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Theorem pntlemo 25853
Description: Lemma for pnt 25860. 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 25843 . . . . . . . . 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 1133 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
181pntrf 25809 . . . . . . . . 9 𝑅:ℝ+⟶ℝ
1918ffvelrni 6706 . . . . . . . 8 (𝑍 ∈ ℝ+ → (𝑅𝑍) ∈ ℝ)
2017, 19syl 17 . . . . . . 7 (𝜑 → (𝑅𝑍) ∈ ℝ)
2120, 17rerpdivcld 12301 . . . . . 6 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℝ)
2221recnd 10504 . . . . 5 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℂ)
2322abscld 14618 . . . 4 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ∈ ℝ)
2417relogcld 24875 . . . 4 (𝜑 → (log‘𝑍) ∈ ℝ)
2523, 24remulcld 10506 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ∈ ℝ)
267rpred 12270 . . . . . 6 (𝜑𝑈 ∈ ℝ)
27 3re 11554 . . . . . . . 8 3 ∈ ℝ
2827a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℝ)
2924, 28readdcld 10505 . . . . . 6 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3026, 29remulcld 10506 . . . . 5 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℝ)
31 2re 11548 . . . . . . 7 2 ∈ ℝ
3231a1i 11 . . . . . 6 (𝜑 → 2 ∈ ℝ)
331, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 25841 . . . . . . . . . . 11 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
3433simp3d 1135 . . . . . . . . . 10 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
3534simp3d 1135 . . . . . . . . 9 (𝜑 → (𝑈𝐸) ∈ ℝ+)
3635rpred 12270 . . . . . . . 8 (𝜑 → (𝑈𝐸) ∈ ℝ)
371, 2, 3, 4, 5, 6pntlemd 25840 . . . . . . . . . . . 12 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
3837simp1d 1133 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℝ+)
3933simp1d 1133 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℝ+)
40 2z 11852 . . . . . . . . . . . 12 2 ∈ ℤ
41 rpexpcl 13286 . . . . . . . . . . . 12 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
4239, 40, 41sylancl 586 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℝ+)
4338, 42rpmulcld 12286 . . . . . . . . . 10 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
44 3nn0 11752 . . . . . . . . . . . . 13 3 ∈ ℕ0
45 2nn 11547 . . . . . . . . . . . . 13 2 ∈ ℕ
4644, 45decnncl 11956 . . . . . . . . . . . 12 32 ∈ ℕ
47 nnrp 12239 . . . . . . . . . . . 12 (32 ∈ ℕ → 32 ∈ ℝ+)
4846, 47ax-mp 5 . . . . . . . . . . 11 32 ∈ ℝ+
49 rpmulcl 12251 . . . . . . . . . . 11 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
5048, 3, 49sylancr 587 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) ∈ ℝ+)
5143, 50rpdivcld 12287 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ+)
5251rpred 12270 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ)
5336, 52remulcld 10506 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℝ)
5453, 24remulcld 10506 . . . . . 6 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℝ)
5532, 54remulcld 10506 . . . . 5 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
5630, 55resubcld 10905 . . . 4 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ∈ ℝ)
5713rpred 12270 . . . 4 (𝜑𝐶 ∈ ℝ)
5856, 57readdcld 10505 . . 3 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ∈ ℝ)
597rpcnd 12272 . . . . . 6 (𝜑𝑈 ∈ ℂ)
6053recnd 10504 . . . . . 6 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℂ)
6124recnd 10504 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
6259, 60, 61subdird 10934 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
6338rpcnd 12272 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℂ)
6442rpcnd 12272 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℂ)
6550rpcnne0d 12279 . . . . . . . . . . 11 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
66 div23 11154 . . . . . . . . . . 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 1362 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = ((𝐿 / (32 · 𝐵)) · (𝐸↑2)))
689oveq1i 7017 . . . . . . . . . . . 12 (𝐸↑2) = ((𝑈 / 𝐷)↑2)
6937simp2d 1134 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ ℝ+)
7069rpcnd 12272 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ ℂ)
7169rpne0d 12275 . . . . . . . . . . . . 13 (𝜑𝐷 ≠ 0)
7259, 70, 71sqdivd 13361 . . . . . . . . . . . 12 (𝜑 → ((𝑈 / 𝐷)↑2) = ((𝑈↑2) / (𝐷↑2)))
7368, 72syl5eq 2841 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) = ((𝑈↑2) / (𝐷↑2)))
7473oveq2d 7023 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · (𝐸↑2)) = ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))))
7538, 50rpdivcld 12287 . . . . . . . . . . . 12 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℝ+)
7675rpcnd 12272 . . . . . . . . . . 11 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℂ)
7759sqcld 13346 . . . . . . . . . . 11 (𝜑 → (𝑈↑2) ∈ ℂ)
78 rpexpcl 13286 . . . . . . . . . . . . 13 ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+)
7969, 40, 78sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝐷↑2) ∈ ℝ+)
8079rpcnne0d 12279 . . . . . . . . . . 11 (𝜑 → ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0))
81 divass 11153 . . . . . . . . . . . 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 11154 . . . . . . . . . . . 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 2831 . . . . . . . . . . 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 1362 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8567, 74, 843eqtrd 2833 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8685oveq2d 7023 . . . . . . . 8 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
87 df-3 11538 . . . . . . . . . . . . 13 3 = (2 + 1)
8887oveq2i 7018 . . . . . . . . . . . 12 (𝑈↑3) = (𝑈↑(2 + 1))
89 2nn0 11751 . . . . . . . . . . . . 13 2 ∈ ℕ0
90 expp1 13274 . . . . . . . . . . . . 13 ((𝑈 ∈ ℂ ∧ 2 ∈ ℕ0) → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9159, 89, 90sylancl 586 . . . . . . . . . . . 12 (𝜑 → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9288, 91syl5eq 2841 . . . . . . . . . . 11 (𝜑 → (𝑈↑3) = ((𝑈↑2) · 𝑈))
9377, 59mulcomd 10497 . . . . . . . . . . 11 (𝜑 → ((𝑈↑2) · 𝑈) = (𝑈 · (𝑈↑2)))
9492, 93eqtrd 2829 . . . . . . . . . 10 (𝜑 → (𝑈↑3) = (𝑈 · (𝑈↑2)))
9594oveq2d 7023 . . . . . . . . 9 (𝜑 → (𝐹 · (𝑈↑3)) = (𝐹 · (𝑈 · (𝑈↑2))))
9637simp3d 1135 . . . . . . . . . . 11 (𝜑𝐹 ∈ ℝ+)
9796rpcnd 12272 . . . . . . . . . 10 (𝜑𝐹 ∈ ℂ)
9897, 59, 77mulassd 10499 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = (𝐹 · (𝑈 · (𝑈↑2))))
99 1cnd 10471 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℂ)
10069rpreccld 12280 . . . . . . . . . . . . . . . 16 (𝜑 → (1 / 𝐷) ∈ ℝ+)
101100rpcnd 12272 . . . . . . . . . . . . . . 15 (𝜑 → (1 / 𝐷) ∈ ℂ)
10299, 101, 59subdird 10934 . . . . . . . . . . . . . 14 (𝜑 → ((1 − (1 / 𝐷)) · 𝑈) = ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)))
10359mulid2d 10494 . . . . . . . . . . . . . . 15 (𝜑 → (1 · 𝑈) = 𝑈)
10459, 70, 71divrec2d 11257 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑈 / 𝐷) = ((1 / 𝐷) · 𝑈))
1059, 104syl5req 2842 . . . . . . . . . . . . . . 15 (𝜑 → ((1 / 𝐷) · 𝑈) = 𝐸)
106103, 105oveq12d 7025 . . . . . . . . . . . . . 14 (𝜑 → ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)) = (𝑈𝐸))
107102, 106eqtr2d 2830 . . . . . . . . . . . . 13 (𝜑 → (𝑈𝐸) = ((1 − (1 / 𝐷)) · 𝑈))
108107oveq1d 7022 . . . . . . . . . . . 12 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
1096oveq1i 7017 . . . . . . . . . . . . 13 (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈)
11099, 101subcld 10834 . . . . . . . . . . . . . 14 (𝜑 → (1 − (1 / 𝐷)) ∈ ℂ)
11175, 79rpdivcld 12287 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+)
112111rpcnd 12272 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℂ)
113110, 112, 59mul32d 10686 . . . . . . . . . . . . 13 (𝜑 → (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
114109, 113syl5eq 2841 . . . . . . . . . . . 12 (𝜑 → (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
115108, 114eqtr4d 2832 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (𝐹 · 𝑈))
116115oveq1d 7022 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝐹 · 𝑈) · (𝑈↑2)))
11735rpcnd 12272 . . . . . . . . . . 11 (𝜑 → (𝑈𝐸) ∈ ℂ)
118117, 112, 77mulassd 10499 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
119116, 118eqtr3d 2831 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12095, 98, 1193eqtr2d 2835 . . . . . . . 8 (𝜑 → (𝐹 · (𝑈↑3)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12186, 120eqtr4d 2832 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = (𝐹 · (𝑈↑3)))
122121oveq2d 7023 . . . . . 6 (𝜑 → (𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) = (𝑈 − (𝐹 · (𝑈↑3))))
123122oveq1d 7022 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12462, 123eqtr3d 2831 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12526, 24remulcld 10506 . . . . 5 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℝ)
126125, 54resubcld 10905 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
127124, 126eqeltrrd 2882 . . 3 (𝜑 → ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)) ∈ ℝ)
12817rpred 12270 . . . . . . . 8 (𝜑𝑍 ∈ ℝ)
12916simp2d 1134 . . . . . . . . 9 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
130129simp1d 1133 . . . . . . . 8 (𝜑 → 1 < 𝑍)
131128, 130rplogcld 24881 . . . . . . 7 (𝜑 → (log‘𝑍) ∈ ℝ+)
13232, 131rerpdivcld 12301 . . . . . 6 (𝜑 → (2 / (log‘𝑍)) ∈ ℝ)
133 fzfid 13179 . . . . . . 7 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
13417adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
135 elfznn 12775 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
136135adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
137136nnrpd 12268 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
138134, 137rpdivcld 12287 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
13918ffvelrni 6706 . . . . . . . . . . . 12 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
140138, 139syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
141140, 134rerpdivcld 12301 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
142141recnd 10504 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
143142abscld 14618 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
144137relogcld 24875 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
145143, 144remulcld 10506 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
146133, 145fsumrecl 14912 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
147132, 146remulcld 10506 . . . . 5 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
148147, 57readdcld 10505 . . . 4 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ∈ ℝ)
14920recnd 10504 . . . . . . . . . . 11 (𝜑 → (𝑅𝑍) ∈ ℂ)
150149abscld 14618 . . . . . . . . . 10 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℝ)
151150recnd 10504 . . . . . . . . 9 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℂ)
152151, 61mulcld 10496 . . . . . . . 8 (𝜑 → ((abs‘(𝑅𝑍)) · (log‘𝑍)) ∈ ℂ)
153132recnd 10504 . . . . . . . . 9 (𝜑 → (2 / (log‘𝑍)) ∈ ℂ)
154140recnd 10504 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℂ)
155154abscld 14618 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℝ)
156155, 144remulcld 10506 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
157133, 156fsumrecl 14912 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
158157recnd 10504 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
159153, 158mulcld 10496 . . . . . . . 8 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
16017rpcnd 12272 . . . . . . . 8 (𝜑𝑍 ∈ ℂ)
16117rpne0d 12275 . . . . . . . 8 (𝜑𝑍 ≠ 0)
162152, 159, 160, 161divsubdird 11292 . . . . . . 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 11290 . . . . . . . . 9 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
164149, 160, 161absdivd 14637 . . . . . . . . . . 11 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / (abs‘𝑍)))
16517rprege0d 12277 . . . . . . . . . . . . 13 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
166 absid 14478 . . . . . . . . . . . . 13 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → (abs‘𝑍) = 𝑍)
167165, 166syl 17 . . . . . . . . . . . 12 (𝜑 → (abs‘𝑍) = 𝑍)
168167oveq2d 7023 . . . . . . . . . . 11 (𝜑 → ((abs‘(𝑅𝑍)) / (abs‘𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
169164, 168eqtrd 2829 . . . . . . . . . 10 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
170169oveq1d 7022 . . . . . . . . 9 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
171163, 170eqtr4d 2832 . . . . . . . 8 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)))
172153, 158, 160, 161divassd 11288 . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℂ)
174161adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ≠ 0)
175154, 173, 174absdivd 14637 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)))
176167adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘𝑍) = 𝑍)
177176oveq2d 7023 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
178175, 177eqtrd 2829 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
179178oveq1d 7022 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
180155recnd 10504 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℂ)
181144recnd 10504 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
18217rpcnne0d 12279 . . . . . . . . . . . . . . 15 (𝜑 → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
183182adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
184 div23 11154 . . . . . . . . . . . . . 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 1362 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
186179, 185eqtr4d 2832 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
187186sumeq2dv 14881 . . . . . . . . . . 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 10504 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
189133, 160, 188, 161fsumdivc 14962 . . . . . . . . . . 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 2832 . . . . . . . . . 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 7023 . . . . . . . . 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 2832 . . . . . . . 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 7025 . . . . . . 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 2829 . . . . . 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 6535 . . . . . . . . . . 11 (𝑧 = 𝑍 → (abs‘(𝑅𝑧)) = (abs‘(𝑅𝑍)))
196 fveq2 6530 . . . . . . . . . . 11 (𝑧 = 𝑍 → (log‘𝑧) = (log‘𝑍))
197195, 196oveq12d 7025 . . . . . . . . . 10 (𝑧 = 𝑍 → ((abs‘(𝑅𝑧)) · (log‘𝑧)) = ((abs‘(𝑅𝑍)) · (log‘𝑍)))
198196oveq2d 7023 . . . . . . . . . . 11 (𝑧 = 𝑍 → (2 / (log‘𝑧)) = (2 / (log‘𝑍)))
199 oveq2 7015 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑧 / 𝑖) = (𝑧 / 𝑛))
200199fveq2d 6534 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (𝑅‘(𝑧 / 𝑖)) = (𝑅‘(𝑧 / 𝑛)))
201200fveq2d 6534 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → (abs‘(𝑅‘(𝑧 / 𝑖))) = (abs‘(𝑅‘(𝑧 / 𝑛))))
202 fveq2 6530 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → (log‘𝑖) = (log‘𝑛))
203201, 202oveq12d 7025 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → ((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)))
204203cbvsumv 14874 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛))
205 fvoveq1 7030 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (⌊‘(𝑧 / 𝑌)) = (⌊‘(𝑍 / 𝑌)))
206205oveq2d 7023 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (1...(⌊‘(𝑧 / 𝑌))) = (1...(⌊‘(𝑍 / 𝑌))))
207 simpl 483 . . . . . . . . . . . . . . . 16 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑧 = 𝑍)
208207fvoveq1d 7029 . . . . . . . . . . . . . . 15 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑧 / 𝑛)) = (𝑅‘(𝑍 / 𝑛)))
209208fveq2d 6534 . . . . . . . . . . . . . 14 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑧 / 𝑛))) = (abs‘(𝑅‘(𝑍 / 𝑛))))
210209oveq1d 7022 . . . . . . . . . . . . 13 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
211206, 210sumeq12rdv 14885 . . . . . . . . . . . 12 (𝑧 = 𝑍 → Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
212204, 211syl5eq 2841 . . . . . . . . . . 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 7025 . . . . . . . . . 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 7025 . . . . . . . . 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 7025 . . . . . . . 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 4966 . . . . . . 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 10476 . . . . . . . . 9 1 ∈ ℝ
220 rexr 10522 . . . . . . . . 9 (1 ∈ ℝ → 1 ∈ ℝ*)
221 elioopnf 12670 . . . . . . . . 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 3560 . . . . . 6 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) ≤ 𝐶)
225194, 224eqbrtrrd 4980 . . . . 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 11076 . . . . 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 233 . . . 4 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶))
228 2cnd 11552 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
229143recnd 10504 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℂ)
230229, 181mulcld 10496 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
231133, 230fsumcl 14911 . . . . . . 7 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
232131rpne0d 12275 . . . . . . 7 (𝜑 → (log‘𝑍) ≠ 0)
233228, 231, 61, 232div23d 11290 . . . . . 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 13449 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
23552, 234remulcld 10506 . . . . . . . . . . 11 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
23636, 235remulcld 10506 . . . . . . . . . 10 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
237 remulcl 10457 . . . . . . . . . 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 10506 . . . . . . . . 9 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) ∈ ℝ)
240 remulcl 10457 . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
243242, 136nndivred 11528 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
244243, 143resubcld 10905 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
245244, 144remulcld 10506 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
246133, 245fsumrecl 14912 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
24732, 246remulcld 10506 . . . . . . . . . 10 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ∈ ℝ)
248239, 241resubcld 10905 . . . . . . . . . 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 25851 . . . . . . . . . . 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 11577 . . . . . . . . . . . . 13 0 < 2
255254a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 2)
256 lemul2 11330 . . . . . . . . . . . 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 1365 . . . . . . . . . . 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 233 . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℂ)
260259, 229, 181subdird 10934 . . . . . . . . . . . . . . 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 14881 . . . . . . . . . . . . . 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 10506 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
263262recnd 10504 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
264133, 263, 230fsumsub 14964 . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . 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 7023 . . . . . . . . . . . 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 14912 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
268267recnd 10504 . . . . . . . . . . . . 13 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
269228, 268, 231subdid 10933 . . . . . . . . . . . 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 2829 . . . . . . . . . . 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 10457 . . . . . . . . . . . . 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 25852 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
274272, 239, 241, 273lesub1dd 11093 . . . . . . . . . . 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 4978 . . . . . . . . . 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 10633 . . . . . . . . 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 11081 . . . . . . . 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 10504 . . . . . . . . . 10 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℂ)
27955recnd 10504 . . . . . . . . . 10 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℂ)
280278, 279, 61subdird 10934 . . . . . . . . 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 10504 . . . . . . . . . . . 12 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℂ)
282228, 281, 61mulassd 10499 . . . . . . . . . . 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 10499 . . . . . . . . . . . . 13 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
28461sqvald 13345 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
285284oveq2d 7023 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
28651rpcnd 12272 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
287234recnd 10504 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
288117, 286, 287mulassd 10499 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
289283, 285, 2883eqtr2d 2835 . . . . . . . . . . . 12 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
290289oveq2d 7023 . . . . . . . . . . 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 2829 . . . . . . . . . 10 (𝜑 → ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))))
292291oveq2d 7023 . . . . . . . . 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 2829 . . . . . . . 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 4984 . . . . . . 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 12324 . . . . . . 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 258 . . . . . 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 4980 . . . . 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 11091 . . . 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 10633 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
300 remulcl 10457 . . . . . . . . 9 ((𝑈 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑈 · 3) ∈ ℝ)
30126, 27, 300sylancl 586 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℝ)
302301, 57readdcld 10505 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ)
30316simp3d 1135 . . . . . . . 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 1135 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))
305302, 54, 125, 304leadd2dd 11092 . . . . . 6 (𝜑 → ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)) ≤ ((𝑈 · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
30628recnd 10504 . . . . . . . . 9 (𝜑 → 3 ∈ ℂ)
30759, 61, 306adddid 10500 . . . . . . . 8 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) = ((𝑈 · (log‘𝑍)) + (𝑈 · 3)))
308307oveq1d 7022 . . . . . . 7 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶))
309125recnd 10504 . . . . . . . 8 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℂ)
31059, 306mulcld 10496 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℂ)
31113rpcnd 12272 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
312309, 310, 311addassd 10498 . . . . . . 7 (𝜑 → (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
313308, 312eqtrd 2829 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
3142812timesd 11717 . . . . . . . 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 7023 . . . . . . 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 10861 . . . . . . 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 2829 . . . . . 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 4988 . . . . 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 10505 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ∈ ℝ)
320319, 55, 126lesubaddd 11074 . . . . 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 258 . . . 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 10855 . . . 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 4987 . . 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 10633 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
325 3z 11853 . . . . . . 7 3 ∈ ℤ
326 rpexpcl 13286 . . . . . . 7 ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℤ) → (𝑈↑3) ∈ ℝ+)
3277, 325, 326sylancl 586 . . . . . 6 (𝜑 → (𝑈↑3) ∈ ℝ+)
32896, 327rpmulcld 12286 . . . . 5 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ+)
329328rpred 12270 . . . 4 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ)
33026, 329resubcld 10905 . . 3 (𝜑 → (𝑈 − (𝐹 · (𝑈↑3))) ∈ ℝ)
33123, 330, 131lemul1d 12313 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))) ↔ ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍))))
332324, 331mpbird 258 1 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1078   = wceq 1520  wcel 2079  wne 2982  wral 3103  wrex 3104   class class class wbr 4956  cmpt 5035  cfv 6217  (class class class)co 7007  cc 10370  cr 10371  0cc0 10372  1c1 10373   + caddc 10375   · cmul 10377  +∞cpnf 10507  *cxr 10509   < clt 10510  cle 10511  cmin 10706   / cdiv 11134  cn 11475  2c2 11529  3c3 11530  4c4 11531  0cn0 11734  cz 11818  cdc 11936  +crp 12228  (,)cioo 12577  [,)cico 12579  [,]cicc 12580  ...cfz 12731  cfl 12998  cexp 13267  csqrt 14414  abscabs 14415  Σcsu 14864  expce 15236  eceu 15237  logclog 24807  ψcchp 25340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-inf2 8939  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450  ax-addf 10451  ax-mulf 10452
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-of 7258  df-om 7428  df-1st 7536  df-2nd 7537  df-supp 7673  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-map 8249  df-pm 8250  df-ixp 8301  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-fsupp 8670  df-fi 8711  df-sup 8742  df-inf 8743  df-oi 8810  df-dju 9165  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-xnn0 11805  df-z 11819  df-dec 11937  df-uz 12083  df-q 12187  df-rp 12229  df-xneg 12346  df-xadd 12347  df-xmul 12348  df-ioo 12581  df-ioc 12582  df-ico 12583  df-icc 12584  df-fz 12732  df-fzo 12873  df-fl 13000  df-mod 13076  df-seq 13208  df-exp 13268  df-fac 13472  df-bc 13501  df-hash 13529  df-shft 14248  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-limsup 14650  df-clim 14667  df-rlim 14668  df-sum 14865  df-ef 15242  df-e 15243  df-sin 15244  df-cos 15245  df-tan 15246  df-pi 15247  df-dvds 15429  df-gcd 15665  df-prm 15833  df-pc 15991  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-sets 16307  df-ress 16308  df-plusg 16395  df-mulr 16396  df-starv 16397  df-sca 16398  df-vsca 16399  df-ip 16400  df-tset 16401  df-ple 16402  df-ds 16404  df-unif 16405  df-hom 16406  df-cco 16407  df-rest 16513  df-topn 16514  df-0g 16532  df-gsum 16533  df-topgen 16534  df-pt 16535  df-prds 16538  df-xrs 16592  df-qtop 16597  df-imas 16598  df-xps 16600  df-mre 16674  df-mrc 16675  df-acs 16677  df-mgm 17669  df-sgrp 17711  df-mnd 17722  df-submnd 17763  df-mulg 17970  df-cntz 18176  df-cmn 18623  df-psmet 20207  df-xmet 20208  df-met 20209  df-bl 20210  df-mopn 20211  df-fbas 20212  df-fg 20213  df-cnfld 20216  df-top 21174  df-topon 21191  df-topsp 21213  df-bases 21226  df-cld 21299  df-ntr 21300  df-cls 21301  df-nei 21378  df-lp 21416  df-perf 21417  df-cn 21507  df-cnp 21508  df-haus 21595  df-cmp 21667  df-tx 21842  df-hmeo 22035  df-fil 22126  df-fm 22218  df-flim 22219  df-flf 22220  df-xms 22601  df-ms 22602  df-tms 22603  df-cncf 23157  df-limc 24135  df-dv 24136  df-ulm 24636  df-log 24809  df-atan 25114  df-em 25240  df-vma 25345  df-chp 25346
This theorem is referenced by:  pntleme  25854
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