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Theorem pntlemf 25011
Description: Lemma for pnt 25020. Add up the pieces in pntlemi 25010 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 25001 . . . . . 6 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
1211simp3d 1067 . . . . 5 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
1312simp3d 1067 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℝ+)
141, 2, 3, 4, 5, 6pntlemd 25000 . . . . . . . 8 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
1514simp1d 1065 . . . . . . 7 (𝜑𝐿 ∈ ℝ+)
1611simp1d 1065 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
17 2z 11242 . . . . . . . 8 2 ∈ ℤ
18 rpexpcl 12696 . . . . . . . 8 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
1916, 17, 18sylancl 692 . . . . . . 7 (𝜑 → (𝐸↑2) ∈ ℝ+)
2015, 19rpmulcld 11720 . . . . . 6 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
21 3nn0 11157 . . . . . . . . 9 3 ∈ ℕ0
22 2nn 11032 . . . . . . . . 9 2 ∈ ℕ
2321, 22decnncl 11350 . . . . . . . 8 32 ∈ ℕ
24 nnrp 11674 . . . . . . . 8 (32 ∈ ℕ → 32 ∈ ℝ+)
2523, 24ax-mp 5 . . . . . . 7 32 ∈ ℝ+
26 rpmulcl 11687 . . . . . . 7 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
2725, 3, 26sylancr 693 . . . . . 6 (𝜑 → (32 · 𝐵) ∈ ℝ+)
2820, 27rpdivcld 11721 . . . . 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 25003 . . . . . . . . 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 1065 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
3635rpred 11704 . . . . . . 7 (𝜑𝑍 ∈ ℝ)
3734simp2d 1066 . . . . . . . 8 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
3837simp1d 1065 . . . . . . 7 (𝜑 → 1 < 𝑍)
3936, 38rplogcld 24096 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ+)
40 rpexpcl 12696 . . . . . 6 (((log‘𝑍) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ+)
4139, 17, 40sylancl 692 . . . . 5 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ+)
4228, 41rpmulcld 11720 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ+)
4313, 42rpmulcld 11720 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ+)
4443rpred 11704 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
4515, 16rpmulcld 11720 . . . . . . 7 (𝜑 → (𝐿 · 𝐸) ∈ ℝ+)
46 8re 10952 . . . . . . . 8 8 ∈ ℝ
47 8pos 10968 . . . . . . . 8 0 < 8
4846, 47elrpii 11667 . . . . . . 7 8 ∈ ℝ+
49 rpdivcl 11688 . . . . . . 7 (((𝐿 · 𝐸) ∈ ℝ+ ∧ 8 ∈ ℝ+) → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5045, 48, 49sylancl 692 . . . . . 6 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5150, 39rpmulcld 11720 . . . . 5 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ+)
5213, 51rpmulcld 11720 . . . 4 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
5352rpred 11704 . . 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 25004 . . . . . . 7 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
5756simp1d 1065 . . . . . 6 (𝜑𝑀 ∈ ℕ)
5856simp2d 1066 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
59 eluznn 11590 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑁 ∈ ℕ)
6057, 58, 59syl2anc 690 . . . . 5 (𝜑𝑁 ∈ ℕ)
6160nnred 10882 . . . 4 (𝜑𝑁 ∈ ℝ)
6257nnred 10882 . . . 4 (𝜑𝑀 ∈ ℝ)
6361, 62resubcld 10309 . . 3 (𝜑 → (𝑁𝑀) ∈ ℝ)
6453, 63remulcld 9926 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ∈ ℝ)
65 fzfid 12589 . . 3 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
667rpred 11704 . . . . . 6 (𝜑𝑈 ∈ ℝ)
67 elfznn 12196 . . . . . 6 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68 nndivre 10903 . . . . . 6 ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
6966, 67, 68syl2an 492 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
7035adantr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
7167adantl 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
7271nnrpd 11702 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
7370, 72rpdivcld 11721 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
741pntrf 24969 . . . . . . . . . 10 𝑅:ℝ+⟶ℝ
7574ffvelrni 6251 . . . . . . . . 9 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7673, 75syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7776, 70rerpdivcld 11735 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
7877recnd 9924 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
7978abscld 13969 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
8069, 79resubcld 10309 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
8172relogcld 24090 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
8280, 81remulcld 9926 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8365, 82fsumrecl 14258 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8445rpcnd 11706 . . . . . . . . 9 (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
8511simp2d 1066 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ+)
8685rpred 11704 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℝ)
8712simp2d 1066 . . . . . . . . . . . 12 (𝜑 → 1 < 𝐾)
8886, 87rplogcld 24096 . . . . . . . . . . 11 (𝜑 → (log‘𝐾) ∈ ℝ+)
8939, 88rpdivcld 11721 . . . . . . . . . 10 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
9089rpcnd 11706 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91 rpcnne0 11682 . . . . . . . . . 10 (8 ∈ ℝ+ → (8 ∈ ℂ ∧ 8 ≠ 0))
9248, 91mp1i 13 . . . . . . . . 9 (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93 4re 10944 . . . . . . . . . . 11 4 ∈ ℝ
94 4pos 10963 . . . . . . . . . . 11 0 < 4
9593, 94elrpii 11667 . . . . . . . . . 10 4 ∈ ℝ+
96 rpcnne0 11682 . . . . . . . . . 10 (4 ∈ ℝ+ → (4 ∈ ℂ ∧ 4 ≠ 0))
9795, 96mp1i 13 . . . . . . . . 9 (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98 divmuldiv 10574 . . . . . . . . 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 1318 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
10010fveq2i 6091 . . . . . . . . . . . . . 14 (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
1013, 16rpdivcld 11721 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 / 𝐸) ∈ ℝ+)
102101rpred 11704 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103102relogefd 24095 . . . . . . . . . . . . . 14 (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104100, 103syl5eq 2655 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105104oveq2d 6543 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
10639rpcnd 11706 . . . . . . . . . . . . 13 (𝜑 → (log‘𝑍) ∈ ℂ)
1073rpcnne0d 11713 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
10816rpcnne0d 11713 . . . . . . . . . . . . 13 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109 divdiv2 10586 . . . . . . . . . . . . 13 (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110106, 107, 108, 109syl3anc 1317 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111105, 110eqtrd 2643 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112111oveq2d 6543 . . . . . . . . . 10 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11316rpcnd 11706 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℂ)
114106, 113mulcld 9916 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115 divass 10552 . . . . . . . . . . 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 1317 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11715rpcnd 11706 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ℂ)
118117, 113, 106, 113mul4d 10099 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119113sqvald 12822 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120119oveq2d 6543 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121113sqcld 12823 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) ∈ ℂ)
122117, 106, 121mul32d 10097 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123118, 120, 1223eqtr2d 2649 . . . . . . . . . . 11 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124123oveq1d 6542 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125112, 116, 1243eqtr2d 2649 . . . . . . . . 9 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126 8t4e32 11488 . . . . . . . . . 10 (8 · 4) = 32
127126a1i 11 . . . . . . . . 9 (𝜑 → (8 · 4) = 32)
128125, 127oveq12d 6545 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32))
12920rpcnd 11706 . . . . . . . . . . 11 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130129, 106mulcld 9916 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131 rpcnne0 11682 . . . . . . . . . . 11 (32 ∈ ℝ+ → (32 ∈ ℂ ∧ 32 ≠ 0))
13225, 131mp1i 13 . . . . . . . . . 10 (𝜑 → (32 ∈ ℂ ∧ 32 ≠ 0))
133 divdiv1 10585 . . . . . . . . . 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 1317 . . . . . . . . 9 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
13523nncni 10877 . . . . . . . . . . 11 32 ∈ ℂ
1363rpcnd 11706 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℂ)
137 mulcom 9878 . . . . . . . . . . 11 ((32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (32 · 𝐵) = (𝐵 · 32))
138135, 136, 137sylancr 693 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) = (𝐵 · 32))
139138oveq2d 6543 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
14027rpcnne0d 11713 . . . . . . . . . 10 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
141 div23 10553 . . . . . . . . . 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 1317 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
143134, 139, 1423eqtr2d 2649 . . . . . . . 8 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
14499, 128, 1433eqtrd 2647 . . . . . . 7 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
145144oveq1d 6542 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
14650rpcnd 11706 . . . . . . 7 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
14789rpred 11704 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148 4nn 11034 . . . . . . . . 9 4 ∈ ℕ
149 nndivre 10903 . . . . . . . . 9 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150147, 148, 149sylancl 692 . . . . . . . 8 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151150recnd 9924 . . . . . . 7 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152146, 106, 151mul32d 10097 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153106sqvald 12822 . . . . . . . 8 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154153oveq2d 6543 . . . . . . 7 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
15528rpcnd 11706 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
156155, 106, 106mulassd 9919 . . . . . . 7 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157154, 156eqtr4d 2646 . . . . . 6 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158145, 152, 1573eqtr4d 2653 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))
15956simp3d 1067 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
160150, 63, 51lemul2d 11748 . . . . . 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 220 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
162158, 161eqbrtrrd 4601 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
16342rpred 11704 . . . . 5 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
16451rpred 11704 . . . . . 6 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165164, 63remulcld 9926 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ∈ ℝ)
166163, 165, 13lemul2d 11748 . . . 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 220 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
16813rpcnd 11706 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℂ)
16951rpcnd 11706 . . . 4 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
17063recnd 9924 . . . 4 (𝜑 → (𝑁𝑀) ∈ ℂ)
171168, 169, 170mulassd 9919 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) = ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
172167, 171breqtrrd 4605 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
173 fzfid 12589 . . . 4 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
17460nnzd 11313 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
17585, 174rpexpcld 12849 . . . . . . . . . . 11 (𝜑 → (𝐾𝑁) ∈ ℝ+)
17635, 175rpdivcld 11721 . . . . . . . . . 10 (𝜑 → (𝑍 / (𝐾𝑁)) ∈ ℝ+)
177176rprege0d 11711 . . . . . . . . 9 (𝜑 → ((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))))
178 flge0nn0 12438 . . . . . . . . 9 (((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))) → (⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0)
179 nn0p1nn 11179 . . . . . . . . 9 ((⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
180177, 178, 1793syl 18 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
181 nnuz 11555 . . . . . . . 8 ℕ = (ℤ‘1)
182180, 181syl6eleq 2697 . . . . . . 7 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1))
183 fzss1 12206 . . . . . . 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 3567 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186185, 82syldan 485 . . . 4 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187173, 186fsumrecl 14258 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188 eluzfz2 12175 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
18958, 188syl 17 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
190 oveq1 6534 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚𝑀) = (𝑀𝑀))
191190oveq2d 6543 . . . . . . 7 (𝑚 = 𝑀 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)))
192 oveq2 6535 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝐾𝑚) = (𝐾𝑀))
193192oveq2d 6543 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑀)))
194193fveq2d 6092 . . . . . . . . . 10 (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑀))))
195194oveq1d 6542 . . . . . . . . 9 (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑀))) + 1))
196195oveq1d 6542 . . . . . . . 8 (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197196sumeq1d 14225 . . . . . . 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 4590 . . . . . 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 328 . . . . 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 6534 . . . . . . . 8 (𝑚 = 𝑗 → (𝑚𝑀) = (𝑗𝑀))
201200oveq2d 6543 . . . . . . 7 (𝑚 = 𝑗 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)))
202 oveq2 6535 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐾𝑚) = (𝐾𝑗))
203202oveq2d 6543 . . . . . . . . . . 11 (𝑚 = 𝑗 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑗)))
204203fveq2d 6092 . . . . . . . . . 10 (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑗))))
205204oveq1d 6542 . . . . . . . . 9 (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
206205oveq1d 6542 . . . . . . . 8 (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207206sumeq1d 14225 . . . . . . 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 4590 . . . . . 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 328 . . . . 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 6534 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (𝑚𝑀) = ((𝑗 + 1) − 𝑀))
211210oveq2d 6543 . . . . . . 7 (𝑚 = (𝑗 + 1) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212 oveq2 6535 . . . . . . . . . . . 12 (𝑚 = (𝑗 + 1) → (𝐾𝑚) = (𝐾↑(𝑗 + 1)))
213212oveq2d 6543 . . . . . . . . . . 11 (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214213fveq2d 6092 . . . . . . . . . 10 (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215214oveq1d 6542 . . . . . . . . 9 (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216215oveq1d 6542 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217216sumeq1d 14225 . . . . . . 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 4590 . . . . . 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 328 . . . . 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 6534 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚𝑀) = (𝑁𝑀))
221220oveq2d 6543 . . . . . . 7 (𝑚 = 𝑁 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
222 oveq2 6535 . . . . . . . . . . . 12 (𝑚 = 𝑁 → (𝐾𝑚) = (𝐾𝑁))
223222oveq2d 6543 . . . . . . . . . . 11 (𝑚 = 𝑁 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑁)))
224223fveq2d 6092 . . . . . . . . . 10 (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑁))))
225224oveq1d 6542 . . . . . . . . 9 (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑁))) + 1))
226225oveq1d 6542 . . . . . . . 8 (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227226sumeq1d 14225 . . . . . . 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 4590 . . . . . 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 328 . . . . 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 10883 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
231230subidd 10231 . . . . . . . . 9 (𝜑 → (𝑀𝑀) = 0)
232231oveq2d 6543 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
23352rpcnd 11706 . . . . . . . . 9 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234233mul01d 10086 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235232, 234eqtrd 2643 . . . . . . 7 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = 0)
236 fzfid 12589 . . . . . . . 8 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
23757nnzd 11313 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
23885, 237rpexpcld 12849 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾𝑀) ∈ ℝ+)
23935, 238rpdivcld 11721 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / (𝐾𝑀)) ∈ ℝ+)
240239rprege0d 11711 . . . . . . . . . . . . 13 (𝜑 → ((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))))
241 flge0nn0 12438 . . . . . . . . . . . . 13 (((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))) → (⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0)
242 nn0p1nn 11179 . . . . . . . . . . . . 13 ((⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
243240, 241, 2423syl 18 . . . . . . . . . . . 12 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
244243, 181syl6eleq 2697 . . . . . . . . . . 11 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1))
245 fzss1 12206 . . . . . . . . . . 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 3567 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248247, 82syldan 485 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249 elfzle2 12171 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250249adantl 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
25129simpld 473 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ ℝ+)
25235, 251rpdivcld 11721 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / 𝑌) ∈ ℝ+)
253252rpred 11704 . . . . . . . . . . . . 13 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254 elfzelz 12168 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255 flge 12423 . . . . . . . . . . . . 13 (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256253, 254, 255syl2an 492 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257250, 256mpbird 245 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
25871, 257jca 552 . . . . . . . . . 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 25006 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261258, 260syldan 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262247, 261syldan 485 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263236, 248, 262fsumge0 14314 . . . . . . 7 (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264235, 263eqbrtrd 4599 . . . . . 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 2609 . . . . . . . . . 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 25010 . . . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
270269rpred 11704 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271 elfzoelz 12294 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272271adantl 480 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273272zred 11314 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
27457adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275274nnred 10882 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276273, 275resubcld 10309 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℝ)
277270, 276remulcld 9926 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ)
278 fzfid 12589 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∈ Fin)
279 ssun1 3737 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
28036adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
28185adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ+)
282272peano2zd 11317 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283281, 282rpexpcld 12849 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ+)
284280, 283rerpdivcld 11735 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285281, 272rpexpcld 12849 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ+)
286280, 285rerpdivcld 11735 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ∈ ℝ)
28786adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288 1re 9895 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
289 ltle 9977 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290288, 86, 289sylancr 693 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
29187, 290mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝐾)
292291adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293 uzid 11534 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
294 peano2uz 11573 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
295272, 293, 2943syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑗))
296287, 292, 295leexp2ad 12858 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)))
29735adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ+)
298285, 283, 297lediv2d 11728 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))))
299296, 298mpbid 220 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗)))
300 flword2 12431 . . . . . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302 eluzp1p1 11545 . . . . . . . . . . . . . . . . 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 12416 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ)
305252adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ+)
306305rpred 11704 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307306flcld 12416 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308251adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ+)
309308rpred 11704 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310285rpred 11704 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ)
31130simpld 473 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ∈ ℝ+)
312311rpred 11704 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 ∈ ℝ)
313312adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
31430simprd 477 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 < 𝑋)
315314adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316 elfzofz 12309 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
3171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemh 25005 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
318316, 317sylan2 489 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
319318simpld 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾𝑗))
320309, 313, 310, 315, 319lttrd 10049 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾𝑗))
321309, 310, 320ltled 10036 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾𝑗))
322308, 285, 297lediv2d 11728 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾𝑗) ↔ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)))
323321, 322mpbid 220 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌))
324 flwordi 12430 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325286, 306, 323, 324syl3anc 1317 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326 eluz2 11525 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327304, 307, 325, 326syl3anbrc 1238 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))))
328 fzsplit2 12192 . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 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 3616 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331297, 283rpdivcld 11721 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ+)
332331rprege0d 11711 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333 flge0nn0 12438 . . . . . . . . . . . . . . . . 17 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0)
334 nn0p1nn 11179 . . . . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1))
337 fzss1 12206 . . . . . . . . . . . . . . 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 3577 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340339sselda 3567 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
34182adantlr 746 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342340, 341syldan 485 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
343278, 342fsumrecl 14258 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344 fzfid 12589 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345 ssun2 3738 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346345, 329syl5sseqr 3616 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347346, 338sstrd 3577 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348347sselda 3567 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349348, 341syldan 485 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
350344, 349fsumrecl 14258 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351 le2add 10359 . . . . . . . . . 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 1318 . . . . . . . . 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 706 . . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355 1cnd 9912 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356272zcnd 11315 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357230adantr 479 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358356, 357subcld 10243 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℂ)
359354, 355, 358adddid 9920 . . . . . . . . . 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 10089 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗𝑀) + 1))
361356, 355, 357addsubd 10264 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗𝑀) + 1))
362360, 361eqtr4d 2646 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗 + 1) − 𝑀))
363362oveq2d 6543 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364354mulid1d 9913 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365364oveq1d 6542 . . . . . . . . . 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 2651 . . . . . . . . 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 12414 . . . . . . . . . . . . 13 ((𝑍 / (𝐾𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
368286, 367syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
369368ltp1d 10803 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
370 fzdisj 12194 . . . . . . . . . . 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 12589 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373338sselda 3567 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374373, 341syldan 485 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
375374recnd 9924 . . . . . . . . . 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 14264 . . . . . . . . 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 4590 . . . . . . . 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 247 . . . . . . 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 449 . . . . . 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 12403 . . . 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 37 . . 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 14315 . . 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 10045 . 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 10045 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cun 3537  cin 3538  wss 3539  c0 3873   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  +∞cpnf 9927   < clt 9930  cle 9931  cmin 10117   / cdiv 10533  cn 10867  2c2 10917  3c3 10918  4c4 10919  8c8 10923  0cn0 11139  cz 11210  cdc 11325  cuz 11519  +crp 11664  (,)cioo 12002  [,)cico 12004  [,]cicc 12005  ...cfz 12152  ..^cfzo 12289  cfl 12408  cexp 12677  csqrt 13767  abscabs 13768  Σcsu 14210  expce 14577  eceu 14578  logclog 24022  ψcchp 24536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ioc 12007  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-seq 12619  df-exp 12678  df-fac 12878  df-bc 12907  df-hash 12935  df-shft 13601  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-limsup 13996  df-clim 14013  df-rlim 14014  df-sum 14211  df-ef 14583  df-e 14584  df-sin 14585  df-cos 14586  df-pi 14588  df-dvds 14768  df-gcd 15001  df-prm 15170  df-pc 15326  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-sca 15730  df-vsca 15731  df-ip 15732  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-hom 15739  df-cco 15740  df-rest 15852  df-topn 15853  df-0g 15871  df-gsum 15872  df-topgen 15873  df-pt 15874  df-prds 15877  df-xrs 15931  df-qtop 15936  df-imas 15937  df-xps 15939  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-mulg 17310  df-cntz 17519  df-cmn 17964  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-cnfld 19514  df-top 20463  df-bases 20464  df-topon 20465  df-topsp 20466  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lp 20692  df-perf 20693  df-cn 20783  df-cnp 20784  df-haus 20871  df-tx 21117  df-hmeo 21310  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-xms 21876  df-ms 21877  df-tms 21878  df-cncf 22420  df-limc 23353  df-dv 23354  df-log 24024  df-vma 24541  df-chp 24542
This theorem is referenced by:  pntlemo  25013
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