14011 . . . 4 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
17460nnzd 12638 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
17585, 174rpexpcld 14283 . . . . . . . . . . 11 (𝜑 → (𝐾𝑁) ∈ ℝ+)
17635, 175rpdivcld 13092 . . . . . . . . . 10 (𝜑 → (𝑍 / (𝐾𝑁)) ∈ ℝ+)
177176rprege0d 13082 . . . . . . . . 9 (𝜑 → ((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))))
178 flge0nn0 13857 . . . . . . . . 9 (((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))) → (⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0)
179 nn0p1nn 12563 . . . . . . . . 9 ((⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
180177, 178, 1793syl 18 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
181 nnuz 12919 . . . . . . . 8 ℕ = (ℤ‘1)
182180, 181eleqtrdi 2849 . . . . . . 7 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1))
183 fzss1 13600 . . . . . . 7 (((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1) → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
184182, 183syl 17 . . . . . 6 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
185184sselda 3995 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186185, 82syldan 591 . . . 4 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187173, 186fsumrecl 15767 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188 eluzfz2 13569 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
18958, 188syl 17 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
190 oveq1 7438 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚𝑀) = (𝑀𝑀))
191190oveq2d 7447 . . . . . . 7 (𝑚 = 𝑀 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)))
192 oveq2 7439 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝐾𝑚) = (𝐾𝑀))
193192oveq2d 7447 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑀)))
194193fveq2d 6911 . . . . . . . . . 10 (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑀))))
195194oveq1d 7446 . . . . . . . . 9 (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑀))) + 1))
196195oveq1d 7446 . . . . . . . 8 (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197196sumeq1d 15733 . . . . . . 7 (𝑚 = 𝑀 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
198191, 197breq12d 5161 . . . . . 6 (𝑚 = 𝑀 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
199198imbi2d 340 . . . . 5 (𝑚 = 𝑀 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
200 oveq1 7438 . . . . . . . 8 (𝑚 = 𝑗 → (𝑚𝑀) = (𝑗𝑀))
201200oveq2d 7447 . . . . . . 7 (𝑚 = 𝑗 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)))
202 oveq2 7439 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐾𝑚) = (𝐾𝑗))
203202oveq2d 7447 . . . . . . . . . . 11 (𝑚 = 𝑗 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑗)))
204203fveq2d 6911 . . . . . . . . . 10 (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑗))))
205204oveq1d 7446 . . . . . . . . 9 (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
206205oveq1d 7446 . . . . . . . 8 (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207206sumeq1d 15733 . . . . . . 7 (𝑚 = 𝑗 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
208201, 207breq12d 5161 . . . . . 6 (𝑚 = 𝑗 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
209208imbi2d 340 . . . . 5 (𝑚 = 𝑗 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
210 oveq1 7438 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (𝑚𝑀) = ((𝑗 + 1) − 𝑀))
211210oveq2d 7447 . . . . . . 7 (𝑚 = (𝑗 + 1) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212 oveq2 7439 . . . . . . . . . . . 12 (𝑚 = (𝑗 + 1) → (𝐾𝑚) = (𝐾↑(𝑗 + 1)))
213212oveq2d 7447 . . . . . . . . . . 11 (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214213fveq2d 6911 . . . . . . . . . 10 (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215214oveq1d 7446 . . . . . . . . 9 (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216215oveq1d 7446 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217216sumeq1d 15733 . . . . . . 7 (𝑚 = (𝑗 + 1) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
218211, 217breq12d 5161 . . . . . 6 (𝑚 = (𝑗 + 1) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
219218imbi2d 340 . . . . 5 (𝑚 = (𝑗 + 1) → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
220 oveq1 7438 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚𝑀) = (𝑁𝑀))
221220oveq2d 7447 . . . . . . 7 (𝑚 = 𝑁 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
222 oveq2 7439 . . . . . . . . . . . 12 (𝑚 = 𝑁 → (𝐾𝑚) = (𝐾𝑁))
223222oveq2d 7447 . . . . . . . . . . 11 (𝑚 = 𝑁 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑁)))
224223fveq2d 6911 . . . . . . . . . 10 (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑁))))
225224oveq1d 7446 . . . . . . . . 9 (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑁))) + 1))
226225oveq1d 7446 . . . . . . . 8 (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227226sumeq1d 15733 . . . . . . 7 (𝑚 = 𝑁 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
228221, 227breq12d 5161 . . . . . 6 (𝑚 = 𝑁 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
229228imbi2d 340 . . . . 5 (𝑚 = 𝑁 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
23057nncnd 12280 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
231230subidd 11606 . . . . . . . . 9 (𝜑 → (𝑀𝑀) = 0)
232231oveq2d 7447 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
23352rpcnd 13077 . . . . . . . . 9 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234233mul01d 11458 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235232, 234eqtrd 2775 . . . . . . 7 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = 0)
236 fzfid 14011 . . . . . . . 8 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
23757nnzd 12638 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
23885, 237rpexpcld 14283 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾𝑀) ∈ ℝ+)
23935, 238rpdivcld 13092 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / (𝐾𝑀)) ∈ ℝ+)
240239rprege0d 13082 . . . . . . . . . . . . 13 (𝜑 → ((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))))
241 flge0nn0 13857 . . . . . . . . . . . . 13 (((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))) → (⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0)
242 nn0p1nn 12563 . . . . . . . . . . . . 13 ((⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
243240, 241, 2423syl 18 . . . . . . . . . . . 12 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
244243, 181eleqtrdi 2849 . . . . . . . . . . 11 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1))
245 fzss1 13600 . . . . . . . . . . 11 (((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1) → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
246244, 245syl 17 . . . . . . . . . 10 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
247246sselda 3995 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248247, 82syldan 591 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249 elfzle2 13565 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250249adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
25129simpld 494 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ ℝ+)
25235, 251rpdivcld 13092 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / 𝑌) ∈ ℝ+)
253252rpred 13075 . . . . . . . . . . . . 13 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254 elfzelz 13561 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255 flge 13842 . . . . . . . . . . . . 13 (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256253, 254, 255syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257250, 256mpbird 257 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
25871, 257jca 511 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌)))
259 pntlem1.U . . . . . . . . . . 11 (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)
2601, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55, 259pntlemn 27659 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261258, 260syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262247, 261syldan 591 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263236, 248, 262fsumge0 15828 . . . . . . 7 (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264235, 263eqbrtrd 5170 . . . . . 6 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
265264a1i 11 . . . . 5 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
266 pntlem1.K . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))
267 eqid 2735 . . . . . . . . . 10 (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))
2681, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55, 259, 266, 267pntlemi 27663 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
26952adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
270269rpred 13075 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271 elfzoelz 13696 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272271adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273272zred 12720 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
27457adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275274nnred 12279 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276273, 275resubcld 11689 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℝ)
277270, 276remulcld 11289 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ)
278 fzfid 14011 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∈ Fin)
279 ssun1 4188 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
28036adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
28185adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ+)
282272peano2zd 12723 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283281, 282rpexpcld 14283 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ+)
284280, 283rerpdivcld 13106 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285281, 272rpexpcld 14283 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ+)
286280, 285rerpdivcld 13106 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ∈ ℝ)
28786adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288 1re 11259 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
289 ltle 11347 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290288, 86, 289sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
29187, 290mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝐾)
292291adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293 uzid 12891 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
294 peano2uz 12941 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
295272, 293, 2943syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑗))
296287, 292, 295leexp2ad 14290 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)))
29735adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ+)
298285, 283, 297lediv2d 13099 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))))
299296, 298mpbid 232 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗)))
300 flword2 13850 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
301284, 286, 299, 300syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302 eluzp1p1 12904 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))) → ((⌊‘(𝑍 / (𝐾𝑗))) + 1) ∈ (ℤ‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
303301, 302syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾𝑗))) + 1) ∈ (ℤ‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
304286flcld 13835 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ)
305252adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ+)
306305rpred 13075 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307306flcld 13835 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308251adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ+)
309308rpred 13075 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310285rpred 13075 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ)
31130simpld 494 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ∈ ℝ+)
312311rpred 13075 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 ∈ ℝ)
313312adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
31430simprd 495 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 < 𝑋)
315314adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316 elfzofz 13712 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
3171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemh 27658 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
318316, 317sylan2 593 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
319318simpld 494 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾𝑗))
320309, 313, 310, 315, 319lttrd 11420 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾𝑗))
321309, 310, 320ltled 11407 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾𝑗))
322308, 285, 297lediv2d 13099 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾𝑗) ↔ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)))
323321, 322mpbid 232 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌))
324 flwordi 13849 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325286, 306, 323, 324syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326 eluz2 12882 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327304, 307, 325, 326syl3anbrc 1342 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))))
328 fzsplit2 13586 . . . . . . . . . . . . . . . 16 ((((⌊‘(𝑍 / (𝐾𝑗))) + 1) ∈ (ℤ‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)) ∧ (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗))))) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
329303, 327, 328syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
330279, 329sseqtrrid 4049 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331297, 283rpdivcld 13092 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ+)
332331rprege0d 13082 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333 flge0nn0 13857 . . . . . . . . . . . . . . . . 17 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0)
334 nn0p1nn 12563 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
335332, 333, 3343syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
336335, 181eleqtrdi 2849 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1))
337 fzss1 13600 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
338336, 337syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
339330, 338sstrd 4006 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340339sselda 3995 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
34182adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342340, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
343278, 342fsumrecl 15767 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344 fzfid 14011 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345 ssun2 4189 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346345, 329sseqtrrid 4049 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347346, 338sstrd 4006 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348347sselda 3995 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349348, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
350344, 349fsumrecl 15767 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351 le2add 11743 . . . . . . . . . 10 (((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ ∧ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ) ∧ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ ∧ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∧ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
352270, 277, 343, 350, 351syl22anc 839 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∧ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
353268, 352mpand 695 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
354233adantr 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355 1cnd 11254 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356272zcnd 12721 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357230adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358356, 357subcld 11618 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℂ)
359354, 355, 358adddid 11283 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))))
360355, 358addcomd 11461 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗𝑀) + 1))
361356, 355, 357addsubd 11639 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗𝑀) + 1))
362360, 361eqtr4d 2778 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗 + 1) − 𝑀))
363362oveq2d 7447 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364354mulridd 11276 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365364oveq1d 7446 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))))
366359, 363, 3653eqtr3d 2783 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))))
367 reflcl 13833 . . . . . . . . . . . . 13 ((𝑍 / (𝐾𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
368286, 367syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
369368ltp1d 12196 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
370 fzdisj 13588 . . . . . . . . . . 11 ((⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1) → ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∩ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) = ∅)
371369, 370syl 17 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∩ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) = ∅)
372 fzfid 14011 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373338sselda 3995 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374373, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
375374recnd 11287 . . . . . . . . . 10 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℂ)
376371, 329, 372, 375fsumsplit 15774 . . . . . . . . 9 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
377366, 376breq12d 5161 . . . . . . . 8 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
378353, 377sylibrd 259 . . . . . . 7 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
379378expcom 413 . . . . . 6 (𝑗 ∈ (𝑀..^𝑁) → (𝜑 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
380379a2d 29 . . . . 5 (𝑗 ∈ (𝑀..^𝑁) → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
381199, 209, 219, 229, 265, 380fzind2 13821 . . . 4 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
382189, 381mpcom 38 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
38365, 82, 261, 184fsumless 15829 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
38464, 187, 83, 382, 383letrd 11416 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
38544, 64, 83, 172, 384letrd 11416 1 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cun 3961  cin 3962  wss 3963  c0 4339   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  cn 12264  2c2 12319  3c3 12320  4c4 12321  8c8 12325  0cn0 12524  cz 12611  cdc 12731  cuz 12876  +crp 13032  (,)cioo 13384  [,)cico 13386  [,]cicc 13387  ...cfz 13544  ..^cfzo 13691  cfl 13827  cexp 14099  csqrt 15269  abscabs 15270  Σcsu 15719  expce 16094  eceu 16095  logclog 26611  ψcchp 27151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-e 16101  df-sin 16102  df-cos 16103  df-pi 16105  df-dvds 16288  df-gcd 16529  df-prm 16706  df-pc 16871  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-log 26613  df-vma 27156  df-chp 27157
This theorem is referenced by:  pntlemo  27666
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