MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemo Structured version   Visualization version   GIF version

Theorem pntlemo 25013
Description: Lemma for pnt 25020. 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 25003 . . . . . . . . 9 (𝜑 → (𝑍 ∈ ℝ+ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))))
1716simp1d 1065 . . . . . . . 8 (𝜑𝑍 ∈ ℝ+)
181pntrf 24969 . . . . . . . . 9 𝑅:ℝ+⟶ℝ
1918ffvelrni 6251 . . . . . . . 8 (𝑍 ∈ ℝ+ → (𝑅𝑍) ∈ ℝ)
2017, 19syl 17 . . . . . . 7 (𝜑 → (𝑅𝑍) ∈ ℝ)
2120, 17rerpdivcld 11735 . . . . . 6 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℝ)
2221recnd 9924 . . . . 5 (𝜑 → ((𝑅𝑍) / 𝑍) ∈ ℂ)
2322abscld 13969 . . . 4 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ∈ ℝ)
2417relogcld 24090 . . . 4 (𝜑 → (log‘𝑍) ∈ ℝ)
2523, 24remulcld 9926 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ∈ ℝ)
267rpred 11704 . . . . . 6 (𝜑𝑈 ∈ ℝ)
27 3re 10941 . . . . . . . 8 3 ∈ ℝ
2827a1i 11 . . . . . . 7 (𝜑 → 3 ∈ ℝ)
2924, 28readdcld 9925 . . . . . 6 (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
3026, 29remulcld 9926 . . . . 5 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℝ)
31 2re 10937 . . . . . . 7 2 ∈ ℝ
3231a1i 11 . . . . . 6 (𝜑 → 2 ∈ ℝ)
331, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 25001 . . . . . . . . . . 11 (𝜑 → (𝐸 ∈ ℝ+𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+)))
3433simp3d 1067 . . . . . . . . . 10 (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈𝐸) ∈ ℝ+))
3534simp3d 1067 . . . . . . . . 9 (𝜑 → (𝑈𝐸) ∈ ℝ+)
3635rpred 11704 . . . . . . . 8 (𝜑 → (𝑈𝐸) ∈ ℝ)
371, 2, 3, 4, 5, 6pntlemd 25000 . . . . . . . . . . . 12 (𝜑 → (𝐿 ∈ ℝ+𝐷 ∈ ℝ+𝐹 ∈ ℝ+))
3837simp1d 1065 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℝ+)
3933simp1d 1065 . . . . . . . . . . . 12 (𝜑𝐸 ∈ ℝ+)
40 2z 11242 . . . . . . . . . . . 12 2 ∈ ℤ
41 rpexpcl 12696 . . . . . . . . . . . 12 ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+)
4239, 40, 41sylancl 692 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℝ+)
4338, 42rpmulcld 11720 . . . . . . . . . 10 (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+)
44 3nn0 11157 . . . . . . . . . . . . 13 3 ∈ ℕ0
45 2nn 11032 . . . . . . . . . . . . 13 2 ∈ ℕ
4644, 45decnncl 11350 . . . . . . . . . . . 12 32 ∈ ℕ
47 nnrp 11674 . . . . . . . . . . . 12 (32 ∈ ℕ → 32 ∈ ℝ+)
4846, 47ax-mp 5 . . . . . . . . . . 11 32 ∈ ℝ+
49 rpmulcl 11687 . . . . . . . . . . 11 ((32 ∈ ℝ+𝐵 ∈ ℝ+) → (32 · 𝐵) ∈ ℝ+)
5048, 3, 49sylancr 693 . . . . . . . . . 10 (𝜑 → (32 · 𝐵) ∈ ℝ+)
5143, 50rpdivcld 11721 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ+)
5251rpred 11704 . . . . . . . 8 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℝ)
5336, 52remulcld 9926 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℝ)
5453, 24remulcld 9926 . . . . . 6 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℝ)
5532, 54remulcld 9926 . . . . 5 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
5630, 55resubcld 10309 . . . 4 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) ∈ ℝ)
5713rpred 11704 . . . 4 (𝜑𝐶 ∈ ℝ)
5856, 57readdcld 9925 . . 3 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ∈ ℝ)
597rpcnd 11706 . . . . . 6 (𝜑𝑈 ∈ ℂ)
6053recnd 9924 . . . . . 6 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) ∈ ℂ)
6124recnd 9924 . . . . . 6 (𝜑 → (log‘𝑍) ∈ ℂ)
6259, 60, 61subdird 10337 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
6338rpcnd 11706 . . . . . . . . . . 11 (𝜑𝐿 ∈ ℂ)
6442rpcnd 11706 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) ∈ ℂ)
6550rpcnne0d 11713 . . . . . . . . . . 11 (𝜑 → ((32 · 𝐵) ∈ ℂ ∧ (32 · 𝐵) ≠ 0))
66 div23 10553 . . . . . . . . . . 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 1317 . . . . . . . . . 10 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = ((𝐿 / (32 · 𝐵)) · (𝐸↑2)))
689oveq1i 6537 . . . . . . . . . . . 12 (𝐸↑2) = ((𝑈 / 𝐷)↑2)
6937simp2d 1066 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ ℝ+)
7069rpcnd 11706 . . . . . . . . . . . . 13 (𝜑𝐷 ∈ ℂ)
7169rpne0d 11709 . . . . . . . . . . . . 13 (𝜑𝐷 ≠ 0)
7259, 70, 71sqdivd 12838 . . . . . . . . . . . 12 (𝜑 → ((𝑈 / 𝐷)↑2) = ((𝑈↑2) / (𝐷↑2)))
7368, 72syl5eq 2655 . . . . . . . . . . 11 (𝜑 → (𝐸↑2) = ((𝑈↑2) / (𝐷↑2)))
7473oveq2d 6543 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · (𝐸↑2)) = ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))))
7538, 50rpdivcld 11721 . . . . . . . . . . . 12 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℝ+)
7675rpcnd 11706 . . . . . . . . . . 11 (𝜑 → (𝐿 / (32 · 𝐵)) ∈ ℂ)
7759sqcld 12823 . . . . . . . . . . 11 (𝜑 → (𝑈↑2) ∈ ℂ)
78 rpexpcl 12696 . . . . . . . . . . . . 13 ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+)
7969, 40, 78sylancl 692 . . . . . . . . . . . 12 (𝜑 → (𝐷↑2) ∈ ℝ+)
8079rpcnne0d 11713 . . . . . . . . . . 11 (𝜑 → ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0))
81 divass 10552 . . . . . . . . . . . 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 10553 . . . . . . . . . . . 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 2645 . . . . . . . . . . 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 1317 . . . . . . . . . 10 (𝜑 → ((𝐿 / (32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8567, 74, 843eqtrd 2647 . . . . . . . . 9 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) = (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
8685oveq2d 6543 . . . . . . . 8 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
87 df-3 10927 . . . . . . . . . . . . 13 3 = (2 + 1)
8887oveq2i 6538 . . . . . . . . . . . 12 (𝑈↑3) = (𝑈↑(2 + 1))
89 2nn0 11156 . . . . . . . . . . . . 13 2 ∈ ℕ0
90 expp1 12684 . . . . . . . . . . . . 13 ((𝑈 ∈ ℂ ∧ 2 ∈ ℕ0) → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9159, 89, 90sylancl 692 . . . . . . . . . . . 12 (𝜑 → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
9288, 91syl5eq 2655 . . . . . . . . . . 11 (𝜑 → (𝑈↑3) = ((𝑈↑2) · 𝑈))
9377, 59mulcomd 9917 . . . . . . . . . . 11 (𝜑 → ((𝑈↑2) · 𝑈) = (𝑈 · (𝑈↑2)))
9492, 93eqtrd 2643 . . . . . . . . . 10 (𝜑 → (𝑈↑3) = (𝑈 · (𝑈↑2)))
9594oveq2d 6543 . . . . . . . . 9 (𝜑 → (𝐹 · (𝑈↑3)) = (𝐹 · (𝑈 · (𝑈↑2))))
9637simp3d 1067 . . . . . . . . . . 11 (𝜑𝐹 ∈ ℝ+)
9796rpcnd 11706 . . . . . . . . . 10 (𝜑𝐹 ∈ ℂ)
9897, 59, 77mulassd 9919 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = (𝐹 · (𝑈 · (𝑈↑2))))
99 1cnd 9912 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℂ)
10069rpreccld 11714 . . . . . . . . . . . . . . . 16 (𝜑 → (1 / 𝐷) ∈ ℝ+)
101100rpcnd 11706 . . . . . . . . . . . . . . 15 (𝜑 → (1 / 𝐷) ∈ ℂ)
10299, 101, 59subdird 10337 . . . . . . . . . . . . . 14 (𝜑 → ((1 − (1 / 𝐷)) · 𝑈) = ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)))
10359mulid2d 9914 . . . . . . . . . . . . . . 15 (𝜑 → (1 · 𝑈) = 𝑈)
10459, 70, 71divrec2d 10654 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑈 / 𝐷) = ((1 / 𝐷) · 𝑈))
1059, 104syl5req 2656 . . . . . . . . . . . . . . 15 (𝜑 → ((1 / 𝐷) · 𝑈) = 𝐸)
106103, 105oveq12d 6545 . . . . . . . . . . . . . 14 (𝜑 → ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)) = (𝑈𝐸))
107102, 106eqtr2d 2644 . . . . . . . . . . . . 13 (𝜑 → (𝑈𝐸) = ((1 − (1 / 𝐷)) · 𝑈))
108107oveq1d 6542 . . . . . . . . . . . 12 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
1096oveq1i 6537 . . . . . . . . . . . . 13 (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈)
11099, 101subcld 10243 . . . . . . . . . . . . . 14 (𝜑 → (1 − (1 / 𝐷)) ∈ ℂ)
11175, 79rpdivcld 11721 . . . . . . . . . . . . . . 15 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+)
112111rpcnd 11706 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 / (32 · 𝐵)) / (𝐷↑2)) ∈ ℂ)
113110, 112, 59mul32d 10097 . . . . . . . . . . . . 13 (𝜑 → (((1 − (1 / 𝐷)) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
114109, 113syl5eq 2655 . . . . . . . . . . . 12 (𝜑 → (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))))
115108, 114eqtr4d 2646 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) = (𝐹 · 𝑈))
116115oveq1d 6542 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝐹 · 𝑈) · (𝑈↑2)))
11735rpcnd 11706 . . . . . . . . . . 11 (𝜑 → (𝑈𝐸) ∈ ℂ)
118117, 112, 77mulassd 9919 . . . . . . . . . 10 (𝜑 → (((𝑈𝐸) · ((𝐿 / (32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
119116, 118eqtr3d 2645 . . . . . . . . 9 (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12095, 98, 1193eqtr2d 2649 . . . . . . . 8 (𝜑 → (𝐹 · (𝑈↑3)) = ((𝑈𝐸) · (((𝐿 / (32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
12186, 120eqtr4d 2646 . . . . . . 7 (𝜑 → ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) = (𝐹 · (𝑈↑3)))
122121oveq2d 6543 . . . . . 6 (𝜑 → (𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) = (𝑈 − (𝐹 · (𝑈↑3))))
123122oveq1d 6542 . . . . 5 (𝜑 → ((𝑈 − ((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12462, 123eqtr3d 2645 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
12526, 24remulcld 9926 . . . . 5 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℝ)
126125, 54resubcld 10309 . . . 4 (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
127124, 126eqeltrrd 2688 . . 3 (𝜑 → ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)) ∈ ℝ)
12817rpred 11704 . . . . . . . 8 (𝜑𝑍 ∈ ℝ)
12916simp2d 1066 . . . . . . . . 9 (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
130129simp1d 1065 . . . . . . . 8 (𝜑 → 1 < 𝑍)
131128, 130rplogcld 24096 . . . . . . 7 (𝜑 → (log‘𝑍) ∈ ℝ+)
13232, 131rerpdivcld 11735 . . . . . 6 (𝜑 → (2 / (log‘𝑍)) ∈ ℝ)
133 fzfid 12589 . . . . . . 7 (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
13417adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ+)
135 elfznn 12196 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
136135adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
137136nnrpd 11702 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ+)
138134, 137rpdivcld 11721 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ+)
13918ffvelrni 6251 . . . . . . . . . . . 12 ((𝑍 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
140138, 139syl 17 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
141140, 134rerpdivcld 11735 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
142141recnd 9924 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
143142abscld 13969 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
144137relogcld 24090 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
145143, 144remulcld 9926 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
146133, 145fsumrecl 14258 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
147132, 146remulcld 9926 . . . . 5 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
148147, 57readdcld 9925 . . . 4 (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ∈ ℝ)
14920recnd 9924 . . . . . . . . . . 11 (𝜑 → (𝑅𝑍) ∈ ℂ)
150149abscld 13969 . . . . . . . . . 10 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℝ)
151150recnd 9924 . . . . . . . . 9 (𝜑 → (abs‘(𝑅𝑍)) ∈ ℂ)
152151, 61mulcld 9916 . . . . . . . 8 (𝜑 → ((abs‘(𝑅𝑍)) · (log‘𝑍)) ∈ ℂ)
153132recnd 9924 . . . . . . . . 9 (𝜑 → (2 / (log‘𝑍)) ∈ ℂ)
154140recnd 9924 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℂ)
155154abscld 13969 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℝ)
156155, 144remulcld 9926 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
157133, 156fsumrecl 14258 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
158157recnd 9924 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
159153, 158mulcld 9916 . . . . . . . 8 (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
16017rpcnd 11706 . . . . . . . 8 (𝜑𝑍 ∈ ℂ)
16117rpne0d 11709 . . . . . . . 8 (𝜑𝑍 ≠ 0)
162152, 159, 160, 161divsubdird 10689 . . . . . . 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 10687 . . . . . . . . 9 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
164149, 160, 161absdivd 13988 . . . . . . . . . . 11 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / (abs‘𝑍)))
16517rprege0d 11711 . . . . . . . . . . . . 13 (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
166 absid 13830 . . . . . . . . . . . . 13 ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → (abs‘𝑍) = 𝑍)
167165, 166syl 17 . . . . . . . . . . . 12 (𝜑 → (abs‘𝑍) = 𝑍)
168167oveq2d 6543 . . . . . . . . . . 11 (𝜑 → ((abs‘(𝑅𝑍)) / (abs‘𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
169164, 168eqtrd 2643 . . . . . . . . . 10 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) = ((abs‘(𝑅𝑍)) / 𝑍))
170169oveq1d 6542 . . . . . . . . 9 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) = (((abs‘(𝑅𝑍)) / 𝑍) · (log‘𝑍)))
171163, 170eqtr4d 2646 . . . . . . . 8 (𝜑 → (((abs‘(𝑅𝑍)) · (log‘𝑍)) / 𝑍) = ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)))
172153, 158, 160, 161divassd 10685 . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℂ)
174161adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ≠ 0)
175154, 173, 174absdivd 13988 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)))
176167adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘𝑍) = 𝑍)
177176oveq2d 6543 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
178175, 177eqtrd 2643 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
179178oveq1d 6542 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
180155recnd 9924 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℂ)
181144recnd 9924 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
18217rpcnne0d 11713 . . . . . . . . . . . . . . 15 (𝜑 → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
183182adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
184 div23 10553 . . . . . . . . . . . . . 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 1317 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
186179, 185eqtr4d 2646 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
187186sumeq2dv 14227 . . . . . . . . . . 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 9924 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
189133, 160, 188, 161fsumdivc 14306 . . . . . . . . . . 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 2646 . . . . . . . . . 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 6543 . . . . . . . . 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 2646 . . . . . . . 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 6545 . . . . . . 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 2643 . . . . . 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 1re 9895 . . . . . . . . 9 1 ∈ ℝ
196 rexr 9941 . . . . . . . . 9 (1 ∈ ℝ → 1 ∈ ℝ*)
197 elioopnf 12094 . . . . . . . . 9 (1 ∈ ℝ* → (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍)))
198195, 196, 197mp2b 10 . . . . . . . 8 (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍))
199128, 130, 198sylanbrc 694 . . . . . . 7 (𝜑𝑍 ∈ (1(,)+∞))
200 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)
201 fveq2 6088 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → (𝑅𝑧) = (𝑅𝑍))
202201fveq2d 6092 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (abs‘(𝑅𝑧)) = (abs‘(𝑅𝑍)))
203 fveq2 6088 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (log‘𝑧) = (log‘𝑍))
204202, 203oveq12d 6545 . . . . . . . . . . 11 (𝑧 = 𝑍 → ((abs‘(𝑅𝑧)) · (log‘𝑧)) = ((abs‘(𝑅𝑍)) · (log‘𝑍)))
205203oveq2d 6543 . . . . . . . . . . . 12 (𝑧 = 𝑍 → (2 / (log‘𝑧)) = (2 / (log‘𝑍)))
206 oveq2 6535 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑛 → (𝑧 / 𝑖) = (𝑧 / 𝑛))
207206fveq2d 6092 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑅‘(𝑧 / 𝑖)) = (𝑅‘(𝑧 / 𝑛)))
208207fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (abs‘(𝑅‘(𝑧 / 𝑖))) = (abs‘(𝑅‘(𝑧 / 𝑛))))
209 fveq2 6088 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (log‘𝑖) = (log‘𝑛))
210208, 209oveq12d 6545 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → ((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)))
211210cbvsumv 14220 . . . . . . . . . . . . 13 Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛))
212 oveq1 6534 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑍 → (𝑧 / 𝑌) = (𝑍 / 𝑌))
213212fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑧 = 𝑍 → (⌊‘(𝑧 / 𝑌)) = (⌊‘(𝑍 / 𝑌)))
214213oveq2d 6543 . . . . . . . . . . . . . 14 (𝑧 = 𝑍 → (1...(⌊‘(𝑧 / 𝑌))) = (1...(⌊‘(𝑍 / 𝑌))))
215 simpl 471 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑧 = 𝑍)
216215oveq1d 6542 . . . . . . . . . . . . . . . . 17 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑧 / 𝑛) = (𝑍 / 𝑛))
217216fveq2d 6092 . . . . . . . . . . . . . . . 16 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑧 / 𝑛)) = (𝑅‘(𝑍 / 𝑛)))
218217fveq2d 6092 . . . . . . . . . . . . . . 15 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑧 / 𝑛))) = (abs‘(𝑅‘(𝑍 / 𝑛))))
219218oveq1d 6542 . . . . . . . . . . . . . 14 ((𝑧 = 𝑍𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
220214, 219sumeq12rdv 14231 . . . . . . . . . . . . 13 (𝑧 = 𝑍 → Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
221211, 220syl5eq 2655 . . . . . . . . . . . 12 (𝑧 = 𝑍 → Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
222205, 221oveq12d 6545 . . . . . . . . . . 11 (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))))
223204, 222oveq12d 6545 . . . . . . . . . 10 (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)))))
224 id 22 . . . . . . . . . 10 (𝑧 = 𝑍𝑧 = 𝑍)
225223, 224oveq12d 6545 . . . . . . . . 9 (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))
226225breq1d 4587 . . . . . . . 8 (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))
227226rspcv 3277 . . . . . . 7 (𝑍 ∈ (1(,)+∞) → (∀𝑧 ∈ (1(,)+∞)((((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))
228199, 200, 227sylc 62 . . . . . 6 (𝜑 → ((((abs‘(𝑅𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) ≤ 𝐶)
229194, 228eqbrtrrd 4601 . . . . 5 (𝜑 → (((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ 𝐶)
23025, 147, 57lesubadd2d 10475 . . . . 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)))
231229, 230mpbid 220 . . . 4 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶))
232 2cnd 10940 . . . . . . 7 (𝜑 → 2 ∈ ℂ)
233143recnd 9924 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℂ)
234233, 181mulcld 9916 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
235133, 234fsumcl 14257 . . . . . . 7 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
236131rpne0d 11709 . . . . . . 7 (𝜑 → (log‘𝑍) ≠ 0)
237232, 235, 61, 236div23d 10687 . . . . . 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))))
23824resqcld 12852 . . . . . . . . . . . 12 (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
23952, 238remulcld 9926 . . . . . . . . . . 11 (𝜑 → (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
24036, 239remulcld 9926 . . . . . . . . . 10 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
241 remulcl 9877 . . . . . . . . . 10 ((2 ∈ ℝ ∧ ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ) → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
24231, 240, 241sylancr 693 . . . . . . . . 9 (𝜑 → (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
24330, 24remulcld 9926 . . . . . . . . 9 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) ∈ ℝ)
244 remulcl 9877 . . . . . . . . . 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)
24531, 146, 244sylancr 693 . . . . . . . . 9 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
24626adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
247246, 136nndivred 10916 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
248247, 143resubcld 10309 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
249248, 144remulcld 9926 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
250133, 249fsumrecl 14258 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
25132, 250remulcld 9926 . . . . . . . . . 10 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ∈ ℝ)
252243, 245resubcld 10309 . . . . . . . . . 10 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ∈ ℝ)
253 pntlem1.m . . . . . . . . . . . 12 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
254 pntlem1.n . . . . . . . . . . . 12 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
255 pntlem1.U . . . . . . . . . . . 12 (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅𝑧) / 𝑧)) ≤ 𝑈)
256 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))
2571, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 253, 254, 255, 256pntlemf 25011 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
258 2pos 10959 . . . . . . . . . . . . 13 0 < 2
259258a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 2)
260 lemul2 10725 . . . . . . . . . . . 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)))))
261240, 250, 32, 259, 260syl112anc 1321 . . . . . . . . . . 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)))))
262257, 261mpbid 220 . . . . . . . . . 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))))
263247recnd 9924 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℂ)
264263, 233, 181subdird 10337 . . . . . . . . . . . . . . 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))))
265264sumeq2dv 14227 . . . . . . . . . . . . . 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))))
266247, 144remulcld 9926 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
267266recnd 9924 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
268133, 267, 234fsumsub 14308 . . . . . . . . . . . . . 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))))
269265, 268eqtrd 2643 . . . . . . . . . . . . 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))))
270269oveq2d 6543 . . . . . . . . . . . 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)))))
271133, 266fsumrecl 14258 . . . . . . . . . . . . . 14 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
272271recnd 9924 . . . . . . . . . . . . 13 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
273232, 272, 235subdid 10336 . . . . . . . . . . . 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)))))
274270, 273eqtrd 2643 . . . . . . . . . . 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)))))
275 remulcl 9877 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
27631, 271, 275sylancr 693 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
2771, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 253, 254, 255, 256pntlemk 25012 . . . . . . . . . . . 12 (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
278276, 243, 245, 277lesub1dd 10492 . . . . . . . . . . 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)))))
279274, 278eqbrtrd 4599 . . . . . . . . . 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)))))
280242, 251, 252, 262, 279letrd 10045 . . . . . . . . 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)))))
281242, 243, 245, 280lesubd 10480 . . . . . . . 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))))))
28230recnd 9924 . . . . . . . . . 10 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℂ)
28355recnd 9924 . . . . . . . . . 10 (𝜑 → (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) ∈ ℂ)
284282, 283, 61subdird 10337 . . . . . . . . 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))))
28554recnd 9924 . . . . . . . . . . . 12 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) ∈ ℂ)
286232, 285, 61mulassd 9919 . . . . . . . . . . 11 (𝜑 → ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))))
28760, 61, 61mulassd 9919 . . . . . . . . . . . . 13 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
28861sqvald 12822 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
289288oveq2d 6543 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
29051rpcnd 11706 . . . . . . . . . . . . . 14 (𝜑 → ((𝐿 · (𝐸↑2)) / (32 · 𝐵)) ∈ ℂ)
291238recnd 9924 . . . . . . . . . . . . . 14 (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
292117, 290, 291mulassd 9919 . . . . . . . . . . . . 13 (𝜑 → (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · ((log‘𝑍)↑2)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
293287, 289, 2923eqtr2d 2649 . . . . . . . . . . . 12 (𝜑 → ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2))))
294293oveq2d 6543 . . . . . . . . . . 11 (𝜑 → (2 · ((((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))) = (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))))
295286, 294eqtrd 2643 . . . . . . . . . 10 (𝜑 → ((2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((𝑈𝐸) · (((𝐿 · (𝐸↑2)) / (32 · 𝐵)) · ((log‘𝑍)↑2)))))
296295oveq2d 6543 . . . . . . . . 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))))))
297284, 296eqtrd 2643 . . . . . . . 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))))))
298281, 297breqtrrd 4605 . . . . . . 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)))
299245, 56, 131ledivmul2d 11758 . . . . . . 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))))
300298, 299mpbird 245 . . . . . 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)))))
301237, 300eqbrtrrd 4601 . . . . 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)))))
302147, 56, 57, 301leadd1dd 10490 . . . 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))
30325, 148, 58, 231, 302letrd 10045 . . 3 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
304 remulcl 9877 . . . . . . . . 9 ((𝑈 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑈 · 3) ∈ ℝ)
30526, 27, 304sylancl 692 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℝ)
306305, 57readdcld 9925 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ)
30716simp3d 1067 . . . . . . . 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))))
308307simp3d 1067 . . . . . . 7 (𝜑 → ((𝑈 · 3) + 𝐶) ≤ (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))
309306, 54, 125, 308leadd2dd 10491 . . . . . 6 (𝜑 → ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)) ≤ ((𝑈 · (log‘𝑍)) + (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍))))
31028recnd 9924 . . . . . . . . 9 (𝜑 → 3 ∈ ℂ)
31159, 61, 310adddid 9920 . . . . . . . 8 (𝜑 → (𝑈 · ((log‘𝑍) + 3)) = ((𝑈 · (log‘𝑍)) + (𝑈 · 3)))
312311oveq1d 6542 . . . . . . 7 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶))
313125recnd 9924 . . . . . . . 8 (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℂ)
31459, 310mulcld 9916 . . . . . . . 8 (𝜑 → (𝑈 · 3) ∈ ℂ)
31513rpcnd 11706 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
316313, 314, 315addassd 9918 . . . . . . 7 (𝜑 → (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
317312, 316eqtrd 2643 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
3182852timesd 11122 . . . . . . . 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))))
319318oveq2d 6543 . . . . . . 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)))))
320313, 285, 285nppcan3d 10270 . . . . . . 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))))
321319, 320eqtrd 2643 . . . . . 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))))
322309, 317, 3213brtr4d 4609 . . . . 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)))))
32330, 57readdcld 9925 . . . . . 6 (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ∈ ℝ)
324323, 55, 126lesubaddd 10473 . . . . 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))))))
325322, 324mpbird 245 . . . 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))))
326282, 315, 283addsubd 10264 . . . 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))
327325, 326, 1243brtr3d 4608 . . 3 (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈𝐸) · ((𝐿 · (𝐸↑2)) / (32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
32825, 58, 127, 303, 327letrd 10045 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
329 3z 11243 . . . . . . 7 3 ∈ ℤ
330 rpexpcl 12696 . . . . . . 7 ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℤ) → (𝑈↑3) ∈ ℝ+)
3317, 329, 330sylancl 692 . . . . . 6 (𝜑 → (𝑈↑3) ∈ ℝ+)
33296, 331rpmulcld 11720 . . . . 5 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ+)
333332rpred 11704 . . . 4 (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ)
33426, 333resubcld 10309 . . 3 (𝜑 → (𝑈 − (𝐹 · (𝑈↑3))) ∈ ℝ)
33523, 334, 131lemul1d 11747 . 2 (𝜑 → ((abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))) ↔ ((abs‘((𝑅𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍))))
336328, 335mpbird 245 1 (𝜑 → (abs‘((𝑅𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797  +∞cpnf 9927  *cxr 9929   < clt 9930  cle 9931  cmin 10117   / cdiv 10533  cn 10867  2c2 10917  3c3 10918  4c4 10919  0cn0 11139  cz 11210  cdc 11325  +crp 11664  (,)cioo 12002  [,)cico 12004  [,]cicc 12005  ...cfz 12152  cfl 12408  cexp 12677  csqrt 13767  abscabs 13768  Σcsu 14210  expce 14577  eceu 14578  logclog 24022  ψcchp 24536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ioc 12007  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-seq 12619  df-exp 12678  df-fac 12878  df-bc 12907  df-hash 12935  df-shft 13601  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-limsup 13996  df-clim 14013  df-rlim 14014  df-sum 14211  df-ef 14583  df-e 14584  df-sin 14585  df-cos 14586  df-pi 14588  df-dvds 14768  df-gcd 15001  df-prm 15170  df-pc 15326  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-sca 15730  df-vsca 15731  df-ip 15732  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-hom 15739  df-cco 15740  df-rest 15852  df-topn 15853  df-0g 15871  df-gsum 15872  df-topgen 15873  df-pt 15874  df-prds 15877  df-xrs 15931  df-qtop 15936  df-imas 15937  df-xps 15939  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-mulg 17310  df-cntz 17519  df-cmn 17964  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-cnfld 19514  df-top 20463  df-bases 20464  df-topon 20465  df-topsp 20466  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lp 20692  df-perf 20693  df-cn 20783  df-cnp 20784  df-haus 20871  df-tx 21117  df-hmeo 21310  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-xms 21876  df-ms 21877  df-tms 21878  df-cncf 22420  df-limc 23353  df-dv 23354  df-log 24024  df-em 24436  df-vma 24541  df-chp 24542
This theorem is referenced by:  pntleme  25014
  Copyright terms: Public domain W3C validator