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Theorem pntlemf 26097
Description: Lemma for pnt 26106. Add up the pieces in pntlemi 26096 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 26087 . . . . . 6 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
1211simp3d 1138 . . . . 5 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
1312simp3d 1138 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℝ+)
141, 2, 3, 4, 5, 6pntlemd 26086 . . . . . . . 8 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
1514simp1d 1136 . . . . . . 7 (𝜑𝐿 ∈ ℝ+)
1611simp1d 1136 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
17 2z 12006 . . . . . . . 8 2 ∈ ℤ
18 rpexpcl 13441 . . . . . . . 8 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
1916, 17, 18sylancl 586 . . . . . . 7 (𝜑 → (𝐸↑2) ∈ ℝ+)
2015, 19rpmulcld 12440 . . . . . 6 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
21 3nn0 11907 . . . . . . . . 9 3 ∈ ℕ0
22 2nn 11702 . . . . . . . . 9 2 ∈ ℕ
2321, 22decnncl 12110 . . . . . . . 8 32 ∈ ℕ
24 nnrp 12393 . . . . . . . 8 (32 ∈ ℕ → 32 ∈ ℝ+)
2523, 24ax-mp 5 . . . . . . 7 32 ∈ ℝ+
26 rpmulcl 12405 . . . . . . 7 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
2725, 3, 26sylancr 587 . . . . . 6 (𝜑 → (32 · 𝐵) ∈ ℝ+)
2820, 27rpdivcld 12441 . . . . 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 26089 . . . . . . . . 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 1136 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
3635rpred 12424 . . . . . . 7 (𝜑𝑍 ∈ ℝ)
3734simp2d 1137 . . . . . . . 8 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
3837simp1d 1136 . . . . . . 7 (𝜑 → 1 < 𝑍)
3936, 38rplogcld 25127 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ+)
40 rpexpcl 13441 . . . . . 6 (((log‘𝑍) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ+)
4139, 17, 40sylancl 586 . . . . 5 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ+)
4228, 41rpmulcld 12440 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ+)
4313, 42rpmulcld 12440 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ+)
4443rpred 12424 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
4515, 16rpmulcld 12440 . . . . . . 7 (𝜑 → (𝐿 · 𝐸) ∈ ℝ+)
46 8re 11725 . . . . . . . 8 8 ∈ ℝ
47 8pos 11741 . . . . . . . 8 0 < 8
4846, 47elrpii 12385 . . . . . . 7 8 ∈ ℝ+
49 rpdivcl 12407 . . . . . . 7 (((𝐿 · 𝐸) ∈ ℝ+ ∧ 8 ∈ ℝ+) → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5045, 48, 49sylancl 586 . . . . . 6 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5150, 39rpmulcld 12440 . . . . 5 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ+)
5213, 51rpmulcld 12440 . . . 4 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
5352rpred 12424 . . 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 26090 . . . . . . 7 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
5756simp1d 1136 . . . . . 6 (𝜑𝑀 ∈ ℕ)
5856simp2d 1137 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
59 eluznn 12310 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑁 ∈ ℕ)
6057, 58, 59syl2anc 584 . . . . 5 (𝜑𝑁 ∈ ℕ)
6160nnred 11645 . . . 4 (𝜑𝑁 ∈ ℝ)
6257nnred 11645 . . . 4 (𝜑𝑀 ∈ ℝ)
6361, 62resubcld 11060 . . 3 (𝜑 → (𝑁𝑀) ∈ ℝ)
6453, 63remulcld 10663 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ∈ ℝ)
65 fzfid 13334 . . 3 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
667rpred 12424 . . . . . 6 (𝜑𝑈 ∈ ℝ)
67 elfznn 12929 . . . . . 6 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68 nndivre 11670 . . . . . 6 ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
6966, 67, 68syl2an 595 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
7035adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
7167adantl 482 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
7271nnrpd 12422 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
7370, 72rpdivcld 12441 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
741pntrf 26055 . . . . . . . . . 10 𝑅:ℝ+⟶ℝ
7574ffvelrni 6845 . . . . . . . . 9 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7673, 75syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7776, 70rerpdivcld 12455 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
7877recnd 10661 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
7978abscld 14789 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
8069, 79resubcld 11060 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
8172relogcld 25121 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
8280, 81remulcld 10663 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8365, 82fsumrecl 15083 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8445rpcnd 12426 . . . . . . . . 9 (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
8511simp2d 1137 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ+)
8685rpred 12424 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℝ)
8712simp2d 1137 . . . . . . . . . . . 12 (𝜑 → 1 < 𝐾)
8886, 87rplogcld 25127 . . . . . . . . . . 11 (𝜑 → (log‘𝐾) ∈ ℝ+)
8939, 88rpdivcld 12441 . . . . . . . . . 10 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
9089rpcnd 12426 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91 rpcnne0 12400 . . . . . . . . . 10 (8 ∈ ℝ+ → (8 ∈ ℂ ∧ 8 ≠ 0))
9248, 91mp1i 13 . . . . . . . . 9 (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93 4re 11713 . . . . . . . . . . 11 4 ∈ ℝ
94 4pos 11736 . . . . . . . . . . 11 0 < 4
9593, 94elrpii 12385 . . . . . . . . . 10 4 ∈ ℝ+
96 rpcnne0 12400 . . . . . . . . . 10 (4 ∈ ℝ+ → (4 ∈ ℂ ∧ 4 ≠ 0))
9795, 96mp1i 13 . . . . . . . . 9 (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98 divmuldiv 11332 . . . . . . . . 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 836 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
10010fveq2i 6669 . . . . . . . . . . . . . 14 (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
1013, 16rpdivcld 12441 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 / 𝐸) ∈ ℝ+)
102101rpred 12424 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103102relogefd 25126 . . . . . . . . . . . . . 14 (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104100, 103syl5eq 2872 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105104oveq2d 7167 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
10639rpcnd 12426 . . . . . . . . . . . . 13 (𝜑 → (log‘𝑍) ∈ ℂ)
1073rpcnne0d 12433 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
10816rpcnne0d 12433 . . . . . . . . . . . . 13 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109 divdiv2 11344 . . . . . . . . . . . . 13 (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110106, 107, 108, 109syl3anc 1365 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111105, 110eqtrd 2860 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112111oveq2d 7167 . . . . . . . . . 10 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11316rpcnd 12426 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℂ)
114106, 113mulcld 10653 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115 divass 11308 . . . . . . . . . . 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 1365 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11715rpcnd 12426 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ℂ)
118117, 113, 106, 113mul4d 10844 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119113sqvald 13500 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120119oveq2d 7167 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121113sqcld 13501 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) ∈ ℂ)
122117, 106, 121mul32d 10842 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123118, 120, 1223eqtr2d 2866 . . . . . . . . . . 11 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124123oveq1d 7166 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125112, 116, 1243eqtr2d 2866 . . . . . . . . 9 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126 8t4e32 12207 . . . . . . . . . 10 (8 · 4) = 32
127126a1i 11 . . . . . . . . 9 (𝜑 → (8 · 4) = 32)
128125, 127oveq12d 7169 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32))
12920rpcnd 12426 . . . . . . . . . . 11 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130129, 106mulcld 10653 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131 rpcnne0 12400 . . . . . . . . . . 11 (32 ∈ ℝ+ → (32 ∈ ℂ ∧ 32 ≠ 0))
13225, 131mp1i 13 . . . . . . . . . 10 (𝜑 → (32 ∈ ℂ ∧ 32 ≠ 0))
133 divdiv1 11343 . . . . . . . . . 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 1365 . . . . . . . . 9 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
13523nncni 11640 . . . . . . . . . . 11 32 ∈ ℂ
1363rpcnd 12426 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℂ)
137 mulcom 10615 . . . . . . . . . . 11 ((32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (32 · 𝐵) = (𝐵 · 32))
138135, 136, 137sylancr 587 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) = (𝐵 · 32))
139138oveq2d 7167 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
14027rpcnne0d 12433 . . . . . . . . . 10 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
141 div23 11309 . . . . . . . . . 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 1365 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
143134, 139, 1423eqtr2d 2866 . . . . . . . 8 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
14499, 128, 1433eqtrd 2864 . . . . . . 7 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
145144oveq1d 7166 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
14650rpcnd 12426 . . . . . . 7 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
14789rpred 12424 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148 4nn 11712 . . . . . . . . 9 4 ∈ ℕ
149 nndivre 11670 . . . . . . . . 9 ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150147, 148, 149sylancl 586 . . . . . . . 8 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151150recnd 10661 . . . . . . 7 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152146, 106, 151mul32d 10842 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153106sqvald 13500 . . . . . . . 8 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154153oveq2d 7167 . . . . . . 7 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
15528rpcnd 12426 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
156155, 106, 106mulassd 10656 . . . . . . 7 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157154, 156eqtr4d 2863 . . . . . 6 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158145, 152, 1573eqtr4d 2870 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))
15956simp3d 1138 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
160150, 63, 51lemul2d 12468 . . . . . 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 233 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
162158, 161eqbrtrrd 5086 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
16342rpred 12424 . . . . 5 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
16451rpred 12424 . . . . . 6 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165164, 63remulcld 10663 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ∈ ℝ)
166163, 165, 13lemul2d 12468 . . . 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 233 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
16813rpcnd 12426 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℂ)
16951rpcnd 12426 . . . 4 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
17063recnd 10661 . . . 4 (𝜑 → (𝑁𝑀) ∈ ℂ)
171168, 169, 170mulassd 10656 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) = ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
172167, 171breqtrrd 5090 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
173 fzfid 13334 . . . 4 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
17460nnzd 12078 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
17585, 174rpexpcld 13601 . . . . . . . . . . 11 (𝜑 → (𝐾𝑁) ∈ ℝ+)
17635, 175rpdivcld 12441 . . . . . . . . . 10 (𝜑 → (𝑍 / (𝐾𝑁)) ∈ ℝ+)
177176rprege0d 12431 . . . . . . . . 9 (𝜑 → ((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))))
178 flge0nn0 13183 . . . . . . . . 9 (((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))) → (⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0)
179 nn0p1nn 11928 . . . . . . . . 9 ((⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
180177, 178, 1793syl 18 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
181 nnuz 12273 . . . . . . . 8 ℕ = (ℤ‘1)
182180, 181syl6eleq 2927 . . . . . . 7 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1))
183 fzss1 12939 . . . . . . 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 3970 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186185, 82syldan 591 . . . 4 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187173, 186fsumrecl 15083 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188 eluzfz2 12908 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
18958, 188syl 17 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
190 oveq1 7158 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚𝑀) = (𝑀𝑀))
191190oveq2d 7167 . . . . . . 7 (𝑚 = 𝑀 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)))
192 oveq2 7159 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝐾𝑚) = (𝐾𝑀))
193192oveq2d 7167 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑀)))
194193fveq2d 6670 . . . . . . . . . 10 (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑀))))
195194oveq1d 7166 . . . . . . . . 9 (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑀))) + 1))
196195oveq1d 7166 . . . . . . . 8 (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197196sumeq1d 15050 . . . . . . 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 5075 . . . . . 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 342 . . . . 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 7158 . . . . . . . 8 (𝑚 = 𝑗 → (𝑚𝑀) = (𝑗𝑀))
201200oveq2d 7167 . . . . . . 7 (𝑚 = 𝑗 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)))
202 oveq2 7159 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐾𝑚) = (𝐾𝑗))
203202oveq2d 7167 . . . . . . . . . . 11 (𝑚 = 𝑗 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑗)))
204203fveq2d 6670 . . . . . . . . . 10 (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑗))))
205204oveq1d 7166 . . . . . . . . 9 (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
206205oveq1d 7166 . . . . . . . 8 (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207206sumeq1d 15050 . . . . . . 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 5075 . . . . . 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 342 . . . . 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 7158 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (𝑚𝑀) = ((𝑗 + 1) − 𝑀))
211210oveq2d 7167 . . . . . . 7 (𝑚 = (𝑗 + 1) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212 oveq2 7159 . . . . . . . . . . . 12 (𝑚 = (𝑗 + 1) → (𝐾𝑚) = (𝐾↑(𝑗 + 1)))
213212oveq2d 7167 . . . . . . . . . . 11 (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214213fveq2d 6670 . . . . . . . . . 10 (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215214oveq1d 7166 . . . . . . . . 9 (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216215oveq1d 7166 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217216sumeq1d 15050 . . . . . . 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 5075 . . . . . 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 342 . . . . 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 7158 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚𝑀) = (𝑁𝑀))
221220oveq2d 7167 . . . . . . 7 (𝑚 = 𝑁 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
222 oveq2 7159 . . . . . . . . . . . 12 (𝑚 = 𝑁 → (𝐾𝑚) = (𝐾𝑁))
223222oveq2d 7167 . . . . . . . . . . 11 (𝑚 = 𝑁 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑁)))
224223fveq2d 6670 . . . . . . . . . 10 (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑁))))
225224oveq1d 7166 . . . . . . . . 9 (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑁))) + 1))
226225oveq1d 7166 . . . . . . . 8 (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227226sumeq1d 15050 . . . . . . 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 5075 . . . . . 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 342 . . . . 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 11646 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
231230subidd 10977 . . . . . . . . 9 (𝜑 → (𝑀𝑀) = 0)
232231oveq2d 7167 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
23352rpcnd 12426 . . . . . . . . 9 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234233mul01d 10831 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235232, 234eqtrd 2860 . . . . . . 7 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = 0)
236 fzfid 13334 . . . . . . . 8 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
23757nnzd 12078 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
23885, 237rpexpcld 13601 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾𝑀) ∈ ℝ+)
23935, 238rpdivcld 12441 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / (𝐾𝑀)) ∈ ℝ+)
240239rprege0d 12431 . . . . . . . . . . . . 13 (𝜑 → ((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))))
241 flge0nn0 13183 . . . . . . . . . . . . 13 (((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))) → (⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0)
242 nn0p1nn 11928 . . . . . . . . . . . . 13 ((⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
243240, 241, 2423syl 18 . . . . . . . . . . . 12 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
244243, 181syl6eleq 2927 . . . . . . . . . . 11 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1))
245 fzss1 12939 . . . . . . . . . . 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 3970 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248247, 82syldan 591 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249 elfzle2 12904 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250249adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
25129simpld 495 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ ℝ+)
25235, 251rpdivcld 12441 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / 𝑌) ∈ ℝ+)
253252rpred 12424 . . . . . . . . . . . . 13 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254 elfzelz 12901 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255 flge 13168 . . . . . . . . . . . . 13 (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256253, 254, 255syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257250, 256mpbird 258 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
25871, 257jca 512 . . . . . . . . . 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 26092 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261258, 260syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262247, 261syldan 591 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263236, 248, 262fsumge0 15142 . . . . . . 7 (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264235, 263eqbrtrd 5084 . . . . . 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 2825 . . . . . . . . . 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 26096 . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
270269rpred 12424 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271 elfzoelz 13031 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272271adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273272zred 12079 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
27457adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275274nnred 11645 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276273, 275resubcld 11060 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℝ)
277270, 276remulcld 10663 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ)
278 fzfid 13334 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∈ Fin)
279 ssun1 4151 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
28036adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
28185adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ+)
282272peano2zd 12082 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283281, 282rpexpcld 13601 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ+)
284280, 283rerpdivcld 12455 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285281, 272rpexpcld 13601 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ+)
286280, 285rerpdivcld 12455 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ∈ ℝ)
28786adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288 1re 10633 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
289 ltle 10721 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290288, 86, 289sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
29187, 290mpd 15 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ≤ 𝐾)
292291adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293 uzid 12250 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
294 peano2uz 12293 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
295272, 293, 2943syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑗))
296287, 292, 295leexp2ad 13610 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)))
29735adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ+)
298285, 283, 297lediv2d 12448 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))))
299296, 298mpbid 233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗)))
300 flword2 13176 . . . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302 eluzp1p1 12262 . . . . . . . . . . . . . . . . 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 13161 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ)
305252adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ+)
306305rpred 12424 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307306flcld 13161 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308251adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ+)
309308rpred 12424 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310285rpred 12424 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ)
31130simpld 495 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ∈ ℝ+)
312311rpred 12424 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 ∈ ℝ)
313312adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
31430simprd 496 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑌 < 𝑋)
315314adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316 elfzofz 13046 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
3171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemh 26091 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
318316, 317sylan2 592 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
319318simpld 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾𝑗))
320309, 313, 310, 315, 319lttrd 10793 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾𝑗))
321309, 310, 320ltled 10780 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾𝑗))
322308, 285, 297lediv2d 12448 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾𝑗) ↔ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)))
323321, 322mpbid 233 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌))
324 flwordi 13175 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325286, 306, 323, 324syl3anc 1365 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326 eluz2 12241 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327304, 307, 325, 326syl3anbrc 1337 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))))
328 fzsplit2 12925 . . . . . . . . . . . . . . . 16 ((((⌊‘(𝑍 / (𝐾𝑗))) + 1) ∈ (ℤ‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)) ∧ (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗))))) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
329303, 327, 328syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
330279, 329sseqtrrid 4023 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331297, 283rpdivcld 12441 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ+)
332331rprege0d 12431 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333 flge0nn0 13183 . . . . . . . . . . . . . . . . 17 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0)
334 nn0p1nn 11928 . . . . . . . . . . . . . . . . 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 2927 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1))
337 fzss1 12939 . . . . . . . . . . . . . . 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 3980 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340339sselda 3970 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
34182adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342340, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
343278, 342fsumrecl 15083 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344 fzfid 13334 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345 ssun2 4152 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346345, 329sseqtrrid 4023 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347346, 338sstrd 3980 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348347sselda 3970 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349348, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
350344, 349fsumrecl 15083 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351 le2add 11114 . . . . . . . . . 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 836 . . . . . . . . 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 691 . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355 1cnd 10628 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356272zcnd 12080 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357230adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358356, 357subcld 10989 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℂ)
359354, 355, 358adddid 10657 . . . . . . . . . 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 10834 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗𝑀) + 1))
361356, 355, 357addsubd 11010 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗𝑀) + 1))
362360, 361eqtr4d 2863 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗 + 1) − 𝑀))
363362oveq2d 7167 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364354mulid1d 10650 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365364oveq1d 7166 . . . . . . . . . 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 2868 . . . . . . . . 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 13159 . . . . . . . . . . . . 13 ((𝑍 / (𝐾𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
368286, 367syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
369368ltp1d 11562 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
370 fzdisj 12927 . . . . . . . . . . 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 13334 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373338sselda 3970 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374373, 341syldan 591 . . . . . . . . . . 11 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
375374recnd 10661 . . . . . . . . . 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 15089 . . . . . . . . 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 5075 . . . . . . . 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 260 . . . . . . 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 414 . . . . . 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 13148 . . . 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 15143 . . 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 10789 . 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 10789 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020  wral 3142  wrex 3143  cun 3937  cin 3938  wss 3939  c0 4294   class class class wbr 5062  cmpt 5142  cfv 6351  (class class class)co 7151  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664   < clt 10667  cle 10668  cmin 10862   / cdiv 11289  cn 11630  2c2 11684  3c3 11685  4c4 11686  8c8 11690  0cn0 11889  cz 11973  cdc 12090  cuz 12235  +crp 12382  (,)cioo 12731  [,)cico 12733  [,]cicc 12734  ...cfz 12885  ..^cfzo 13026  cfl 13153  cexp 13422  csqrt 14585  abscabs 14586  Σcsu 15035  expce 15407  eceu 15408  logclog 25053  ψcchp 25586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-ioo 12735  df-ioc 12736  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-fac 13627  df-bc 13656  df-hash 13684  df-shft 14419  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-limsup 14821  df-clim 14838  df-rlim 14839  df-sum 15036  df-ef 15413  df-e 15414  df-sin 15415  df-cos 15416  df-pi 15418  df-dvds 15600  df-gcd 15836  df-prm 16008  df-pc 16166  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947  df-mulg 18157  df-cntz 18379  df-cmn 18830  df-psmet 20455  df-xmet 20456  df-met 20457  df-bl 20458  df-mopn 20459  df-fbas 20460  df-fg 20461  df-cnfld 20464  df-top 21420  df-topon 21437  df-topsp 21459  df-bases 21472  df-cld 21545  df-ntr 21546  df-cls 21547  df-nei 21624  df-lp 21662  df-perf 21663  df-cn 21753  df-cnp 21754  df-haus 21841  df-tx 22088  df-hmeo 22281  df-fil 22372  df-fm 22464  df-flim 22465  df-flf 22466  df-xms 22847  df-ms 22848  df-tms 22849  df-cncf 23403  df-limc 24381  df-dv 24382  df-log 25055  df-vma 25591  df-chp 25592
This theorem is referenced by:  pntlemo  26099
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