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Theorem pntlemo 26955

Description: Lemma for pnt 26962. 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	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
pntlem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pntlem1.l	⊢ (𝜑 → 𝐿 ∈ (0(,)1))
pntlem1.d	⊢ 𝐷 = (𝐴 + 1)
pntlem1.f	⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
pntlem1.u	⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
pntlem1.u2	⊢ (𝜑 → 𝑈 ≤ 𝐴)
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	⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
pntlem1.w	⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
pntlem1.z	⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
pntlem1.m	⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
pntlem1.n	⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
pntlem1.U	⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
pntlem1.K	⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
pntlem1.C	⊢ (𝜑 → ∀𝑧 ∈ (1(,)+∞)((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)

**Assertion**
Ref	Expression
pntlemo	⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 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.

Step	Hyp	Ref	Expression
1		pntlem1.r	. . . . . . . . . 10 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
2		pntlem1.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		pntlem1.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
4		pntlem1.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (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 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
8		pntlem1.u2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
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 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
14		pntlem1.w	. . . . . . . . . 10 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
15		pntlem1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	pntlemb 26945	. . . . . . . . 9 ⊢ (𝜑 → (𝑍 ∈ ℝ⁺ ∧ (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)) ∧ ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
17	16	simp1d 1142	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
18	1	pntrf 26911	. . . . . . . . 9 ⊢ 𝑅:ℝ⁺⟶ℝ
19	18	ffvelcdmi 7034	. . . . . . . 8 ⊢ (𝑍 ∈ ℝ⁺ → (𝑅‘𝑍) ∈ ℝ)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝑍) ∈ ℝ)
21	20, 17	rerpdivcld 12988	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝑍) / 𝑍) ∈ ℝ)
22	21	recnd 11183	. . . . 5 ⊢ (𝜑 → ((𝑅‘𝑍) / 𝑍) ∈ ℂ)
23	22	abscld 15321	. . . 4 ⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 𝑍)) ∈ ℝ)
24	17	relogcld 25978	. . . 4 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ)
25	23, 24	remulcld 11185	. . 3 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ∈ ℝ)
26	7	rpred 12957	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
27		3re 12233	. . . . . . . 8 ⊢ 3 ∈ ℝ
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 3 ∈ ℝ)
29	24, 28	readdcld 11184	. . . . . 6 ⊢ (𝜑 → ((log‘𝑍) + 3) ∈ ℝ)
30	26, 29	remulcld 11185	. . . . 5 ⊢ (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℝ)
31		2re 12227	. . . . . . 7 ⊢ 2 ∈ ℝ
32	31	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
33	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 26943	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
34	33	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
35	34	simp3d 1144	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
36	35	rpred 12957	. . . . . . . 8 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ)
37	1, 2, 3, 4, 5, 6	pntlemd 26942	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
38	37	simp1d 1142	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
39	33	simp1d 1142	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
40		2z 12535	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
41		rpexpcl 13986	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ⁺)
42	39, 40, 41	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ⁺)
43	38, 42	rpmulcld 12973	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ⁺)
44		3nn0 12431	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
45		2nn 12226	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
46	44, 45	decnncl 12638	. . . . . . . . . . . 12 ⊢ ;32 ∈ ℕ
47		nnrp 12926	. . . . . . . . . . . 12 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ⁺)
48	46, 47	ax-mp 5	. . . . . . . . . . 11 ⊢ ;32 ∈ ℝ⁺
49		rpmulcl 12938	. . . . . . . . . . 11 ⊢ ((;32 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (;32 · 𝐵) ∈ ℝ⁺)
50	48, 3, 49	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ⁺)
51	43, 50	rpdivcld 12974	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℝ⁺)
52	51	rpred 12957	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℝ)
53	36, 52	remulcld 11185	. . . . . . 7 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) ∈ ℝ)
54	53, 24	remulcld 11185	. . . . . 6 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) ∈ ℝ)
55	32, 54	remulcld 11185	. . . . 5 ⊢ (𝜑 → (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
56	30, 55	resubcld 11583	. . . 4 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) ∈ ℝ)
57	13	rpred 12957	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	56, 57	readdcld 11184	. . 3 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ∈ ℝ)
59	7	rpcnd 12959	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℂ)
60	53	recnd 11183	. . . . . 6 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) ∈ ℂ)
61	24	recnd 11183	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
62	59, 60, 61	subdird 11612	. . . . 5 ⊢ (𝜑 → ((𝑈 − ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
63	38	rpcnd 12959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ ℂ)
64	42	rpcnd 12959	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
65	50	rpcnne0d 12966	. . . . . . . . . . 11 ⊢ (𝜑 → ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0))
66		div23 11832	. . . . . . . . . . 11 ⊢ ((𝐿 ∈ ℂ ∧ (𝐸↑2) ∈ ℂ ∧ ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0)) → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) = ((𝐿 / (;32 · 𝐵)) · (𝐸↑2)))
67	63, 64, 65, 66	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) = ((𝐿 / (;32 · 𝐵)) · (𝐸↑2)))
68	9	oveq1i 7367	. . . . . . . . . . . 12 ⊢ (𝐸↑2) = ((𝑈 / 𝐷)↑2)
69	37	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
70	69	rpcnd 12959	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℂ)
71	69	rpne0d 12962	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ≠ 0)
72	59, 70, 71	sqdivd 14064	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑈 / 𝐷)↑2) = ((𝑈↑2) / (𝐷↑2)))
73	68, 72	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸↑2) = ((𝑈↑2) / (𝐷↑2)))
74	73	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) · (𝐸↑2)) = ((𝐿 / (;32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))))
75	38, 50	rpdivcld 12974	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ⁺)
76	75	rpcnd 12959	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℂ)
77	59	sqcld 14049	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈↑2) ∈ ℂ)
78		rpexpcl 13986	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ⁺)
79	69, 40, 78	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ⁺)
80	79	rpcnne0d 12966	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐷↑2) ∈ ℂ ∧ (𝐷↑2) ≠ 0))
81		divass 11831	. . . . . . . . . . . 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 11832	. . . . . . . . . . . 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)))
83	81, 82	eqtr3d 2778	. . . . . . . . . . 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)))
84	76, 77, 80, 83	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) · ((𝑈↑2) / (𝐷↑2))) = (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
85	67, 74, 84	3eqtrd 2780	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) = (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2)))
86	85	oveq2d 7373	. . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) = ((𝑈 − 𝐸) · (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
87		df-3 12217	. . . . . . . . . . . . 13 ⊢ 3 = (2 + 1)
88	87	oveq2i 7368	. . . . . . . . . . . 12 ⊢ (𝑈↑3) = (𝑈↑(2 + 1))
89		2nn0 12430	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ₀
90		expp1 13974	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ ℂ ∧ 2 ∈ ℕ₀) → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
91	59, 89, 90	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈↑(2 + 1)) = ((𝑈↑2) · 𝑈))
92	88, 91	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈↑3) = ((𝑈↑2) · 𝑈))
93	77, 59	mulcomd 11176	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈↑2) · 𝑈) = (𝑈 · (𝑈↑2)))
94	92, 93	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈↑3) = (𝑈 · (𝑈↑2)))
95	94	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 · (𝑈↑3)) = (𝐹 · (𝑈 · (𝑈↑2))))
96	37	simp3d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ ℝ⁺)
97	96	rpcnd 12959	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℂ)
98	97, 59, 77	mulassd 11178	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = (𝐹 · (𝑈 · (𝑈↑2))))
99		1cnd 11150	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
100	69	rpreccld 12967	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ⁺)
101	100	rpcnd 12959	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 / 𝐷) ∈ ℂ)
102	99, 101, 59	subdird 11612	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · 𝑈) = ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)))
103	59	mulid2d 11173	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 · 𝑈) = 𝑈)
104	59, 70, 71	divrec2d 11935	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑈 / 𝐷) = ((1 / 𝐷) · 𝑈))
105	9, 104	eqtr2id 2789	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 / 𝐷) · 𝑈) = 𝐸)
106	103, 105	oveq12d 7375	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1 · 𝑈) − ((1 / 𝐷) · 𝑈)) = (𝑈 − 𝐸))
107	102, 106	eqtr2d 2777	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈 − 𝐸) = ((1 − (1 / 𝐷)) · 𝑈))
108	107	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))))
109	6	oveq1i 7367	. . . . . . . . . . . . 13 ⊢ (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) · 𝑈)
110	99, 101	subcld 11512	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℂ)
111	75, 79	rpdivcld 12974	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ⁺)
112	111	rpcnd 12959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℂ)
113	110, 112, 59	mul32d 11365	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))))
114	109, 113	eqtrid 2788	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 · 𝑈) = (((1 − (1 / 𝐷)) · 𝑈) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))))
115	108, 114	eqtr4d 2779	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) = (𝐹 · 𝑈))
116	115	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝐹 · 𝑈) · (𝑈↑2)))
117	35	rpcnd 12959	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℂ)
118	117, 112, 77	mulassd 11178	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) · (𝑈↑2)) = ((𝑈 − 𝐸) · (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
119	116, 118	eqtr3d 2778	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 · 𝑈) · (𝑈↑2)) = ((𝑈 − 𝐸) · (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
120	95, 98, 119	3eqtr2d 2782	. . . . . . . 8 ⊢ (𝜑 → (𝐹 · (𝑈↑3)) = ((𝑈 − 𝐸) · (((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) · (𝑈↑2))))
121	86, 120	eqtr4d 2779	. . . . . . 7 ⊢ (𝜑 → ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) = (𝐹 · (𝑈↑3)))
122	121	oveq2d 7373	. . . . . 6 ⊢ (𝜑 → (𝑈 − ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)))) = (𝑈 − (𝐹 · (𝑈↑3))))
123	122	oveq1d 7372	. . . . 5 ⊢ (𝜑 → ((𝑈 − ((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)))) · (log‘𝑍)) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
124	62, 123	eqtr3d 2778	. . . 4 ⊢ (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) = ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
125	26, 24	remulcld 11185	. . . . 5 ⊢ (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℝ)
126	125, 54	resubcld 11583	. . . 4 ⊢ (𝜑 → ((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) ∈ ℝ)
127	124, 126	eqeltrrd 2839	. . 3 ⊢ (𝜑 → ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)) ∈ ℝ)
128	17	rpred 12957	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ)
129	16	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
130	129	simp1d 1142	. . . . . . . 8 ⊢ (𝜑 → 1 < 𝑍)
131	128, 130	rplogcld 25984	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
132	32, 131	rerpdivcld 12988	. . . . . 6 ⊢ (𝜑 → (2 / (log‘𝑍)) ∈ ℝ)
133		fzfid 13878	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
134	17	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ⁺)
135		elfznn 13470	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
136	135	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
137	136	nnrpd 12955	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
138	134, 137	rpdivcld 12974	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ⁺)
139	18	ffvelcdmi 7034	. . . . . . . . . . . 12 ⊢ ((𝑍 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
140	138, 139	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
141	140, 134	rerpdivcld 12988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
142	141	recnd 11183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
143	142	abscld 15321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
144	137	relogcld 25978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
145	143, 144	remulcld 11185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
146	133, 145	fsumrecl 15619	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℝ)
147	132, 146	remulcld 11185	. . . . 5 ⊢ (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
148	147, 57	readdcld 11184	. . . 4 ⊢ (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ∈ ℝ)
149	20	recnd 11183	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅‘𝑍) ∈ ℂ)
150	149	abscld 15321	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝑅‘𝑍)) ∈ ℝ)
151	150	recnd 11183	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(𝑅‘𝑍)) ∈ ℂ)
152	151, 61	mulcld 11175	. . . . . . . 8 ⊢ (𝜑 → ((abs‘(𝑅‘𝑍)) · (log‘𝑍)) ∈ ℂ)
153	132	recnd 11183	. . . . . . . . 9 ⊢ (𝜑 → (2 / (log‘𝑍)) ∈ ℂ)
154	140	recnd 11183	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℂ)
155	154	abscld 15321	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℝ)
156	155, 144	remulcld 11185	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
157	133, 156	fsumrecl 15619	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℝ)
158	157	recnd 11183	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
159	153, 158	mulcld 11175	. . . . . . . 8 ⊢ (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) ∈ ℂ)
160	17	rpcnd 12959	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℂ)
161	17	rpne0d 12962	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ≠ 0)
162	152, 159, 160, 161	divsubdird 11970	. . . . . . 7 ⊢ (𝜑 → ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) = ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) / 𝑍) − (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍)))
163	151, 61, 160, 161	div23d 11968	. . . . . . . . 9 ⊢ (𝜑 → (((abs‘(𝑅‘𝑍)) · (log‘𝑍)) / 𝑍) = (((abs‘(𝑅‘𝑍)) / 𝑍) · (log‘𝑍)))
164	149, 160, 161	absdivd 15340	. . . . . . . . . . 11 ⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 𝑍)) = ((abs‘(𝑅‘𝑍)) / (abs‘𝑍)))
165	17	rprege0d 12964	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 ∈ ℝ ∧ 0 ≤ 𝑍))
166		absid 15181	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ ℝ ∧ 0 ≤ 𝑍) → (abs‘𝑍) = 𝑍)
167	165, 166	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (abs‘𝑍) = 𝑍)
168	167	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘(𝑅‘𝑍)) / (abs‘𝑍)) = ((abs‘(𝑅‘𝑍)) / 𝑍))
169	164, 168	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 𝑍)) = ((abs‘(𝑅‘𝑍)) / 𝑍))
170	169	oveq1d 7372	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) = (((abs‘(𝑅‘𝑍)) / 𝑍) · (log‘𝑍)))
171	163, 170	eqtr4d 2779	. . . . . . . 8 ⊢ (𝜑 → (((abs‘(𝑅‘𝑍)) · (log‘𝑍)) / 𝑍) = ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)))
172	153, 158, 160, 161	divassd 11966	. . . . . . . . 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)))
173	160	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℂ)
174	161	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ≠ 0)
175	154, 173, 174	absdivd 15340	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)))
176	167	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘𝑍) = 𝑍)
177	176	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) / (abs‘𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
178	175, 177	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍))
179	178	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
180	155	recnd 11183	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑍 / 𝑛))) ∈ ℂ)
181	144	recnd 11183	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℂ)
182	17	rpcnne0d 12966	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
183	182	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 ∈ ℂ ∧ 𝑍 ≠ 0))
184		div23 11832	. . . . . . . . . . . . . 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)))
185	180, 181, 183, 184	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = (((abs‘(𝑅‘(𝑍 / 𝑛))) / 𝑍) · (log‘𝑛)))
186	179, 185	eqtr4d 2779	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
187	186	sumeq2dv 15588	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
188	156	recnd 11183	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) ∈ ℂ)
189	133, 160, 188, 161	fsumdivc 15671	. . . . . . . . . . 11 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
190	187, 189	eqtr4d 2779	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) = (Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)) / 𝑍))
191	190	oveq2d 7373	. . . . . . . . 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)))
192	172, 191	eqtr4d 2779	. . . . . . . 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))))
193	171, 192	oveq12d 7375	. . . . . . 7 ⊢ (𝜑 → ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) / 𝑍) − (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛))) / 𝑍)) = (((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
194	162, 193	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) = (((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))))
195		2fveq3 6847	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (abs‘(𝑅‘𝑧)) = (abs‘(𝑅‘𝑍)))
196		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (log‘𝑧) = (log‘𝑍))
197	195, 196	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → ((abs‘(𝑅‘𝑧)) · (log‘𝑧)) = ((abs‘(𝑅‘𝑍)) · (log‘𝑍)))
198	196	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (2 / (log‘𝑧)) = (2 / (log‘𝑍)))
199		oveq2 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑧 / 𝑖) = (𝑧 / 𝑛))
200	199	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → (𝑅‘(𝑧 / 𝑖)) = (𝑅‘(𝑧 / 𝑛)))
201	200	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (abs‘(𝑅‘(𝑧 / 𝑖))) = (abs‘(𝑅‘(𝑧 / 𝑛))))
202		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (log‘𝑖) = (log‘𝑛))
203	201, 202	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑛 → ((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)))
204	203	cbvsumv 15581	. . . . . . . . . . . 12 ⊢ Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛))
205		fvoveq1 7380	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑍 → (⌊‘(𝑧 / 𝑌)) = (⌊‘(𝑍 / 𝑌)))
206	205	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑍 → (1...(⌊‘(𝑧 / 𝑌))) = (1...(⌊‘(𝑍 / 𝑌))))
207		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 𝑍 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑧 = 𝑍)
208	207	fvoveq1d 7379	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 𝑍 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑧 / 𝑛)) = (𝑅‘(𝑍 / 𝑛)))
209	208	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 𝑍 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘(𝑅‘(𝑧 / 𝑛))) = (abs‘(𝑅‘(𝑍 / 𝑛))))
210	209	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑍 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = ((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
211	206, 210	sumeq12rdv 15592	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → Σ𝑛 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑛))) · (log‘𝑛)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
212	204, 211	eqtrid 2788	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)) = Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))
213	198, 212	oveq12d 7375	. . . . . . . . . 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))))
214	197, 213	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) = (((abs‘(𝑅‘𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))))
215		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
216	214, 215	oveq12d 7375	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → ((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) = ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍))
217	216	breq1d 5115	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (((((abs‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶 ↔ ((((abs‘(𝑅‘𝑍)) · (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‘(𝑅‘𝑧)) · (log‘𝑧)) − ((2 / (log‘𝑧)) · Σ𝑖 ∈ (1...(⌊‘(𝑧 / 𝑌)))((abs‘(𝑅‘(𝑧 / 𝑖))) · (log‘𝑖)))) / 𝑧) ≤ 𝐶)
219		1re 11155	. . . . . . . . 9 ⊢ 1 ∈ ℝ
220		rexr 11201	. . . . . . . . 9 ⊢ (1 ∈ ℝ → 1 ∈ ℝ^*)
221		elioopnf 13360	. . . . . . . . 9 ⊢ (1 ∈ ℝ^* → (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍)))
222	219, 220, 221	mp2b 10	. . . . . . . 8 ⊢ (𝑍 ∈ (1(,)+∞) ↔ (𝑍 ∈ ℝ ∧ 1 < 𝑍))
223	128, 130, 222	sylanbrc 583	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (1(,)+∞))
224	217, 218, 223	rspcdva 3582	. . . . . 6 ⊢ (𝜑 → ((((abs‘(𝑅‘𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘(𝑅‘(𝑍 / 𝑛))) · (log‘𝑛)))) / 𝑍) ≤ 𝐶)
225	194, 224	eqbrtrrd 5129	. . . . 5 ⊢ (𝜑 → (((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ 𝐶)
226	25, 147, 57	lesubadd2d 11754	. . . . 5 ⊢ (𝜑 → ((((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) − ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)))) ≤ 𝐶 ↔ ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶)))
227	225, 226	mpbid 231	. . . 4 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶))
228		2cnd 12231	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
229	143	recnd 11183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℂ)
230	229, 181	mulcld 11175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
231	133, 230	fsumcl 15618	. . . . . . 7 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛)) ∈ ℂ)
232	131	rpne0d 12962	. . . . . . 7 ⊢ (𝜑 → (log‘𝑍) ≠ 0)
233	228, 231, 61, 232	div23d 11968	. . . . . 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))))
234	24	resqcld 14030	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ)
235	52, 234	remulcld 11185	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
236	36, 235	remulcld 11185	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
237		remulcl 11136	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ) → (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
238	31, 236, 237	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))) ∈ ℝ)
239	30, 24	remulcld 11185	. . . . . . . . 9 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) ∈ ℝ)
240		remulcl 11136	. . . . . . . . . 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)
241	31, 146, 240	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ∈ ℝ)
242	26	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑈 ∈ ℝ)
243	242, 136	nndivred 12207	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
244	243, 143	resubcld 11583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
245	244, 144	remulcld 11185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
246	133, 245	fsumrecl 15619	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
247	32, 246	remulcld 11185	. . . . . . . . . 10 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ∈ ℝ)
248	239, 241	resubcld 11583	. . . . . . . . . 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‘((𝑅‘𝑧) / 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‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
253	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 249, 250, 251, 252	pntlemf 26953	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
254		2pos 12256	. . . . . . . . . . . . 13 ⊢ 0 < 2
255	254	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 2)
256		lemul2 12008	. . . . . . . . . . . 12 ⊢ ((((𝑈 − 𝐸) · (((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)) → (((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
257	236, 246, 32, 255, 256	syl112anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
258	253, 257	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))) ≤ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
259	243	recnd 11183	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℂ)
260	259, 229, 181	subdird 11612	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = (((𝑈 / 𝑛) · (log‘𝑛)) − ((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))))
261	260	sumeq2dv 15588	. . . . . . . . . . . . . 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))))
262	243, 144	remulcld 11185	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
263	262	recnd 11183	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
264	133, 263, 230	fsumsub 15673	. . . . . . . . . . . . . 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))))
265	261, 264	eqtrd 2776	. . . . . . . . . . . . 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))))
266	265	oveq2d 7373	. . . . . . . . . . . 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)))))
267	133, 262	fsumrecl 15619	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ)
268	267	recnd 11183	. . . . . . . . . . . . 13 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℂ)
269	228, 268, 231	subdid 11611	. . . . . . . . . . . 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)))))
270	266, 269	eqtrd 2776	. . . . . . . . . . 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 11136	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛)) ∈ ℝ) → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
272	31, 267, 271	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ∈ ℝ)
273	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 249, 250, 251, 252	pntlemk 26954	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((𝑈 / 𝑛) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)))
274	272, 239, 241, 273	lesub1dd 11771	. . . . . . . . . . 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)))))
275	270, 274	eqbrtrd 5127	. . . . . . . . . 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)))))
276	238, 247, 248, 258, 275	letrd 11312	. . . . . . . . 9 ⊢ (𝜑 → (2 · ((𝑈 − 𝐸) · (((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)))))
277	238, 239, 241, 276	lesubd 11759	. . . . . . . 8 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))))))
278	30	recnd 11183	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 · ((log‘𝑍) + 3)) ∈ ℂ)
279	55	recnd 11183	. . . . . . . . . 10 ⊢ (𝜑 → (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) ∈ ℂ)
280	278, 279, 61	subdird 11612	. . . . . . . . 9 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − ((2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍))))
281	54	recnd 11183	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) ∈ ℂ)
282	228, 281, 61	mulassd 11178	. . . . . . . . . . 11 ⊢ (𝜑 → ((2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))))
283	60, 61, 61	mulassd 11178	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
284	61	sqvald 14048	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
285	284	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · ((log‘𝑍)↑2)) = (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · ((log‘𝑍) · (log‘𝑍))))
286	51	rpcnd 12959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℂ)
287	234	recnd 11183	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℂ)
288	117, 286, 287	mulassd 11178	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · ((log‘𝑍)↑2)) = ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))))
289	283, 285, 288	3eqtr2d 2782	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍)) = ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))))
290	289	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝜑 → (2 · ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) · (log‘𝑍))) = (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))))
291	282, 290	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍)) = (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))))
292	291	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − ((2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) · (log‘𝑍))) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))))))
293	280, 292	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)) = (((𝑈 · ((log‘𝑍) + 3)) · (log‘𝑍)) − (2 · ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))))))
294	277, 293	breqtrrd 5133	. . . . . . 7 ⊢ (𝜑 → (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍)))
295	241, 56, 131	ledivmul2d 13011	. . . . . . 7 ⊢ (𝜑 → (((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) / (log‘𝑍)) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) ↔ (2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) · (log‘𝑍))))
296	294, 295	mpbird 256	. . . . . 6 ⊢ (𝜑 → ((2 · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) / (log‘𝑍)) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
297	233, 296	eqbrtrrd 5129	. . . . 5 ⊢ (𝜑 → ((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) ≤ ((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
298	147, 56, 57, 297	leadd1dd 11769	. . . 4 ⊢ (𝜑 → (((2 / (log‘𝑍)) · Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))((abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) · (log‘𝑛))) + 𝐶) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
299	25, 148, 58, 227, 298	letrd 11312	. . 3 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ≤ (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
300		remulcl 11136	. . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ ∧ 3 ∈ ℝ) → (𝑈 · 3) ∈ ℝ)
301	26, 27, 300	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ)
302	301, 57	readdcld 11184	. . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ)
303	16	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → ((4 / (𝐿 · 𝐸)) ≤ (√‘𝑍) ∧ (((log‘𝑋) / (log‘𝐾)) + 2) ≤ (((log‘𝑍) / (log‘𝐾)) / 4) ∧ ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
304	303	simp3d 1144	. . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ≤ (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))
305	302, 54, 125, 304	leadd2dd 11770	. . . . . 6 ⊢ (𝜑 → ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)) ≤ ((𝑈 · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
306	28	recnd 11183	. . . . . . . . 9 ⊢ (𝜑 → 3 ∈ ℂ)
307	59, 61, 306	adddid 11179	. . . . . . . 8 ⊢ (𝜑 → (𝑈 · ((log‘𝑍) + 3)) = ((𝑈 · (log‘𝑍)) + (𝑈 · 3)))
308	307	oveq1d 7372	. . . . . . 7 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶))
309	125	recnd 11183	. . . . . . . 8 ⊢ (𝜑 → (𝑈 · (log‘𝑍)) ∈ ℂ)
310	59, 306	mulcld 11175	. . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℂ)
311	13	rpcnd 12959	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
312	309, 310, 311	addassd 11177	. . . . . . 7 ⊢ (𝜑 → (((𝑈 · (log‘𝑍)) + (𝑈 · 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
313	308, 312	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) = ((𝑈 · (log‘𝑍)) + ((𝑈 · 3) + 𝐶)))
314	281	2timesd 12396	. . . . . . . 8 ⊢ (𝜑 → (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) = ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
315	314	oveq2d 7373	. . . . . . 7 ⊢ (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) = (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
316	309, 281, 281	nppcan3d 11539	. . . . . . 7 ⊢ (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + ((((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) = ((𝑈 · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
317	315, 316	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) = ((𝑈 · (log‘𝑍)) + (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
318	305, 313, 317	3brtr4d 5137	. . . . 5 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ≤ (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))))
319	30, 57	readdcld 11184	. . . . . 6 ⊢ (𝜑 → ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ∈ ℝ)
320	319, 55, 126	lesubaddd 11752	. . . . 5 ⊢ (𝜑 → ((((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) ≤ ((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) ↔ ((𝑈 · ((log‘𝑍) + 3)) + 𝐶) ≤ (((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))) + (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))))
321	318, 320	mpbird 256	. . . 4 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) ≤ ((𝑈 · (log‘𝑍)) − (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍))))
322	278, 311, 279	addsubd 11533	. . . 4 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) + 𝐶) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) = (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) + 𝐶))
323	321, 322, 124	3brtr3d 5136	. . 3 ⊢ (𝜑 → (((𝑈 · ((log‘𝑍) + 3)) − (2 · (((𝑈 − 𝐸) · ((𝐿 · (𝐸↑2)) / (;32 · 𝐵))) · (log‘𝑍)))) + 𝐶) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
324	25, 58, 127, 299, 323	letrd 11312	. 2 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍)))
325		3z 12536	. . . . . . 7 ⊢ 3 ∈ ℤ
326		rpexpcl 13986	. . . . . . 7 ⊢ ((𝑈 ∈ ℝ⁺ ∧ 3 ∈ ℤ) → (𝑈↑3) ∈ ℝ⁺)
327	7, 325, 326	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑈↑3) ∈ ℝ⁺)
328	96, 327	rpmulcld 12973	. . . . 5 ⊢ (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ⁺)
329	328	rpred 12957	. . . 4 ⊢ (𝜑 → (𝐹 · (𝑈↑3)) ∈ ℝ)
330	26, 329	resubcld 11583	. . 3 ⊢ (𝜑 → (𝑈 − (𝐹 · (𝑈↑3))) ∈ ℝ)
331	23, 330, 131	lemul1d 13000	. 2 ⊢ (𝜑 → ((abs‘((𝑅‘𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))) ↔ ((abs‘((𝑅‘𝑍) / 𝑍)) · (log‘𝑍)) ≤ ((𝑈 − (𝐹 · (𝑈↑3))) · (log‘𝑍))))
332	324, 331	mpbird 256	1 ⊢ (𝜑 → (abs‘((𝑅‘𝑍) / 𝑍)) ≤ (𝑈 − (𝐹 · (𝑈↑3))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 +∞cpnf 11186 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 3c3 12209 4c4 12210 ℕ₀cn0 12413 ℤcz 12499 cdc 12618 ℝ⁺crp 12915 (,)cioo 13264 [,)cico 13266 [,]cicc 13267 ...cfz 13424 ⌊cfl 13695 ↑cexp 13967 √csqrt 15118 abscabs 15119 Σcsu 15570 expce 15944 eceu 15945 logclog 25910 ψcchp 26442

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-e 15951 df-sin 15952 df-cos 15953 df-tan 15954 df-pi 15955 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-ulm 25736 df-log 25912 df-atan 26217 df-em 26342 df-vma 26447 df-chp 26448

This theorem is referenced by: pntleme 26956

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