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Theorem pntlemo 27665
Description: Lemma for pnt 27672. Combine all the estimates to establish a smaller eventual bound on 𝑅(𝑍) / 𝑍. (Contributed by Mario Carneiro, 14-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))
pntlem1.C (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)
Assertion
Ref Expression
pntlemo (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
Distinct variable groups:   𝑧,𝐶   𝑦,𝑧,𝑢,𝐿   𝑦,𝐾,𝑧   𝑧,𝑀   𝑧,𝑁   𝑢,𝑖,𝑦,𝑧,𝑅   𝑧,𝑈   𝑧,𝑊   𝑦,𝑋,𝑧   𝑖,𝑌,𝑧   𝑢,𝑎,𝑦,𝑧,𝐸   𝑢,𝑍,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑢,𝑖,𝑎)   𝐴(𝑦,𝑧,𝑢,𝑖,𝑎)   𝐵(𝑦,𝑧,𝑢,𝑖,𝑎)   𝐶(𝑦,𝑢,𝑖,𝑎)   𝐷(𝑦,𝑧,𝑢,𝑖,𝑎)   𝑅(𝑎)   𝑈(𝑦,𝑢,𝑖,𝑎)   𝐸(𝑖)   𝐹(𝑦,𝑧,𝑢,𝑖,𝑎)   𝐾(𝑢,𝑖,𝑎)   𝐿(𝑖,𝑎)   𝑀(𝑦,𝑢,𝑖,𝑎)   𝑁(𝑦,𝑢,𝑖,𝑎)   𝑊(𝑦,𝑢,𝑖,𝑎)   𝑋(𝑢,𝑖,𝑎)   𝑌(𝑦,𝑢,𝑎)   𝑍(𝑦,𝑖,𝑎)

Proof of Theorem pntlemo
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 pntlem1.r . . . . . . . . . 10 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
2 pntlem1.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
3 pntlem1.b . . . . . . . . . 10 (𝜑𝐵 ∈ ℝ+)
4 pntlem1.l . . . . . . . . . 10 (𝜑𝐿 ∈ (0(,)1))
5 pntlem1.d . . . . . . . . . 10 𝐷 = (𝐴 + 1)
6 pntlem1.f . . . . . . . . . 10 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2)))
7 pntlem1.u . . . . . . . . . 10 (𝜑𝑈 ∈ ℝ+)
8 pntlem1.u2 . . . . . . . . . 10 (𝜑𝑈𝐴)
9 pntlem1.e . . . . . . . . . 10 𝐸 = (𝑈 / 𝐷)
10 pntlem1.k . . . . . . . . . 10 𝐾 = (exp‘(𝐵 / 𝐸))
11 pntlem1.y . . . . . . . . . 10 (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌))
12 pntlem1.x . . . . . . . . . 10 (𝜑 → (𝑋 ∈ ℝ+𝑌 < 𝑋))
13 pntlem1.c . . . . . . . . . 10 (𝜑𝐶 ∈ ℝ+)
14 pntlem1.w . . . . . . . . . 10 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((32 · 𝐵) / ((𝑈𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
15 pntlem1.z . . . . . . . . . 10 (𝜑𝑍 ∈ (𝑊[,)+∞))
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15pntlemb 27655 . . . . . . . . 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)))))
1716simp1d 1141 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
181pntrf 27621 . . . . . . . . 9 𝑅:ℝ+⟶ℝ
1918ffvelcdmi 7102 . . . . . . . 8 (𝑍 ∈ ℝ+ → (𝑅𝑍) ∈ ℝ)
2017, 19syl 17 . . . . . . 7 (𝜑 → (𝑅𝑍) ∈ ℝ)
2120, 17rerpdivcld 13105 . . . . . 6 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℝ)
2221recnd 11286 . . . . 5 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℂ)
2322abscld 15471 . . . 4 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ∈ ℝ)
2417relogcld 26679 . . . 4 (𝜑 → (log‘𝑍) ∈ ℝ)
2523, 24remulcld 11288 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ∈ ℝ)
267rpred 13074 . . . . . 6 (𝜑𝑈 ∈ ℝ)
27 3re 12343 . . . . . . . 8 3 ∈ ℝ
2827a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℝ)
2924, 28readdcld 11287 . . . . . 6 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3026, 29remulcld 11288 . . . . 5 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℝ)
31 2re 12337 . . . . . . 7 2 ∈ ℝ
3231a1i 11 . . . . . 6 (𝜑 → 2 ∈ ℝ)
331, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 27653 . . . . . . . . . . 11 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
3433simp3d 1143 . . . . . . . . . 10 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
3534simp3d 1143 . . . . . . . . 9 (𝜑 → (𝑈𝐸) ∈ ℝ+)
3635rpred 13074 . . . . . . . 8 (𝜑 → (𝑈𝐸) ∈ ℝ)
371, 2, 3, 4, 5, 6pntlemd 27652 . . . . . . . . . . . 12 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
3837simp1d 1141 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℝ+)
3933simp1d 1141 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℝ+)
40 2z 12646 . . . . . . . . . . . 12 2 ∈ ℤ
41 rpexpcl 14117 . . . . . . . . . . . 12 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
4239, 40, 41sylancl 586 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℝ+)
4338, 42rpmulcld 13090 . . . . . . . . . 10 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
44 3nn0 12541 . . . . . . . . . . . . 13 3 ∈ ℕ0
45 2nn 12336 . . . . . . . . . . . . 13 2 ∈ ℕ
4644, 45decnncl 12750 . . . . . . . . . . . 12 32 ∈ ℕ
47 nnrp 13043 . . . . . . . . . . . 12 (32 ∈ ℕ → 32 ∈ ℝ+)
4846, 47ax-mp 5 . . . . . . . . . . 11 32 ∈ ℝ+
49 rpmulcl 13055 . . . . . . . . . . 11 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
5048, 3, 49sylancr 587 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) ∈ ℝ+)
5143, 50rpdivcld 13091 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ+)
5251rpred 13074 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ)
5336, 52remulcld 11288 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℝ)
5453, 24remulcld 11288 . . . . . 6 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℝ)
5532, 54remulcld 11288 . . . . 5 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
5630, 55resubcld 11688 . . . 4 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ∈ ℝ)
5713rpred 13074 . . . 4 (𝜑𝐶 ∈ ℝ)
5856, 57readdcld 11287 . . 3 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ∈ ℝ)
597rpcnd 13076 . . . . . 6 (𝜑𝑈 ∈ ℂ)
6053recnd 11286 . . . . . 6 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℂ)
6124recnd 11286 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
6259, 60, 61subdird 11717 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
6338rpcnd 13076 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℂ)
6442rpcnd 13076 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℂ)
6550rpcnne0d 13083 . . . . . . . . . . 11 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
66 div23 11938 . . . . . . . . . . 11 ((𝐿 ∈ ℂ ∧ (𝐸↑2) ∈ ℂ ∧ ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0)) → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = ((𝐿 / (32 · 𝐵)) · (𝐸↑2)))
6763, 64, 65, 66syl3anc 1370 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = ((𝐿 / (32 · 𝐵)) · (𝐸↑2)))
689oveq1i 7440 . . . . . . . . . . . 12 (𝐸↑2) = ((𝑈 / 𝐷)↑2)
6937simp2d 1142 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ ℝ+)
7069rpcnd 13076 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ ℂ)
7169rpne0d 13079 . . . . . . . . . . . . 13 (𝜑𝐷 ≠ 0)
7259, 70, 71sqdivd 14195 . . . . . . . . . . . 12 (𝜑 → ((𝑈 / 𝐷)↑2) = ((𝑈↑2) / (𝐷↑2)))
7368, 72eqtrid 2786 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) = ((𝑈↑2) / (𝐷↑2)))
7473oveq2d 7446 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · (𝐸↑2)) = ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))))
7538, 50rpdivcld 13091 . . . . . . . . . . . 12 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℝ+)
7675rpcnd 13076 . . . . . . . . . . 11 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℂ)
7759sqcld 14180 . . . . . . . . . . 11 (𝜑 → (𝑈↑2) ∈ ℂ)
78 rpexpcl 14117 . . . . . . . . . . . . 13 ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+)
7969, 40, 78sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐷↑2) ∈ ℝ+)
8079rpcnne0d 13083 . . . . . . . . . . 11 (𝜑 → ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0))
81 divass 11937 . . . . . . . . . . . 12 (((𝐿 / (32 · 𝐵)) ∈ ℂ ∧ (𝑈↑2) ∈ ℂ ∧ ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0)) → (((𝐿 / (32 · 𝐵)) · (𝑈↑2)) / (𝐷↑2)) = ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))))
82 div23 11938 . . . . . . . . . . . 12 (((𝐿 / (32 · 𝐵)) ∈ ℂ ∧ (𝑈↑2) ∈ ℂ ∧ ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0)) → (((𝐿 / (32 · 𝐵)) · (𝑈↑2)) / (𝐷↑2)) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8381, 82eqtr3d 2776 . . . . . . . . . . 11 (((𝐿 / (32 · 𝐵)) ∈ ℂ ∧ (𝑈↑2) ∈ ℂ ∧ ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0)) → ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8476, 77, 80, 83syl3anc 1370 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8567, 74, 843eqtrd 2778 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8685oveq2d 7446 . . . . . . . 8 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
87 df-3 12327 . . . . . . . . . . . . 13 3 = (2 + 1)
8887oveq2i 7441 . . . . . . . . . . . 12 (𝑈↑3) = (𝑈↑(2 + 1))
89 2nn0 12540 . . . . . . . . . . . . 13 2 ∈ ℕ0
90 expp1 14105 . . . . . . . . . . . . 13 ((𝑈 ∈ ℂ ∧ 2 ∈ ℕ0) → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9159, 89, 90sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9288, 91eqtrid 2786 . . . . . . . . . . 11 (𝜑 → (𝑈↑3) = ((𝑈↑2) · 𝑈))
9377, 59mulcomd 11279 . . . . . . . . . . 11 (𝜑 → ((𝑈↑2) · 𝑈) = (𝑈 · (𝑈↑2)))
9492, 93eqtrd 2774 . . . . . . . . . 10 (𝜑 → (𝑈↑3) = (𝑈 · (𝑈↑2)))
9594oveq2d 7446 . . . . . . . . 9 (𝜑 → (𝐹 · (𝑈↑3)) = (𝐹 · (𝑈 · (𝑈↑2))))
9637simp3d 1143 . . . . . . . . . . 11 (𝜑𝐹 ∈ ℝ+)
9796rpcnd 13076 . . . . . . . . . 10 (𝜑𝐹 ∈ ℂ)
9897, 59, 77mulassd 11281 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = (𝐹 · (𝑈 · (𝑈↑2))))
99 1cnd 11253 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℂ)
10069rpreccld 13084 . . . . . . . . . . . . . . . 16 (𝜑 → (1 / 𝐷) ∈ ℝ+)
101100rpcnd 13076 . . . . . . . . . . . . . . 15 (𝜑 → (1 / 𝐷) ∈ ℂ)
10299, 101, 59subdird 11717 . . . . . . . . . . . . . 14 (𝜑 → ((1 − (1 / 𝐷)) · 𝑈) = ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)))
10359mullidd 11276 . . . . . . . . . . . . . . 15 (𝜑 → (1 · 𝑈) = 𝑈)
10459, 70, 71divrec2d 12044 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑈 / 𝐷) = ((1 / 𝐷) · 𝑈))
1059, 104eqtr2id 2787 . . . . . . . . . . . . . . 15 (𝜑 → ((1 / 𝐷) · 𝑈) = 𝐸)
106103, 105oveq12d 7448 . . . . . . . . . . . . . 14 (𝜑 → ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)) = (𝑈𝐸))
107102, 106eqtr2d 2775 . . . . . . . . . . . . 13 (𝜑 → (𝑈𝐸) = ((1 − (1 / 𝐷)) · 𝑈))
108107oveq1d 7445 . . . . . . . . . . . 12 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
1096oveq1i 7440 . . . . . . . . . . . . 13 (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈)
11099, 101subcld 11617 . . . . . . . . . . . . . 14 (𝜑 → (1 − (1 / 𝐷)) ∈ ℂ)
11175, 79rpdivcld 13091 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+)
112111rpcnd 13076 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℂ)
113110, 112, 59mul32d 11468 . . . . . . . . . . . . 13 (𝜑 → (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
114109, 113eqtrid 2786 . . . . . . . . . . . 12 (𝜑 → (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
115108, 114eqtr4d 2777 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (𝐹 · 𝑈))
116115oveq1d 7445 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝐹 · 𝑈) · (𝑈↑2)))
11735rpcnd 13076 . . . . . . . . . . 11 (𝜑 → (𝑈𝐸) ∈ ℂ)
118117, 112, 77mulassd 11281 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
119116, 118eqtr3d 2776 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12095, 98, 1193eqtr2d 2780 . . . . . . . 8 (𝜑 → (𝐹 · (𝑈↑3)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12186, 120eqtr4d 2777 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = (𝐹 · (𝑈↑3)))
122121oveq2d 7446 . . . . . 6 (𝜑 → (𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) = (𝑈 − (𝐹 · (𝑈↑3))))
123122oveq1d 7445 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12462, 123eqtr3d 2776 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12526, 24remulcld 11288 . . . . 5 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℝ)
126125, 54resubcld 11688 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
127124, 126eqeltrrd 2839 . . 3 (𝜑 → ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)) ∈ ℝ)
12817rpred 13074 . . . . . . . 8 (𝜑𝑍 ∈ ℝ)
12916simp2d 1142 . . . . . . . . 9 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
130129simp1d 1141 . . . . . . . 8 (𝜑 → 1 < 𝑍)
131128, 130rplogcld 26685 . . . . . . 7 (𝜑 → (log‘𝑍) ∈ ℝ+)
13232, 131rerpdivcld 13105 . . . . . 6 (𝜑 → (2 / (log‘𝑍)) ∈ ℝ)
133 fzfid 14010 . . . . . . 7 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
13417adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
135 elfznn 13589 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
136135adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
137136nnrpd 13072 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
138134, 137rpdivcld 13091 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
13918ffvelcdmi 7102 . . . . . . . . . . . 12 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
140138, 139syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
141140, 134rerpdivcld 13105 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
142141recnd 11286 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
143142abscld 15471 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
144137relogcld 26679 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
145143, 144remulcld 11288 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
146133, 145fsumrecl 15766 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
147132, 146remulcld 11288 . . . . 5 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
148147, 57readdcld 11287 . . . 4 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ∈ ℝ)
14920recnd 11286 . . . . . . . . . . 11 (𝜑 → (𝑅𝑍) ∈ ℂ)
150149abscld 15471 . . . . . . . . . 10 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℝ)
151150recnd 11286 . . . . . . . . 9 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℂ)
152151, 61mulcld 11278 . . . . . . . 8 (𝜑 → ((abs‘(𝑅𝑍)) · (log‘𝑍)) ∈ ℂ)
153132recnd 11286 . . . . . . . . 9 (𝜑 → (2 / (log‘𝑍)) ∈ ℂ)
154140recnd 11286 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℂ)
155154abscld 15471 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℝ)
156155, 144remulcld 11288 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
157133, 156fsumrecl 15766 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
158157recnd 11286 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
159153, 158mulcld 11278 . . . . . . . 8 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
16017rpcnd 13076 . . . . . . . 8 (𝜑𝑍 ∈ ℂ)
16117rpne0d 13079 . . . . . . . 8 (𝜑𝑍 ≠ 0)
162152, 159, 160, 161divsubdird 12079 . . . . . . 7 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) = ((((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) − (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍)))
163151, 61, 160, 161div23d 12077 . . . . . . . . 9 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
164149, 160, 161absdivd 15490 . . . . . . . . . . 11 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / (abs‘𝑍)))
16517rprege0d 13081 . . . . . . . . . . . . 13 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
166 absid 15331 . . . . . . . . . . . . 13 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → (abs‘𝑍) = 𝑍)
167165, 166syl 17 . . . . . . . . . . . 12 (𝜑 → (abs‘𝑍) = 𝑍)
168167oveq2d 7446 . . . . . . . . . . 11 (𝜑 → ((abs‘(𝑅𝑍)) / (abs‘𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
169164, 168eqtrd 2774 . . . . . . . . . 10 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
170169oveq1d 7445 . . . . . . . . 9 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
171163, 170eqtr4d 2777 . . . . . . . 8 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)))
172153, 158, 160, 161divassd 12075 . . . . . . . . 9 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍) = ((2 / (log‘𝑍)) · (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍)))
173160adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℂ)
174161adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ≠ 0)
175154, 173, 174absdivd 15490 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)))
176167adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘𝑍) = 𝑍)
177176oveq2d 7446 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
178175, 177eqtrd 2774 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
179178oveq1d 7445 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
180155recnd 11286 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℂ)
181144recnd 11286 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
18217rpcnne0d 13083 . . . . . . . . . . . . . . 15 (𝜑 → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
183182adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
184 div23 11938 . . . . . . . . . . . . . 14 (((abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℂ ∧ (log‘𝑛) ∈ ℂ ∧ (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0)) → (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
185180, 181, 183, 184syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
186179, 185eqtr4d 2777 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
187186sumeq2dv 15734 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
188156recnd 11286 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
189133, 160, 188, 161fsumdivc 15818 . . . . . . . . . . 11 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
190187, 189eqtr4d 2777 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
191190oveq2d 7446 . . . . . . . . 9 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) = ((2 / (log‘𝑍)) · (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍)))
192172, 191eqtr4d 2777 . . . . . . . 8 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍) = ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
193171, 192oveq12d 7448 . . . . . . 7 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) − (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍)) = (((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
194162, 193eqtrd 2774 . . . . . 6 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) = (((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
195 2fveq3 6911 . . . . . . . . . . 11 (𝑧 = 𝑍 → (abs‘(𝑅𝑧)) = (abs‘(𝑅𝑍)))
196 fveq2 6906 . . . . . . . . . . 11 (𝑧 = 𝑍 → (log‘𝑧) = (log‘𝑍))
197195, 196oveq12d 7448 . . . . . . . . . 10 (𝑧 = 𝑍 → ((abs‘(𝑅𝑧)) · (log‘𝑧)) = ((abs‘(𝑅𝑍)) · (log‘𝑍)))
198196oveq2d 7446 . . . . . . . . . . 11 (𝑧 = 𝑍 → (2 / (log‘𝑧)) = (2 / (log‘𝑍)))
199 oveq2 7438 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑧 / 𝑖) = (𝑧 / 𝑛))
200199fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (𝑅‘(𝑧 / 𝑖)) = (𝑅‘(𝑧 / 𝑛)))
201200fveq2d 6910 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → (abs‘(𝑅‘(𝑧 / 𝑖))) = (abs‘(𝑅‘(𝑧 / 𝑛))))
202 fveq2 6906 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → (log‘𝑖) = (log‘𝑛))
203201, 202oveq12d 7448 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → ((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)))
204203cbvsumv 15728 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛))
205 fvoveq1 7453 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (⌊‘(𝑧 / 𝑌)) = (⌊‘(𝑍 / 𝑌)))
206205oveq2d 7446 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (1...(⌊‘(𝑧 / 𝑌))) = (1...(⌊‘(𝑍 / 𝑌))))
207 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑧 = 𝑍)
208207fvoveq1d 7452 . . . . . . . . . . . . . . 15 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑧 / 𝑛)) = (𝑅‘(𝑍 / 𝑛)))
209208fveq2d 6910 . . . . . . . . . . . . . 14 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑧 / 𝑛))) = (abs‘(𝑅‘(𝑍 / 𝑛))))
210209oveq1d 7445 . . . . . . . . . . . . 13 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
211206, 210sumeq12rdv 15739 . . . . . . . . . . . 12 (𝑧 = 𝑍 → Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
212204, 211eqtrid 2786 . . . . . . . . . . 11 (𝑧 = 𝑍 → Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
213198, 212oveq12d 7448 . . . . . . . . . 10 (𝑧 = 𝑍 → ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖))) = ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))))
214197, 213oveq12d 7448 . . . . . . . . 9 (𝑧 = 𝑍 → (((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) = (((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))))
215 id 22 . . . . . . . . 9 (𝑧 = 𝑍𝑧 = 𝑍)
216214, 215oveq12d 7448 . . . . . . . 8 (𝑧 = 𝑍 → ((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) = ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍))
217216breq1d 5157 . . . . . . 7 (𝑧 = 𝑍 → (((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶 ↔ ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) ≤ 𝐶))
218 pntlem1.C . . . . . . 7 (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)
219 1re 11258 . . . . . . . . 9 1 ∈ ℝ
220 rexr 11304 . . . . . . . . 9 (1 ∈ ℝ → 1 ∈ ℝ*)
221 elioopnf 13479 . . . . . . . . 9 (1 ∈ ℝ* → (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍)))
222219, 220, 221mp2b 10 . . . . . . . 8 (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍))
223128, 130, 222sylanbrc 583 . . . . . . 7 (𝜑𝑍 ∈ (1(,)+∞))
224217, 218, 223rspcdva 3622 . . . . . 6 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) ≤ 𝐶)
225194, 224eqbrtrrd 5171 . . . . 5 (𝜑 → (((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ 𝐶)
22625, 147, 57lesubadd2d 11859 . . . . 5 (𝜑 → ((((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ 𝐶 ↔ ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶)))
227225, 226mpbid 232 . . . 4 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶))
228 2cnd 12341 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
229143recnd 11286 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℂ)
230229, 181mulcld 11278 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
231133, 230fsumcl 15765 . . . . . . 7 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
232131rpne0d 13079 . . . . . . 7 (𝜑 → (log‘𝑍) ≠ 0)
233228, 231, 61, 232div23d 12077 . . . . . 6 (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) / (log‘𝑍)) = ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
23424resqcld 14161 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
23552, 234remulcld 11288 . . . . . . . . . . 11 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
23636, 235remulcld 11288 . . . . . . . . . 10 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
237 remulcl 11237 . . . . . . . . . 10 ((2 ∈ ℝ ∧ ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ) → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
23831, 236, 237sylancr 587 . . . . . . . . 9 (𝜑 → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
23930, 24remulcld 11288 . . . . . . . . 9 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) ∈ ℝ)
240 remulcl 11237 . . . . . . . . . 10 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
24131, 146, 240sylancr 587 . . . . . . . . 9 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
24226adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
243242, 136nndivred 12317 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
244243, 143resubcld 11688 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
245244, 144remulcld 11288 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
246133, 245fsumrecl 15766 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
24732, 246remulcld 11288 . . . . . . . . . 10 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ∈ ℝ)
248239, 241resubcld 11688 . . . . . . . . . 10 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ∈ ℝ)
249 pntlem1.m . . . . . . . . . . . 12 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
250 pntlem1.n . . . . . . . . . . . 12 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
251 pntlem1.U . . . . . . . . . . . 12 (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)
252 pntlem1.K . . . . . . . . . . . 12 (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ+ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅𝑢) / 𝑢)) ≤ 𝐸))
2531, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 249, 250, 251, 252pntlemf 27663 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
254 2pos 12366 . . . . . . . . . . . . 13 0 < 2
255254a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 2)
256 lemul2 12117 . . . . . . . . . . . 12 ((((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
257236, 246, 32, 255, 256syl112anc 1373 . . . . . . . . . . 11 (𝜑 → (((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
258253, 257mpbid 232 . . . . . . . . . 10 (𝜑 → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
259243recnd 11286 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℂ)
260259, 229, 181subdird 11717 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = (((𝑈 / 𝑛) · (log‘𝑛)) − ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
261260sumeq2dv 15734 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) · (log‘𝑛)) − ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
262243, 144remulcld 11288 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
263262recnd 11286 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
264133, 263, 230fsumsub 15820 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) · (log‘𝑛)) − ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
265261, 264eqtrd 2774 . . . . . . . . . . . . 13 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
266265oveq2d 7446 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) = (2 · (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
267133, 262fsumrecl 15766 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
268267recnd 11286 . . . . . . . . . . . . 13 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
269228, 268, 231subdid 11716 . . . . . . . . . . . 12 (𝜑 → (2 · (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) = ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
270266, 269eqtrd 2774 . . . . . . . . . . 11 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) = ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
271 remulcl 11237 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
27231, 267, 271sylancr 587 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
2731, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 249, 250, 251, 252pntlemk 27664 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
274272, 239, 241, 273lesub1dd 11876 . . . . . . . . . . 11 (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
275270, 274eqbrtrd 5169 . . . . . . . . . 10 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
276238, 247, 248, 258, 275letrd 11415 . . . . . . . . 9 (𝜑 → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
277238, 239, 241, 276lesubd 11864 . . . . . . . 8 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))))
27830recnd 11286 . . . . . . . . . 10 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℂ)
27955recnd 11286 . . . . . . . . . 10 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℂ)
280278, 279, 61subdird 11717 . . . . . . . . 9 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍))))
28154recnd 11286 . . . . . . . . . . . 12 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℂ)
282228, 281, 61mulassd 11281 . . . . . . . . . . 11 (𝜑 → ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))))
28360, 61, 61mulassd 11281 . . . . . . . . . . . . 13 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
28461sqvald 14179 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
285284oveq2d 7446 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
28651rpcnd 13076 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
287234recnd 11286 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
288117, 286, 287mulassd 11281 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
289283, 285, 2883eqtr2d 2780 . . . . . . . . . . . 12 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
290289oveq2d 7446 . . . . . . . . . . 11 (𝜑 → (2 · ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))) = (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))))
291282, 290eqtrd 2774 . . . . . . . . . 10 (𝜑 → ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))))
292291oveq2d 7446 . . . . . . . . 9 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍))) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))))
293280, 292eqtrd 2774 . . . . . . . 8 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))))
294277, 293breqtrrd 5175 . . . . . . 7 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)))
295241, 56, 131ledivmul2d 13128 . . . . . . 7 (𝜑 → (((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) / (log‘𝑍)) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ↔ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍))))
296294, 295mpbird 257 . . . . . 6 (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) / (log‘𝑍)) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
297233, 296eqbrtrrd 5171 . . . . 5 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
298147, 56, 57, 297leadd1dd 11874 . . . 4 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
29925, 148, 58, 227, 298letrd 11415 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
300 remulcl 11237 . . . . . . . . 9 ((𝑈 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑈 · 3) ∈ ℝ)
30126, 27, 300sylancl 586 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℝ)
302301, 57readdcld 11287 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ)
30316simp3d 1143 . . . . . . . 8 (𝜑 → ((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))))
304303simp3d 1143 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))
305302, 54, 125, 304leadd2dd 11875 . . . . . 6 (𝜑 → ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)) ≤ ((𝑈 · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
30628recnd 11286 . . . . . . . . 9 (𝜑 → 3 ∈ ℂ)
30759, 61, 306adddid 11282 . . . . . . . 8 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) = ((𝑈 · (log‘𝑍)) + (𝑈 · 3)))
308307oveq1d 7445 . . . . . . 7 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶))
309125recnd 11286 . . . . . . . 8 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℂ)
31059, 306mulcld 11278 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℂ)
31113rpcnd 13076 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
312309, 310, 311addassd 11280 . . . . . . 7 (𝜑 → (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
313308, 312eqtrd 2774 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
3142812timesd 12506 . . . . . . . 8 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) = ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
315314oveq2d 7446 . . . . . . 7 (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) = (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
316309, 281, 281nppcan3d 11644 . . . . . . 7 (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) = ((𝑈 · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
317315, 316eqtrd 2774 . . . . . 6 (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) = ((𝑈 · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
318305, 313, 3173brtr4d 5179 . . . . 5 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ≤ (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
31930, 57readdcld 11287 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ∈ ℝ)
320319, 55, 126lesubaddd 11857 . . . . 5 (𝜑 → ((((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ≤ ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ↔ ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ≤ (((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))))
321318, 320mpbird 257 . . . 4 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ≤ ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
322278, 311, 279addsubd 11638 . . . 4 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) = (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
323321, 322, 1243brtr3d 5178 . . 3 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
32425, 58, 127, 299, 323letrd 11415 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
325 3z 12647 . . . . . . 7 3 ∈ ℤ
326 rpexpcl 14117 . . . . . . 7 ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℤ) → (𝑈↑3) ∈ ℝ+)
3277, 325, 326sylancl 586 . . . . . 6 (𝜑 → (𝑈↑3) ∈ ℝ+)
32896, 327rpmulcld 13090 . . . . 5 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ+)
329328rpred 13074 . . . 4 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ)
33026, 329resubcld 11688 . . 3 (𝜑 → (𝑈 − (𝐹 · (𝑈↑3))) ∈ ℝ)
33123, 330, 131lemul1d 13117 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))) ↔ ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍))))
332324, 331mpbird 257 1 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  3c3 12319  4c4 12320  0cn0 12523  cz 12610  cdc 12730  +crp 13031  (,)cioo 13383  [,)cico 13385  [,]cicc 13386  ...cfz 13543  cfl 13826  cexp 14098  csqrt 15268  abscabs 15269  Σcsu 15718  expce 16093  eceu 16094  logclog 26610  ψcchp 27150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-ef 16099  df-e 16100  df-sin 16101  df-cos 16102  df-tan 16103  df-pi 16104  df-dvds 16287  df-gcd 16528  df-prm 16705  df-pc 16870  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-ulm 26434  df-log 26612  df-atan 26924  df-em 27050  df-vma 27155  df-chp 27156
This theorem is referenced by:  pntleme  27666
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