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Theorem pntlemf 25339
Description: Lemma for pnt 25348. Add up the pieces in pntlemi 25338 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 25329 . . . . . 6 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
1211simp3d 1095 . . . . 5 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
1312simp3d 1095 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℝ+)
141, 2, 3, 4, 5, 6pntlemd 25328 . . . . . . . 8 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
1514simp1d 1093 . . . . . . 7 (𝜑𝐿 ∈ ℝ+)
1611simp1d 1093 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
17 2z 11447 . . . . . . . 8 2 ∈ ℤ
18 rpexpcl 12919 . . . . . . . 8 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
1916, 17, 18sylancl 695 . . . . . . 7 (𝜑 → (𝐸↑2) ∈ ℝ+)
2015, 19rpmulcld 11926 . . . . . 6 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
21 3nn0 11348 . . . . . . . . 9 3 ∈ ℕ0
22 2nn 11223 . . . . . . . . 9 2 ∈ ℕ
2321, 22decnncl 11556 . . . . . . . 8 32 ∈ ℕ
24 nnrp 11880 . . . . . . . 8 (32 ∈ ℕ → 32 ∈ ℝ+)
2523, 24ax-mp 5 . . . . . . 7 32 ∈ ℝ+
26 rpmulcl 11893 . . . . . . 7 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
2725, 3, 26sylancr 696 . . . . . 6 (𝜑 → (32 · 𝐵) ∈ ℝ+)
2820, 27rpdivcld 11927 . . . . 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 25331 . . . . . . . . 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 1093 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
3635rpred 11910 . . . . . . 7 (𝜑𝑍 ∈ ℝ)
3734simp2d 1094 . . . . . . . 8 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
3837simp1d 1093 . . . . . . 7 (𝜑 → 1 < 𝑍)
3936, 38rplogcld 24420 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ+)
40 rpexpcl 12919 . . . . . 6 (((log‘𝑍) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ+)
4139, 17, 40sylancl 695 . . . . 5 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ+)
4228, 41rpmulcld 11926 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ+)
4313, 42rpmulcld 11926 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ+)
4443rpred 11910 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
4515, 16rpmulcld 11926 . . . . . . 7 (𝜑 → (𝐿 · 𝐸) ∈ ℝ+)
46 8re 11143 . . . . . . . 8 8 ∈ ℝ
47 8pos 11159 . . . . . . . 8 0 < 8
4846, 47elrpii 11873 . . . . . . 7 8 ∈ ℝ+
49 rpdivcl 11894 . . . . . . 7 (((𝐿 · 𝐸) ∈ ℝ+ ∧ 8 ∈ ℝ+) → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5045, 48, 49sylancl 695 . . . . . 6 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5150, 39rpmulcld 11926 . . . . 5 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ+)
5213, 51rpmulcld 11926 . . . 4 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
5352rpred 11910 . . 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 25332 . . . . . . 7 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
5756simp1d 1093 . . . . . 6 (𝜑𝑀 ∈ ℕ)
5856simp2d 1094 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
59 eluznn 11796 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑁 ∈ ℕ)
6057, 58, 59syl2anc 694 . . . . 5 (𝜑𝑁 ∈ ℕ)
6160nnred 11073 . . . 4 (𝜑𝑁 ∈ ℝ)
6257nnred 11073 . . . 4 (𝜑𝑀 ∈ ℝ)
6361, 62resubcld 10496 . . 3 (𝜑 → (𝑁𝑀) ∈ ℝ)
6453, 63remulcld 10108 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ∈ ℝ)
65 fzfid 12812 . . 3 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
667rpred 11910 . . . . . 6 (𝜑𝑈 ∈ ℝ)
67 elfznn 12408 . . . . . 6 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68 nndivre 11094 . . . . . 6 ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
6966, 67, 68syl2an 493 . . . . 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 11908 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
7370, 72rpdivcld 11927 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
741pntrf 25297 . . . . . . . . . 10 𝑅:ℝ+⟶ℝ
7574ffvelrni 6398 . . . . . . . . 9 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7673, 75syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7776, 70rerpdivcld 11941 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
7877recnd 10106 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
7978abscld 14219 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
8069, 79resubcld 10496 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
8172relogcld 24414 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
8280, 81remulcld 10108 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8365, 82fsumrecl 14509 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8445rpcnd 11912 . . . . . . . . 9 (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
8511simp2d 1094 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ+)
8685rpred 11910 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℝ)
8712simp2d 1094 . . . . . . . . . . . 12 (𝜑 → 1 < 𝐾)
8886, 87rplogcld 24420 . . . . . . . . . . 11 (𝜑 → (log‘𝐾) ∈ ℝ+)
8939, 88rpdivcld 11927 . . . . . . . . . 10 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
9089rpcnd 11912 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91 rpcnne0 11888 . . . . . . . . . 10 (8 ∈ ℝ+ → (8 ∈ ℂ ∧ 8 ≠ 0))
9248, 91mp1i 13 . . . . . . . . 9 (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93 4re 11135 . . . . . . . . . . 11 4 ∈ ℝ
94 4pos 11154 . . . . . . . . . . 11 0 < 4
9593, 94elrpii 11873 . . . . . . . . . 10 4 ∈ ℝ+
96 rpcnne0 11888 . . . . . . . . . 10 (4 ∈ ℝ+ → (4 ∈ ℂ ∧ 4 ≠ 0))
9795, 96mp1i 13 . . . . . . . . 9 (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98 divmuldiv 10763 . . . . . . . . 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 1367 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
10010fveq2i 6232 . . . . . . . . . . . . . 14 (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
1013, 16rpdivcld 11927 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 / 𝐸) ∈ ℝ+)
102101rpred 11910 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103102relogefd 24419 . . . . . . . . . . . . . 14 (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104100, 103syl5eq 2697 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105104oveq2d 6706 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
10639rpcnd 11912 . . . . . . . . . . . . 13 (𝜑 → (log‘𝑍) ∈ ℂ)
1073rpcnne0d 11919 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
10816rpcnne0d 11919 . . . . . . . . . . . . 13 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109 divdiv2 10775 . . . . . . . . . . . . 13 (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110106, 107, 108, 109syl3anc 1366 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111105, 110eqtrd 2685 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112111oveq2d 6706 . . . . . . . . . 10 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11316rpcnd 11912 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℂ)
114106, 113mulcld 10098 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115 divass 10741 . . . . . . . . . . 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 1366 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11715rpcnd 11912 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ℂ)
118117, 113, 106, 113mul4d 10286 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119113sqvald 13045 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120119oveq2d 6706 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121113sqcld 13046 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) ∈ ℂ)
122117, 106, 121mul32d 10284 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123118, 120, 1223eqtr2d 2691 . . . . . . . . . . 11 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124123oveq1d 6705 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125112, 116, 1243eqtr2d 2691 . . . . . . . . 9 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126 8t4e32 11694 . . . . . . . . . 10 (8 · 4) = 32
127126a1i 11 . . . . . . . . 9 (𝜑 → (8 · 4) = 32)
128125, 127oveq12d 6708 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32))
12920rpcnd 11912 . . . . . . . . . . 11 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130129, 106mulcld 10098 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131 rpcnne0 11888 . . . . . . . . . . 11 (32 ∈ ℝ+ → (32 ∈ ℂ ∧ 32 ≠ 0))
13225, 131mp1i 13 . . . . . . . . . 10 (𝜑 → (32 ∈ ℂ ∧ 32 ≠ 0))
133 divdiv1 10774 . . . . . . . . . 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 1366 . . . . . . . . 9 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
13523nncni 11068 . . . . . . . . . . 11 32 ∈ ℂ
1363rpcnd 11912 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℂ)
137 mulcom 10060 . . . . . . . . . . 11 ((32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (32 · 𝐵) = (𝐵 · 32))
138135, 136, 137sylancr 696 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) = (𝐵 · 32))
139138oveq2d 6706 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
14027rpcnne0d 11919 . . . . . . . . . 10 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
141 div23 10742 . . . . . . . . . 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 1366 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
143134, 139, 1423eqtr2d 2691 . . . . . . . 8 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
14499, 128, 1433eqtrd 2689 . . . . . . 7 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
145144oveq1d 6705 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
14650rpcnd 11912 . . . . . . 7 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
14789rpred 11910 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148 4nn 11225 . . . . . . . . 9 4 ∈ ℕ
149 nndivre 11094 . . . . . . . . 9 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150147, 148, 149sylancl 695 . . . . . . . 8 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151150recnd 10106 . . . . . . 7 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152146, 106, 151mul32d 10284 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153106sqvald 13045 . . . . . . . 8 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154153oveq2d 6706 . . . . . . 7 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
15528rpcnd 11912 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
156155, 106, 106mulassd 10101 . . . . . . 7 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157154, 156eqtr4d 2688 . . . . . 6 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158145, 152, 1573eqtr4d 2695 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))
15956simp3d 1095 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
160150, 63, 51lemul2d 11954 . . . . . 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 222 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
162158, 161eqbrtrrd 4709 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
16342rpred 11910 . . . . 5 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
16451rpred 11910 . . . . . 6 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165164, 63remulcld 10108 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ∈ ℝ)
166163, 165, 13lemul2d 11954 . . . 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 222 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
16813rpcnd 11912 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℂ)
16951rpcnd 11912 . . . 4 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
17063recnd 10106 . . . 4 (𝜑 → (𝑁𝑀) ∈ ℂ)
171168, 169, 170mulassd 10101 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) = ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
172167, 171breqtrrd 4713 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
173 fzfid 12812 . . . 4 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
17460nnzd 11519 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
17585, 174rpexpcld 13072 . . . . . . . . . . 11 (𝜑 → (𝐾𝑁) ∈ ℝ+)
17635, 175rpdivcld 11927 . . . . . . . . . 10 (𝜑 → (𝑍 / (𝐾𝑁)) ∈ ℝ+)
177176rprege0d 11917 . . . . . . . . 9 (𝜑 → ((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))))
178 flge0nn0 12661 . . . . . . . . 9 (((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))) → (⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0)
179 nn0p1nn 11370 . . . . . . . . 9 ((⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
180177, 178, 1793syl 18 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
181 nnuz 11761 . . . . . . . 8 ℕ = (ℤ‘1)
182180, 181syl6eleq 2740 . . . . . . 7 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1))
183 fzss1 12418 . . . . . . 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 3636 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186185, 82syldan 486 . . . 4 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187173, 186fsumrecl 14509 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188 eluzfz2 12387 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
18958, 188syl 17 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
190 oveq1 6697 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚𝑀) = (𝑀𝑀))
191190oveq2d 6706 . . . . . . 7 (𝑚 = 𝑀 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)))
192 oveq2 6698 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝐾𝑚) = (𝐾𝑀))
193192oveq2d 6706 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑀)))
194193fveq2d 6233 . . . . . . . . . 10 (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑀))))
195194oveq1d 6705 . . . . . . . . 9 (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑀))) + 1))
196195oveq1d 6705 . . . . . . . 8 (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197196sumeq1d 14475 . . . . . . 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 4698 . . . . . 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 329 . . . . 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 6697 . . . . . . . 8 (𝑚 = 𝑗 → (𝑚𝑀) = (𝑗𝑀))
201200oveq2d 6706 . . . . . . 7 (𝑚 = 𝑗 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)))
202 oveq2 6698 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐾𝑚) = (𝐾𝑗))
203202oveq2d 6706 . . . . . . . . . . 11 (𝑚 = 𝑗 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑗)))
204203fveq2d 6233 . . . . . . . . . 10 (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑗))))
205204oveq1d 6705 . . . . . . . . 9 (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
206205oveq1d 6705 . . . . . . . 8 (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207206sumeq1d 14475 . . . . . . 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 4698 . . . . . 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 329 . . . . 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 6697 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (𝑚𝑀) = ((𝑗 + 1) − 𝑀))
211210oveq2d 6706 . . . . . . 7 (𝑚 = (𝑗 + 1) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212 oveq2 6698 . . . . . . . . . . . 12 (𝑚 = (𝑗 + 1) → (𝐾𝑚) = (𝐾↑(𝑗 + 1)))
213212oveq2d 6706 . . . . . . . . . . 11 (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214213fveq2d 6233 . . . . . . . . . 10 (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215214oveq1d 6705 . . . . . . . . 9 (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216215oveq1d 6705 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217216sumeq1d 14475 . . . . . . 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 4698 . . . . . 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 329 . . . . 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 6697 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚𝑀) = (𝑁𝑀))
221220oveq2d 6706 . . . . . . 7 (𝑚 = 𝑁 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
222 oveq2 6698 . . . . . . . . . . . 12 (𝑚 = 𝑁 → (𝐾𝑚) = (𝐾𝑁))
223222oveq2d 6706 . . . . . . . . . . 11 (𝑚 = 𝑁 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑁)))
224223fveq2d 6233 . . . . . . . . . 10 (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑁))))
225224oveq1d 6705 . . . . . . . . 9 (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑁))) + 1))
226225oveq1d 6705 . . . . . . . 8 (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227226sumeq1d 14475 . . . . . . 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 4698 . . . . . 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 329 . . . . 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 11074 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
231230subidd 10418 . . . . . . . . 9 (𝜑 → (𝑀𝑀) = 0)
232231oveq2d 6706 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
23352rpcnd 11912 . . . . . . . . 9 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234233mul01d 10273 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235232, 234eqtrd 2685 . . . . . . 7 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = 0)
236 fzfid 12812 . . . . . . . 8 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
23757nnzd 11519 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
23885, 237rpexpcld 13072 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾𝑀) ∈ ℝ+)
23935, 238rpdivcld 11927 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / (𝐾𝑀)) ∈ ℝ+)
240239rprege0d 11917 . . . . . . . . . . . . 13 (𝜑 → ((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))))
241 flge0nn0 12661 . . . . . . . . . . . . 13 (((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))) → (⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0)
242 nn0p1nn 11370 . . . . . . . . . . . . 13 ((⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
243240, 241, 2423syl 18 . . . . . . . . . . . 12 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
244243, 181syl6eleq 2740 . . . . . . . . . . 11 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1))
245 fzss1 12418 . . . . . . . . . . 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 3636 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248247, 82syldan 486 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249 elfzle2 12383 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250249adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
25129simpld 474 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ ℝ+)
25235, 251rpdivcld 11927 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / 𝑌) ∈ ℝ+)
253252rpred 11910 . . . . . . . . . . . . 13 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254 elfzelz 12380 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255 flge 12646 . . . . . . . . . . . . 13 (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256253, 254, 255syl2an 493 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257250, 256mpbird 247 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
25871, 257jca 553 . . . . . . . . . 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 25334 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261258, 260syldan 486 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262247, 261syldan 486 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263236, 248, 262fsumge0 14571 . . . . . . 7 (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264235, 263eqbrtrd 4707 . . . . . 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 2651 . . . . . . . . . 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 25338 . . . . . . . . 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 11910 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271 elfzoelz 12509 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272271adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273272zred 11520 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
27457adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275274nnred 11073 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276273, 275resubcld 10496 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℝ)
277270, 276remulcld 10108 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ)
278 fzfid 12812 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∈ Fin)
279 ssun1 3809 . . . . . . . . . . . . . . 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 11523 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283281, 282rpexpcld 13072 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ+)
284280, 283rerpdivcld 11941 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285281, 272rpexpcld 13072 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ+)
286280, 285rerpdivcld 11941 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ∈ ℝ)
28786adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288 1re 10077 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
289 ltle 10164 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290288, 86, 289sylancr 696 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
29187, 290mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝐾)
292291adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293 uzid 11740 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
294 peano2uz 11779 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
295272, 293, 2943syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑗))
296287, 292, 295leexp2ad 13081 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)))
29735adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ+)
298285, 283, 297lediv2d 11934 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))))
299296, 298mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗)))
300 flword2 12654 . . . . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302 eluzp1p1 11751 . . . . . . . . . . . . . . . . 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 12639 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ)
305252adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ+)
306305rpred 11910 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307306flcld 12639 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308251adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ+)
309308rpred 11910 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310285rpred 11910 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ)
31130simpld 474 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ∈ ℝ+)
312311rpred 11910 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 ∈ ℝ)
313312adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
31430simprd 478 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 < 𝑋)
315314adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316 elfzofz 12524 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
3171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemh 25333 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
318316, 317sylan2 490 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
319318simpld 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾𝑗))
320309, 313, 310, 315, 319lttrd 10236 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾𝑗))
321309, 310, 320ltled 10223 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾𝑗))
322308, 285, 297lediv2d 11934 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾𝑗) ↔ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)))
323321, 322mpbid 222 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌))
324 flwordi 12653 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325286, 306, 323, 324syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326 eluz2 11731 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327304, 307, 325, 326syl3anbrc 1265 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))))
328 fzsplit2 12404 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
330279, 329syl5sseqr 3687 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331297, 283rpdivcld 11927 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ+)
332331rprege0d 11917 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333 flge0nn0 12661 . . . . . . . . . . . . . . . . 17 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0)
334 nn0p1nn 11370 . . . . . . . . . . . . . . . . 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, 181syl6eleq 2740 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1))
337 fzss1 12418 . . . . . . . . . . . . . . 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 3646 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340339sselda 3636 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
34182adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342340, 341syldan 486 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
343278, 342fsumrecl 14509 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344 fzfid 12812 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345 ssun2 3810 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346345, 329syl5sseqr 3687 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347346, 338sstrd 3646 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348347sselda 3636 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349348, 341syldan 486 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
350344, 349fsumrecl 14509 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351 le2add 10548 . . . . . . . . . 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 1367 . . . . . . . . 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 711 . . . . . . . 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 10094 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356272zcnd 11521 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357230adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358356, 357subcld 10430 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℂ)
359354, 355, 358adddid 10102 . . . . . . . . . 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 10276 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗𝑀) + 1))
361356, 355, 357addsubd 10451 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗𝑀) + 1))
362360, 361eqtr4d 2688 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗 + 1) − 𝑀))
363362oveq2d 6706 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364354mulid1d 10095 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365364oveq1d 6705 . . . . . . . . . 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 2693 . . . . . . . . 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 12637 . . . . . . . . . . . . 13 ((𝑍 / (𝐾𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
368286, 367syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
369368ltp1d 10992 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
370 fzdisj 12406 . . . . . . . . . . 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 12812 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373338sselda 3636 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374373, 341syldan 486 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
375374recnd 10106 . . . . . . . . . 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 14515 . . . . . . . . 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 4698 . . . . . . . 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 249 . . . . . . 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 450 . . . . . 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 12626 . . . 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 14572 . . 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 10232 . 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 10232 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cun 3605  cin 3606  wss 3607  c0 3948   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  8c8 11114  0cn0 11330  cz 11415  cdc 11531  cuz 11725  +crp 11870  (,)cioo 12213  [,)cico 12215  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  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-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  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-vma 24869  df-chp 24870
This theorem is referenced by:  pntlemo  25341
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