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Theorem pntlemf 26953
Description: Lemma for pnt 26962. Add up the pieces in pntlemi 26952 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 26943 . . . . . 6 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
1211simp3d 1144 . . . . 5 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
1312simp3d 1144 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℝ+)
141, 2, 3, 4, 5, 6pntlemd 26942 . . . . . . . 8 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
1514simp1d 1142 . . . . . . 7 (𝜑𝐿 ∈ ℝ+)
1611simp1d 1142 . . . . . . . 8 (𝜑𝐸 ∈ ℝ+)
17 2z 12535 . . . . . . . 8 2 ∈ ℤ
18 rpexpcl 13986 . . . . . . . 8 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
1916, 17, 18sylancl 586 . . . . . . 7 (𝜑 → (𝐸↑2) ∈ ℝ+)
2015, 19rpmulcld 12973 . . . . . 6 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
21 3nn0 12431 . . . . . . . . 9 3 ∈ ℕ0
22 2nn 12226 . . . . . . . . 9 2 ∈ ℕ
2321, 22decnncl 12638 . . . . . . . 8 32 ∈ ℕ
24 nnrp 12926 . . . . . . . 8 (32 ∈ ℕ → 32 ∈ ℝ+)
2523, 24ax-mp 5 . . . . . . 7 32 ∈ ℝ+
26 rpmulcl 12938 . . . . . . 7 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
2725, 3, 26sylancr 587 . . . . . 6 (𝜑 → (32 · 𝐵) ∈ ℝ+)
2820, 27rpdivcld 12974 . . . . 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 26945 . . . . . . . . 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 1142 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
3635rpred 12957 . . . . . . 7 (𝜑𝑍 ∈ ℝ)
3734simp2d 1143 . . . . . . . 8 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
3837simp1d 1142 . . . . . . 7 (𝜑 → 1 < 𝑍)
3936, 38rplogcld 25984 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℝ+)
40 rpexpcl 13986 . . . . . 6 (((log‘𝑍) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ+)
4139, 17, 40sylancl 586 . . . . 5 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ+)
4228, 41rpmulcld 12973 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ+)
4313, 42rpmulcld 12973 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ+)
4443rpred 12957 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
4515, 16rpmulcld 12973 . . . . . . 7 (𝜑 → (𝐿 · 𝐸) ∈ ℝ+)
46 8re 12249 . . . . . . . 8 8 ∈ ℝ
47 8pos 12265 . . . . . . . 8 0 < 8
4846, 47elrpii 12918 . . . . . . 7 8 ∈ ℝ+
49 rpdivcl 12940 . . . . . . 7 (((𝐿 · 𝐸) ∈ ℝ+ ∧ 8 ∈ ℝ+) → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5045, 48, 49sylancl 586 . . . . . 6 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ+)
5150, 39rpmulcld 12973 . . . . 5 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ+)
5213, 51rpmulcld 12973 . . . 4 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ+)
5352rpred 12957 . . 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 26946 . . . . . . 7 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀)))
5756simp1d 1142 . . . . . 6 (𝜑𝑀 ∈ ℕ)
5856simp2d 1143 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
59 eluznn 12843 . . . . . 6 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ𝑀)) → 𝑁 ∈ ℕ)
6057, 58, 59syl2anc 584 . . . . 5 (𝜑𝑁 ∈ ℕ)
6160nnred 12168 . . . 4 (𝜑𝑁 ∈ ℝ)
6257nnred 12168 . . . 4 (𝜑𝑀 ∈ ℝ)
6361, 62resubcld 11583 . . 3 (𝜑 → (𝑁𝑀) ∈ ℝ)
6453, 63remulcld 11185 . 2 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ∈ ℝ)
65 fzfid 13878 . . 3 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
667rpred 12957 . . . . . 6 (𝜑𝑈 ∈ ℝ)
67 elfznn 13470 . . . . . 6 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68 nndivre 12194 . . . . . 6 ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
6966, 67, 68syl2an 596 . . . . 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 12955 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
7370, 72rpdivcld 12974 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
741pntrf 26911 . . . . . . . . . 10 𝑅:ℝ+⟶ℝ
7574ffvelcdmi 7034 . . . . . . . . 9 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7673, 75syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
7776, 70rerpdivcld 12988 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
7877recnd 11183 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
7978abscld 15321 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
8069, 79resubcld 11583 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
8172relogcld 25978 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
8280, 81remulcld 11185 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8365, 82fsumrecl 15619 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
8445rpcnd 12959 . . . . . . . . 9 (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
8511simp2d 1143 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ+)
8685rpred 12957 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℝ)
8712simp2d 1143 . . . . . . . . . . . 12 (𝜑 → 1 < 𝐾)
8886, 87rplogcld 25984 . . . . . . . . . . 11 (𝜑 → (log‘𝐾) ∈ ℝ+)
8939, 88rpdivcld 12974 . . . . . . . . . 10 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ+)
9089rpcnd 12959 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91 rpcnne0 12933 . . . . . . . . . 10 (8 ∈ ℝ+ → (8 ∈ ℂ ∧ 8 ≠ 0))
9248, 91mp1i 13 . . . . . . . . 9 (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93 4re 12237 . . . . . . . . . . 11 4 ∈ ℝ
94 4pos 12260 . . . . . . . . . . 11 0 < 4
9593, 94elrpii 12918 . . . . . . . . . 10 4 ∈ ℝ+
96 rpcnne0 12933 . . . . . . . . . 10 (4 ∈ ℝ+ → (4 ∈ ℂ ∧ 4 ≠ 0))
9795, 96mp1i 13 . . . . . . . . 9 (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98 divmuldiv 11855 . . . . . . . . 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 837 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
10010fveq2i 6845 . . . . . . . . . . . . . 14 (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
1013, 16rpdivcld 12974 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐵 / 𝐸) ∈ ℝ+)
102101rpred 12957 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103102relogefd 25983 . . . . . . . . . . . . . 14 (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104100, 103eqtrid 2788 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105104oveq2d 7373 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
10639rpcnd 12959 . . . . . . . . . . . . 13 (𝜑 → (log‘𝑍) ∈ ℂ)
1073rpcnne0d 12966 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
10816rpcnne0d 12966 . . . . . . . . . . . . 13 (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109 divdiv2 11867 . . . . . . . . . . . . 13 (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110106, 107, 108, 109syl3anc 1371 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111105, 110eqtrd 2776 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112111oveq2d 7373 . . . . . . . . . 10 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11316rpcnd 12959 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℂ)
114106, 113mulcld 11175 . . . . . . . . . . 11 (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115 divass 11831 . . . . . . . . . . 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 1371 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
11715rpcnd 12959 . . . . . . . . . . . . 13 (𝜑𝐿 ∈ ℂ)
118117, 113, 106, 113mul4d 11367 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119113sqvald 14048 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120119oveq2d 7373 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121113sqcld 14049 . . . . . . . . . . . . 13 (𝜑 → (𝐸↑2) ∈ ℂ)
122117, 106, 121mul32d 11365 . . . . . . . . . . . 12 (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123118, 120, 1223eqtr2d 2782 . . . . . . . . . . 11 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124123oveq1d 7372 . . . . . . . . . 10 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125112, 116, 1243eqtr2d 2782 . . . . . . . . 9 (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126 8t4e32 12735 . . . . . . . . . 10 (8 · 4) = 32
127126a1i 11 . . . . . . . . 9 (𝜑 → (8 · 4) = 32)
128125, 127oveq12d 7375 . . . . . . . 8 (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32))
12920rpcnd 12959 . . . . . . . . . . 11 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130129, 106mulcld 11175 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131 rpcnne0 12933 . . . . . . . . . . 11 (32 ∈ ℝ+ → (32 ∈ ℂ ∧ 32 ≠ 0))
13225, 131mp1i 13 . . . . . . . . . 10 (𝜑 → (32 ∈ ℂ ∧ 32 ≠ 0))
133 divdiv1 11866 . . . . . . . . . 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 1371 . . . . . . . . 9 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
13523nncni 12163 . . . . . . . . . . 11 32 ∈ ℂ
1363rpcnd 12959 . . . . . . . . . . 11 (𝜑𝐵 ∈ ℂ)
137 mulcom 11137 . . . . . . . . . . 11 ((32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (32 · 𝐵) = (𝐵 · 32))
138135, 136, 137sylancr 587 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) = (𝐵 · 32))
139138oveq2d 7373 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · 32)))
14027rpcnne0d 12966 . . . . . . . . . 10 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
141 div23 11832 . . . . . . . . . 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 1371 . . . . . . . . 9 (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
143134, 139, 1423eqtr2d 2782 . . . . . . . 8 (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / 32) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
14499, 128, 1433eqtrd 2780 . . . . . . 7 (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)))
145144oveq1d 7372 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
14650rpcnd 12959 . . . . . . 7 (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
14789rpred 12957 . . . . . . . . 9 (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148 4nn 12236 . . . . . . . . 9 4 ∈ ℕ
149 nndivre 12194 . . . . . . . . 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 11183 . . . . . . 7 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152146, 106, 151mul32d 11365 . . . . . 6 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153106sqvald 14048 . . . . . . . 8 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154153oveq2d 7373 . . . . . . 7 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
15528rpcnd 12959 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
156155, 106, 106mulassd 11178 . . . . . . 7 (𝜑 → ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157154, 156eqtr4d 2779 . . . . . 6 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158145, 152, 1573eqtr4d 2786 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))
15956simp3d 1144 . . . . . 6 (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁𝑀))
160150, 63, 51lemul2d 13001 . . . . . 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 231 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
162158, 161eqbrtrrd 5129 . . . 4 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)))
16342rpred 12957 . . . . 5 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
16451rpred 12957 . . . . . 6 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165164, 63remulcld 11185 . . . . 5 (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀)) ∈ ℝ)
166163, 165, 13lemul2d 13001 . . . 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 231 . . 3 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
16813rpcnd 12959 . . . 4 (𝜑 → (𝑈𝐸) ∈ ℂ)
16951rpcnd 12959 . . . 4 (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
17063recnd 11183 . . . 4 (𝜑 → (𝑁𝑀) ∈ ℂ)
171168, 169, 170mulassd 11178 . . 3 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) = ((𝑈𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁𝑀))))
172167, 171breqtrrd 5133 . 2 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
173 fzfid 13878 . . . 4 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
17460nnzd 12526 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
17585, 174rpexpcld 14150 . . . . . . . . . . 11 (𝜑 → (𝐾𝑁) ∈ ℝ+)
17635, 175rpdivcld 12974 . . . . . . . . . 10 (𝜑 → (𝑍 / (𝐾𝑁)) ∈ ℝ+)
177176rprege0d 12964 . . . . . . . . 9 (𝜑 → ((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))))
178 flge0nn0 13725 . . . . . . . . 9 (((𝑍 / (𝐾𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑁))) → (⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0)
179 nn0p1nn 12452 . . . . . . . . 9 ((⌊‘(𝑍 / (𝐾𝑁))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
180177, 178, 1793syl 18 . . . . . . . 8 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ ℕ)
181 nnuz 12806 . . . . . . . 8 ℕ = (ℤ‘1)
182180, 181eleqtrdi 2848 . . . . . . 7 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑁))) + 1) ∈ (ℤ‘1))
183 fzss1 13480 . . . . . . 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 3944 . . . . 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 15619 . . 3 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188 eluzfz2 13449 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
18958, 188syl 17 . . . 4 (𝜑𝑁 ∈ (𝑀...𝑁))
190 oveq1 7364 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚𝑀) = (𝑀𝑀))
191190oveq2d 7373 . . . . . . 7 (𝑚 = 𝑀 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)))
192 oveq2 7365 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝐾𝑚) = (𝐾𝑀))
193192oveq2d 7373 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑀)))
194193fveq2d 6846 . . . . . . . . . 10 (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑀))))
195194oveq1d 7372 . . . . . . . . 9 (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑀))) + 1))
196195oveq1d 7372 . . . . . . . 8 (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197196sumeq1d 15586 . . . . . . 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 5118 . . . . . 6 (𝑚 = 𝑀 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
199198imbi2d 340 . . . . 5 (𝑚 = 𝑀 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
200 oveq1 7364 . . . . . . . 8 (𝑚 = 𝑗 → (𝑚𝑀) = (𝑗𝑀))
201200oveq2d 7373 . . . . . . 7 (𝑚 = 𝑗 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)))
202 oveq2 7365 . . . . . . . . . . . 12 (𝑚 = 𝑗 → (𝐾𝑚) = (𝐾𝑗))
203202oveq2d 7373 . . . . . . . . . . 11 (𝑚 = 𝑗 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑗)))
204203fveq2d 6846 . . . . . . . . . 10 (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑗))))
205204oveq1d 7372 . . . . . . . . 9 (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
206205oveq1d 7372 . . . . . . . 8 (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207206sumeq1d 15586 . . . . . . 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 5118 . . . . . 6 (𝑚 = 𝑗 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
209208imbi2d 340 . . . . 5 (𝑚 = 𝑗 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
210 oveq1 7364 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (𝑚𝑀) = ((𝑗 + 1) − 𝑀))
211210oveq2d 7373 . . . . . . 7 (𝑚 = (𝑗 + 1) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212 oveq2 7365 . . . . . . . . . . . 12 (𝑚 = (𝑗 + 1) → (𝐾𝑚) = (𝐾↑(𝑗 + 1)))
213212oveq2d 7373 . . . . . . . . . . 11 (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214213fveq2d 6846 . . . . . . . . . 10 (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215214oveq1d 7372 . . . . . . . . 9 (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216215oveq1d 7372 . . . . . . . 8 (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217216sumeq1d 15586 . . . . . . 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 5118 . . . . . 6 (𝑚 = (𝑗 + 1) → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
219218imbi2d 340 . . . . 5 (𝑚 = (𝑗 + 1) → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
220 oveq1 7364 . . . . . . . 8 (𝑚 = 𝑁 → (𝑚𝑀) = (𝑁𝑀))
221220oveq2d 7373 . . . . . . 7 (𝑚 = 𝑁 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)))
222 oveq2 7365 . . . . . . . . . . . 12 (𝑚 = 𝑁 → (𝐾𝑚) = (𝐾𝑁))
223222oveq2d 7373 . . . . . . . . . . 11 (𝑚 = 𝑁 → (𝑍 / (𝐾𝑚)) = (𝑍 / (𝐾𝑁)))
224223fveq2d 6846 . . . . . . . . . 10 (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾𝑚))) = (⌊‘(𝑍 / (𝐾𝑁))))
225224oveq1d 7372 . . . . . . . . 9 (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾𝑁))) + 1))
226225oveq1d 7372 . . . . . . . 8 (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227226sumeq1d 15586 . . . . . . 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 5118 . . . . . 6 (𝑚 = 𝑁 → ((((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
229228imbi2d 340 . . . . 5 (𝑚 = 𝑁 → ((𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
23057nncnd 12169 . . . . . . . . . 10 (𝜑𝑀 ∈ ℂ)
231230subidd 11500 . . . . . . . . 9 (𝜑 → (𝑀𝑀) = 0)
232231oveq2d 7373 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
23352rpcnd 12959 . . . . . . . . 9 (𝜑 → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234233mul01d 11354 . . . . . . . 8 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235232, 234eqtrd 2776 . . . . . . 7 (𝜑 → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀𝑀)) = 0)
236 fzfid 13878 . . . . . . . 8 (𝜑 → (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
23757nnzd 12526 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℤ)
23885, 237rpexpcld 14150 . . . . . . . . . . . . . . 15 (𝜑 → (𝐾𝑀) ∈ ℝ+)
23935, 238rpdivcld 12974 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / (𝐾𝑀)) ∈ ℝ+)
240239rprege0d 12964 . . . . . . . . . . . . 13 (𝜑 → ((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))))
241 flge0nn0 13725 . . . . . . . . . . . . 13 (((𝑍 / (𝐾𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾𝑀))) → (⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0)
242 nn0p1nn 12452 . . . . . . . . . . . . 13 ((⌊‘(𝑍 / (𝐾𝑀))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
243240, 241, 2423syl 18 . . . . . . . . . . . 12 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ ℕ)
244243, 181eleqtrdi 2848 . . . . . . . . . . 11 (𝜑 → ((⌊‘(𝑍 / (𝐾𝑀))) + 1) ∈ (ℤ‘1))
245 fzss1 13480 . . . . . . . . . . 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 3944 . . . . . . . . 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 13445 . . . . . . . . . . . . 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 12974 . . . . . . . . . . . . . 14 (𝜑 → (𝑍 / 𝑌) ∈ ℝ+)
253252rpred 12957 . . . . . . . . . . . . 13 (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254 elfzelz 13441 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255 flge 13710 . . . . . . . . . . . . 13 (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256253, 254, 255syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257250, 256mpbird 256 . . . . . . . . . . 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 26948 . . . . . . . . . 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 15680 . . . . . . 7 (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264235, 263eqbrtrd 5127 . . . . . 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 2736 . . . . . . . . . 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 26952 . . . . . . . . 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 12957 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271 elfzoelz 13572 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272271adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273272zred 12607 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
27457adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275274nnred 12168 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276273, 275resubcld 11583 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℝ)
277270, 276remulcld 11185 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗𝑀)) ∈ ℝ)
278 fzfid 13878 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∈ Fin)
279 ssun1 4132 . . . . . . . . . . . . . . 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 12610 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283281, 282rpexpcld 14150 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ+)
284280, 283rerpdivcld 12988 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285281, 272rpexpcld 14150 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ+)
286280, 285rerpdivcld 12988 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ∈ ℝ)
28786adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288 1re 11155 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ
289 ltle 11243 . . . . . . . . . . . . . . . . . . . . . . 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 12778 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ𝑗))
294 peano2uz 12826 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (ℤ𝑗) → (𝑗 + 1) ∈ (ℤ𝑗))
295272, 293, 2943syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ𝑗))
296287, 292, 295leexp2ad 14157 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)))
29735adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ+)
298285, 283, 297lediv2d 12981 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝐾𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗))))
299296, 298mpbid 231 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾𝑗)))
300 flword2 13718 . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302 eluzp1p1 12791 . . . . . . . . . . . . . . . . 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 13703 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ)
305252adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ+)
306305rpred 12957 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307306flcld 13703 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308251adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ+)
309308rpred 12957 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310285rpred 12957 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝐾𝑗) ∈ ℝ)
31130simpld 495 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ∈ ℝ+)
312311rpred 12957 . . . . . . . . . . . . . . . . . . . . . 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 13588 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
3171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55pntlemh 26947 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
318316, 317sylan2 593 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾𝑗) ∧ (𝐾𝑗) ≤ (√‘𝑍)))
319318simpld 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾𝑗))
320309, 313, 310, 315, 319lttrd 11316 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾𝑗))
321309, 310, 320ltled 11303 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾𝑗))
322308, 285, 297lediv2d 12981 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾𝑗) ↔ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)))
323321, 322mpbid 231 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌))
324 flwordi 13717 . . . . . . . . . . . . . . . . . 18 (((𝑍 / (𝐾𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325286, 306, 323, 324syl3anc 1371 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326 eluz2 12769 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327304, 307, 325, 326syl3anbrc 1343 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ‘(⌊‘(𝑍 / (𝐾𝑗)))))
328 fzsplit2 13466 . . . . . . . . . . . . . . . 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 3997 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331297, 283rpdivcld 12974 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ+)
332331rprege0d 12964 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333 flge0nn0 13725 . . . . . . . . . . . . . . . . 17 (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0)
334 nn0p1nn 12452 . . . . . . . . . . . . . . . . 17 ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ0 → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
335332, 333, 3343syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
336335, 181eleqtrdi 2848 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ‘1))
337 fzss1 13480 . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340339sselda 3944 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
34182adantlr 713 . . . . . . . . . . . 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 15619 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344 fzfid 13878 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345 ssun2 4133 . . . . . . . . . . . . . . 15 (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346345, 329sseqtrrid 3997 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347346, 338sstrd 3954 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348347sselda 3944 . . . . . . . . . . . 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 15619 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351 le2add 11637 . . . . . . . . . 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 837 . . . . . . . . 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 693 . . . . . . . 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 11150 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356272zcnd 12608 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357230adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358356, 357subcld 11512 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (𝑗𝑀) ∈ ℂ)
359354, 355, 358adddid 11179 . . . . . . . . . 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 11357 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗𝑀) + 1))
361356, 355, 357addsubd 11533 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗𝑀) + 1))
362360, 361eqtr4d 2779 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗𝑀)) = ((𝑗 + 1) − 𝑀))
363362oveq2d 7373 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗𝑀))) = (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364354mulid1d 11172 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365364oveq1d 7372 . . . . . . . . . 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 2784 . . . . . . . . 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 13701 . . . . . . . . . . . . 13 ((𝑍 / (𝐾𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
368286, 367syl 17 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) ∈ ℝ)
369368ltp1d 12085 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾𝑗))) < ((⌊‘(𝑍 / (𝐾𝑗))) + 1))
370 fzdisj 13468 . . . . . . . . . . 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 13878 . . . . . . . . . 10 ((𝜑𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373338sselda 3944 . . . . . . . . . . . 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 11183 . . . . . . . . . 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 15626 . . . . . . . . 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 5118 . . . . . . . 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 258 . . . . . . 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 13690 . . . 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 15681 . . 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 11312 . 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 11312 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cun 3908  cin 3909  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  2c2 12208  3c3 12209  4c4 12210  8c8 12214  0cn0 12413  cz 12499  cdc 12618  cuz 12763  +crp 12915  (,)cioo 13264  [,)cico 13266  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  cfl 13695  cexp 13967  csqrt 15118  abscabs 15119  Σcsu 15570  expce 15944  eceu 15945  logclog 25910  ψcchp 26442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-ef 15950  df-e 15951  df-sin 15952  df-cos 15953  df-pi 15955  df-dvds 16137  df-gcd 16375  df-prm 16548  df-pc 16709  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231  df-log 25912  df-vma 26447  df-chp 26448
This theorem is referenced by:  pntlemo  26955
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