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Theorem pntlemf 26183

Description: Lemma for pnt 26192. Add up the pieces in pntlemi 26182 to get an estimate slightly better than the naive lower bound 0. (Contributed by Mario Carneiro, 13-Apr-2016.)

**Hypotheses**
Ref	Expression
pntlem1.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
pntlem1.a	⊢ (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))

**Assertion**
Ref	Expression
pntlemf	⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))

Distinct variable groups: 𝑧,𝐶 𝑦,𝑛,𝑧,𝑢,𝐿 𝑛,𝐾,𝑦,𝑧 𝑛,𝑀,𝑧 𝜑,𝑛 𝑛,𝑁,𝑧 𝑅,𝑛,𝑢,𝑦,𝑧 𝑈,𝑛,𝑧 𝑛,𝑊,𝑧 𝑛,𝑋,𝑦,𝑧 𝑛,𝑌,𝑧 𝑛,𝑎,𝑢,𝑦,𝑧,𝐸 𝑛,𝑍,𝑢,𝑧 Allowed substitution hints: 𝜑(𝑦,𝑧,𝑢,𝑎) 𝐴(𝑦,𝑧,𝑢,𝑛,𝑎) 𝐵(𝑦,𝑧,𝑢,𝑛,𝑎) 𝐶(𝑦,𝑢,𝑛,𝑎) 𝐷(𝑦,𝑧,𝑢,𝑛,𝑎) 𝑅(𝑎) 𝑈(𝑦,𝑢,𝑎) 𝐹(𝑦,𝑧,𝑢,𝑛,𝑎) 𝐾(𝑢,𝑎) 𝐿(𝑎) 𝑀(𝑦,𝑢,𝑎) 𝑁(𝑦,𝑢,𝑎) 𝑊(𝑦,𝑢,𝑎) 𝑋(𝑢,𝑎) 𝑌(𝑦,𝑢,𝑎) 𝑍(𝑦,𝑎)

Proof of Theorem pntlemf Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pntlem1.r	. . . . . . 7 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
2		pntlem1.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
3		pntlem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
4		pntlem1.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (0(,)1))
5		pntlem1.d	. . . . . . 7 ⊢ 𝐷 = (𝐴 + 1)
6		pntlem1.f	. . . . . . 7 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)))
7		pntlem1.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ⁺)
8		pntlem1.u2	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≤ 𝐴)
9		pntlem1.e	. . . . . . 7 ⊢ 𝐸 = (𝑈 / 𝐷)
10		pntlem1.k	. . . . . . 7 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸))
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	pntlemc 26173	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
12	11	simp3d 1140	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
13	12	simp3d 1140	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
14	1, 2, 3, 4, 5, 6	pntlemd 26172	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
15	14	simp1d 1138	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
16	11	simp1d 1138	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17		2z 12017	. . . . . . . 8 ⊢ 2 ∈ ℤ
18		rpexpcl 13451	. . . . . . . 8 ⊢ ((𝐸 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ⁺)
19	16, 17, 18	sylancl 588	. . . . . . 7 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ⁺)
20	15, 19	rpmulcld 12450	. . . . . 6 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ⁺)
21		3nn0 11918	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
22		2nn 11713	. . . . . . . . 9 ⊢ 2 ∈ ℕ
23	21, 22	decnncl 12121	. . . . . . . 8 ⊢ ;32 ∈ ℕ
24		nnrp 12403	. . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ⁺)
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ ;32 ∈ ℝ⁺
26		rpmulcl 12415	. . . . . . 7 ⊢ ((;32 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (;32 · 𝐵) ∈ ℝ⁺)
27	25, 3, 26	sylancr 589	. . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ⁺)
28	20, 27	rpdivcld 12451	. . . . 5 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℝ⁺)
29		pntlem1.y	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ ℝ⁺ ∧ 1 ≤ 𝑌))
30		pntlem1.x	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ ℝ⁺ ∧ 𝑌 < 𝑋))
31		pntlem1.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
32		pntlem1.w	. . . . . . . . . 10 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))))
33		pntlem1.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ (𝑊[,)+∞))
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33	pntlemb 26175	. . . . . . . . 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)))))
35	34	simp1d 1138	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
36	35	rpred 12434	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	34	simp2d 1139	. . . . . . . 8 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
38	37	simp1d 1138	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑍)
39	36, 38	rplogcld 25214	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
40		rpexpcl 13451	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ⁺)
41	39, 17, 40	sylancl 588	. . . . 5 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ⁺)
42	28, 41	rpmulcld 12450	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ⁺)
43	13, 42	rpmulcld 12450	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ⁺)
44	43	rpred 12434	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
45	15, 16	rpmulcld 12450	. . . . . . 7 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
46		8re 11736	. . . . . . . 8 ⊢ 8 ∈ ℝ
47		8pos 11752	. . . . . . . 8 ⊢ 0 < 8
48	46, 47	elrpii 12395	. . . . . . 7 ⊢ 8 ∈ ℝ⁺
49		rpdivcl 12417	. . . . . . 7 ⊢ (((𝐿 · 𝐸) ∈ ℝ⁺ ∧ 8 ∈ ℝ⁺) → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
50	45, 48, 49	sylancl 588	. . . . . 6 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
51	50, 39	rpmulcld 12450	. . . . 5 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ⁺)
52	13, 51	rpmulcld 12450	. . . 4 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
53	52	rpred 12434	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
54		pntlem1.m	. . . . . . . 8 ⊢ 𝑀 = ((⌊‘((log‘𝑋) / (log‘𝐾))) + 1)
55		pntlem1.n	. . . . . . . 8 ⊢ 𝑁 = (⌊‘(((log‘𝑍) / (log‘𝐾)) / 2))
56	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55	pntlemg 26176	. . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
57	56	simp1d 1138	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
58	56	simp2d 1139	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
59		eluznn 12321	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
60	57, 58, 59	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
61	60	nnred 11655	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
62	57	nnred 11655	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	61, 62	resubcld 11070	. . 3 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
64	53, 63	remulcld 10673	. 2 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ∈ ℝ)
65		fzfid 13344	. . 3 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
66	7	rpred 12434	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
67		elfznn 12939	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68		nndivre 11681	. . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
69	66, 67, 68	syl2an 597	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
70	35	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ⁺)
71	67	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
72	71	nnrpd 12432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
73	70, 72	rpdivcld 12451	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ⁺)
74	1	pntrf 26141	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
75	74	ffvelrni 6852	. . . . . . . . 9 ⊢ ((𝑍 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
76	73, 75	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
77	76, 70	rerpdivcld 12465	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
78	77	recnd 10671	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
79	78	abscld 14798	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
80	69, 79	resubcld 11070	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
81	72	relogcld 25208	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
82	80, 81	remulcld 10673	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
83	65, 82	fsumrecl 15093	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
84	45	rpcnd 12436	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
85	11	simp2d 1139	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
86	85	rpred 12434	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
87	12	simp2d 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝐾)
88	86, 87	rplogcld 25214	. . . . . . . . . . 11 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
89	39, 88	rpdivcld 12451	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
90	89	rpcnd 12436	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91		rpcnne0 12410	. . . . . . . . . 10 ⊢ (8 ∈ ℝ⁺ → (8 ∈ ℂ ∧ 8 ≠ 0))
92	48, 91	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93		4re 11724	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
94		4pos 11747	. . . . . . . . . . 11 ⊢ 0 < 4
95	93, 94	elrpii 12395	. . . . . . . . . 10 ⊢ 4 ∈ ℝ⁺
96		rpcnne0 12410	. . . . . . . . . 10 ⊢ (4 ∈ ℝ⁺ → (4 ∈ ℂ ∧ 4 ≠ 0))
97	95, 96	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98		divmuldiv 11342	. . . . . . . . 9 ⊢ ((((𝐿 · 𝐸) ∈ ℂ ∧ ((log‘𝑍) / (log‘𝐾)) ∈ ℂ) ∧ ((8 ∈ ℂ ∧ 8 ≠ 0) ∧ (4 ∈ ℂ ∧ 4 ≠ 0))) → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
99	84, 90, 92, 97, 98	syl22anc 836	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
100	10	fveq2i 6675	. . . . . . . . . . . . . 14 ⊢ (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
101	3, 16	rpdivcld 12451	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
102	101	rpred 12434	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103	102	relogefd 25213	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104	100, 103	syl5eq 2870	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105	104	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
106	39	rpcnd 12436	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
107	3	rpcnne0d 12443	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
108	16	rpcnne0d 12443	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109		divdiv2 11354	. . . . . . . . . . . . 13 ⊢ (((log‘𝑍) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0)) → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
110	106, 107, 108, 109	syl3anc 1367	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111	105, 110	eqtrd 2858	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112	111	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
113	16	rpcnd 12436	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
114	106, 113	mulcld 10663	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115		divass 11318	. . . . . . . . . . 11 ⊢ (((𝐿 · 𝐸) ∈ ℂ ∧ ((log‘𝑍) · 𝐸) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
116	84, 114, 107, 115	syl3anc 1367	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
117	15	rpcnd 12436	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
118	117, 113, 106, 113	mul4d 10854	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119	113	sqvald 13510	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120	119	oveq2d 7174	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121	113	sqcld 13511	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
122	117, 106, 121	mul32d 10852	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123	118, 120, 122	3eqtr2d 2864	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124	123	oveq1d 7173	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125	112, 116, 124	3eqtr2d 2864	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126		8t4e32 12218	. . . . . . . . . 10 ⊢ (8 · 4) = ;32
127	126	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (8 · 4) = ;32)
128	125, 127	oveq12d 7176	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32))
129	20	rpcnd 12436	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130	129, 106	mulcld 10663	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131		rpcnne0 12410	. . . . . . . . . . 11 ⊢ (;32 ∈ ℝ⁺ → (;32 ∈ ℂ ∧ ;32 ≠ 0))
132	25, 131	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → (;32 ∈ ℂ ∧ ;32 ≠ 0))
133		divdiv1 11353	. . . . . . . . . 10 ⊢ ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (;32 ∈ ℂ ∧ ;32 ≠ 0)) → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
134	130, 107, 132, 133	syl3anc 1367	. . . . . . . . 9 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
135	23	nncni 11650	. . . . . . . . . . 11 ⊢ ;32 ∈ ℂ
136	3	rpcnd 12436	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
137		mulcom 10625	. . . . . . . . . . 11 ⊢ ((;32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;32 · 𝐵) = (𝐵 · ;32))
138	135, 136, 137	sylancr 589	. . . . . . . . . 10 ⊢ (𝜑 → (;32 · 𝐵) = (𝐵 · ;32))
139	138	oveq2d 7174	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
140	27	rpcnne0d 12443	. . . . . . . . . 10 ⊢ (𝜑 → ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0))
141		div23 11319	. . . . . . . . . 10 ⊢ (((𝐿 · (𝐸↑2)) ∈ ℂ ∧ (log‘𝑍) ∈ ℂ ∧ ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0)) → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
142	129, 106, 140, 141	syl3anc 1367	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
143	134, 139, 142	3eqtr2d 2864	. . . . . . . 8 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
144	99, 128, 143	3eqtrd 2862	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
145	144	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
146	50	rpcnd 12436	. . . . . . 7 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
147	89	rpred 12434	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148		4nn 11723	. . . . . . . . 9 ⊢ 4 ∈ ℕ
149		nndivre 11681	. . . . . . . . 9 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150	147, 148, 149	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151	150	recnd 10671	. . . . . . 7 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152	146, 106, 151	mul32d 10852	. . . . . 6 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)))
153	106	sqvald 13510	. . . . . . . 8 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154	153	oveq2d 7174	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
155	28	rpcnd 12436	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℂ)
156	155, 106, 106	mulassd 10666	. . . . . . 7 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157	154, 156	eqtr4d 2861	. . . . . 6 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158	145, 152, 157	3eqtr4d 2868	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))
159	56	simp3d 1140	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
160	150, 63, 51	lemul2d 12478	. . . . . 6 ⊢ (𝜑 → ((((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀) ↔ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
161	159, 160	mpbid 234	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))
162	158, 161	eqbrtrrd 5092	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))
163	42	rpred 12434	. . . . 5 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
164	51	rpred 12434	. . . . . 6 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165	164, 63	remulcld 10673	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)) ∈ ℝ)
166	163, 165, 13	lemul2d 12478	. . . 4 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)) ↔ ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))))
167	162, 166	mpbid 234	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
168	13	rpcnd 12436	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℂ)
169	51	rpcnd 12436	. . . 4 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
170	63	recnd 10671	. . . 4 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
171	168, 169, 170	mulassd 10666	. . 3 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) = ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
172	167, 171	breqtrrd 5096	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
173		fzfid 13344	. . . 4 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
174	60	nnzd 12089	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
175	85, 174	rpexpcld 13611	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ⁺)
176	35, 175	rpdivcld 12451	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑁)) ∈ ℝ⁺)
177	176	rprege0d 12441	. . . . . . . . 9 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))))
178		flge0nn0 13193	. . . . . . . . 9 ⊢ (((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))) → (⌊‘(𝑍 / (𝐾↑𝑁))) ∈ ℕ₀)
179		nn0p1nn 11939	. . . . . . . . 9 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑁))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ ℕ)
180	177, 178, 179	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ ℕ)
181		nnuz 12284	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
182	180, 181	eleqtrdi 2925	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ (ℤ_≥‘1))
183		fzss1 12949	. . . . . . 7 ⊢ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
184	182, 183	syl 17	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
185	184	sselda 3969	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186	185, 82	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187	173, 186	fsumrecl 15093	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188		eluzfz2 12918	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
189	58, 188	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
190		oveq1 7165	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 − 𝑀) = (𝑀 − 𝑀))
191	190	oveq2d 7174	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)))
192		oveq2 7166	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝐾↑𝑚) = (𝐾↑𝑀))
193	192	oveq2d 7174	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑀)))
194	193	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑀))))
195	194	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1))
196	195	oveq1d 7173	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197	196	sumeq1d 15060	. . . . . . 7 ⊢ (𝑚 = 𝑀 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
198	191, 197	breq12d 5081	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
199	198	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
200		oveq1 7165	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝑚 − 𝑀) = (𝑗 − 𝑀))
201	200	oveq2d 7174	. . . . . . 7 ⊢ (𝑚 = 𝑗 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)))
202		oveq2 7166	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐾↑𝑚) = (𝐾↑𝑗))
203	202	oveq2d 7174	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑗)))
204	203	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑗))))
205	204	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
206	205	oveq1d 7173	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207	206	sumeq1d 15060	. . . . . . 7 ⊢ (𝑚 = 𝑗 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
208	201, 207	breq12d 5081	. . . . . 6 ⊢ (𝑚 = 𝑗 → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
209	208	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑗 → ((𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
210		oveq1 7165	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (𝑚 − 𝑀) = ((𝑗 + 1) − 𝑀))
211	210	oveq2d 7174	. . . . . . 7 ⊢ (𝑚 = (𝑗 + 1) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
212		oveq2 7166	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑗 + 1) → (𝐾↑𝑚) = (𝐾↑(𝑗 + 1)))
213	212	oveq2d 7174	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214	213	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215	214	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216	215	oveq1d 7173	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217	216	sumeq1d 15060	. . . . . . 7 ⊢ (𝑚 = (𝑗 + 1) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
218	211, 217	breq12d 5081	. . . . . 6 ⊢ (𝑚 = (𝑗 + 1) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
219	218	imbi2d 343	. . . . 5 ⊢ (𝑚 = (𝑗 + 1) → ((𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
220		oveq1 7165	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑀) = (𝑁 − 𝑀))
221	220	oveq2d 7174	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
222		oveq2 7166	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝐾↑𝑚) = (𝐾↑𝑁))
223	222	oveq2d 7174	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑁)))
224	223	fveq2d 6676	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑁))))
225	224	oveq1d 7173	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1))
226	225	oveq1d 7173	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227	226	sumeq1d 15060	. . . . . . 7 ⊢ (𝑚 = 𝑁 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
228	221, 227	breq12d 5081	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
229	228	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑁 → ((𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) ↔ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
230	57	nncnd 11656	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
231	230	subidd 10987	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
232	231	oveq2d 7174	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
233	52	rpcnd 12436	. . . . . . . . 9 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234	233	mul01d 10841	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235	232, 234	eqtrd 2858	. . . . . . 7 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = 0)
236		fzfid 13344	. . . . . . . 8 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
237	57	nnzd 12089	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
238	85, 237	rpexpcld 13611	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾↑𝑀) ∈ ℝ⁺)
239	35, 238	rpdivcld 12451	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑀)) ∈ ℝ⁺)
240	239	rprege0d 12441	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))))
241		flge0nn0 13193	. . . . . . . . . . . . 13 ⊢ (((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))) → (⌊‘(𝑍 / (𝐾↑𝑀))) ∈ ℕ₀)
242		nn0p1nn 11939	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑀))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ ℕ)
243	240, 241, 242	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ ℕ)
244	243, 181	eleqtrdi 2925	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ (ℤ_≥‘1))
245		fzss1 12949	. . . . . . . . . . 11 ⊢ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
246	244, 245	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
247	246	sselda 3969	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248	247, 82	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249		elfzle2 12914	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250	249	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
251	29	simpld 497	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
252	35, 251	rpdivcld 12451	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ⁺)
253	252	rpred 12434	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254		elfzelz 12911	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255		flge 13178	. . . . . . . . . . . . 13 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256	253, 254, 255	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257	250, 256	mpbird 259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
258	71, 257	jca 514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌)))
259		pntlem1.U	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ (𝑌[,)+∞)(abs‘((𝑅‘𝑧) / 𝑧)) ≤ 𝑈)
260	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55, 259	pntlemn 26178	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261	258, 260	syldan 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262	247, 261	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263	236, 248, 262	fsumge0 15152	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264	235, 263	eqbrtrd 5090	. . . . . 6 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
265	264	a1i 11	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
266		pntlem1.K	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ (𝑋(,)+∞)∃𝑧 ∈ ℝ⁺ ((𝑦 < 𝑧 ∧ ((1 + (𝐿 · 𝐸)) · 𝑧) < (𝐾 · 𝑦)) ∧ ∀𝑢 ∈ (𝑧[,]((1 + (𝐿 · 𝐸)) · 𝑧))(abs‘((𝑅‘𝑢) / 𝑢)) ≤ 𝐸))
267		eqid 2823	. . . . . . . . . 10 ⊢ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))
268	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55, 259, 266, 267	pntlemi 26182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
269	52	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
270	269	rpred 12434	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271		elfzoelz 13041	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272	271	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273	272	zred 12090	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
274	57	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275	274	nnred 11655	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276	273, 275	resubcld 11070	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℝ)
277	270, 276	remulcld 10673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ∈ ℝ)
278		fzfid 13344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∈ Fin)
279		ssun1 4150	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
280	36	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
281	85	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ⁺)
282	272	peano2zd 12093	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283	281, 282	rpexpcld 13611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ⁺)
284	280, 283	rerpdivcld 12465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285	281, 272	rpexpcld 13611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ⁺)
286	280, 285	rerpdivcld 12465	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑𝑗)) ∈ ℝ)
287	86	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288		1re 10643	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
289		ltle 10731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290	288, 86, 289	sylancr 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
291	87, 290	mpd 15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ≤ 𝐾)
292	291	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293		uzid 12261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
294		peano2uz 12304	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
295	272, 293, 294	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
296	287, 292, 295	leexp2ad 13620	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ≤ (𝐾↑(𝑗 + 1)))
297	35	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ⁺)
298	285, 283, 297	lediv2d 12458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝐾↑𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾↑𝑗))))
299	296, 298	mpbid 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾↑𝑗)))
300		flword2 13186	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ (𝑍 / (𝐾↑𝑗)) ∈ ℝ ∧ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾↑𝑗))) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
301	284, 286, 299, 300	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302		eluzp1p1 12273	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))) → ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
303	301, 302	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
304	286	flcld 13171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℤ)
305	252	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ⁺)
306	305	rpred 12434	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307	306	flcld 13171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308	251	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ⁺)
309	308	rpred 12434	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310	285	rpred 12434	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ)
311	30	simpld 497	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
312	311	rpred 12434	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 ∈ ℝ)
313	312	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
314	30	simprd 498	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 < 𝑋)
315	314	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316		elfzofz 13056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ (𝑀...𝑁))
317	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 29, 30, 31, 32, 33, 54, 55	pntlemh 26177	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
318	316, 317	sylan2 594	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
319	318	simpld 497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾↑𝑗))
320	309, 313, 310, 315, 319	lttrd 10803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾↑𝑗))
321	309, 310, 320	ltled 10790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾↑𝑗))
322	308, 285, 297	lediv2d 12458	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾↑𝑗) ↔ (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌)))
323	321, 322	mpbid 234	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌))
324		flwordi 13185	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑍 / (𝐾↑𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325	286, 306, 323, 324	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326		eluz2 12252	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑍 / 𝑌)) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑𝑗)))) ↔ ((⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℤ ∧ (⌊‘(𝑍 / 𝑌)) ∈ ℤ ∧ (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌))))
327	304, 307, 325, 326	syl3anbrc 1339	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑𝑗)))))
328		fzsplit2 12935	. . . . . . . . . . . . . . . 16 ⊢ ((((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)) ∧ (⌊‘(𝑍 / 𝑌)) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑𝑗))))) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
329	303, 327, 328	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) = ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))))
330	279, 329	sseqtrrid 4022	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331	297, 283	rpdivcld 12451	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ⁺)
332	331	rprege0d 12441	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333		flge0nn0 13193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ₀)
334		nn0p1nn 11939	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
335	332, 333, 334	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
336	335, 181	eleqtrdi 2925	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ_≥‘1))
337		fzss1 12949	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
338	336, 337	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
339	330, 338	sstrd 3979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340	339	sselda 3969	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
341	82	adantlr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342	340, 341	syldan 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
343	278, 342	fsumrecl 15093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
344		fzfid 13344	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345		ssun2 4151	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∪ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
346	345, 329	sseqtrrid 4022	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347	346, 338	sstrd 3979	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348	347	sselda 3969	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349	348, 341	syldan 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
350	344, 349	fsumrecl 15093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351		le2add 11124	. . . . . . . . . 10 ⊢ (((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ ∧ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ∈ ℝ) ∧ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ ∧ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∧ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
352	270, 277, 343, 350, 351	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∧ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
353	268, 352	mpand 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
354	233	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355		1cnd 10638	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356	272	zcnd 12091	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357	230	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358	356, 357	subcld 10999	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℂ)
359	354, 355, 358	adddid 10667	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗 − 𝑀))) = ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))))
360	355, 358	addcomd 10844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 − 𝑀) + 1))
361	356, 355, 357	addsubd 11020	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗 − 𝑀) + 1))
362	360, 361	eqtr4d 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 + 1) − 𝑀))
363	362	oveq2d 7174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (1 + (𝑗 − 𝑀))) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)))
364	354	mulid1d 10660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365	364	oveq1d 7173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))))
366	359, 363, 365	3eqtr3d 2866	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))))
367		reflcl 13169	. . . . . . . . . . . . 13 ⊢ ((𝑍 / (𝐾↑𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℝ)
368	286, 367	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℝ)
369	368	ltp1d 11572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) < ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
370		fzdisj 12937	. . . . . . . . . . 11 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑗))) < ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) → ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∩ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) = ∅)
371	369, 370	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∩ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) = ∅)
372		fzfid 13344	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373	338	sselda 3969	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374	373, 341	syldan 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
375	374	recnd 10671	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℂ)
376	371, 329, 372, 375	fsumsplit 15099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) = (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
377	366, 376	breq12d 5081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ↔ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) + (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀))) ≤ (Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) + Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
378	353, 377	sylibrd 261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
379	378	expcom 416	. . . . . 6 ⊢ (𝑗 ∈ (𝑀..^𝑁) → (𝜑 → ((((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
380	379	a2d 29	. . . . 5 ⊢ (𝑗 ∈ (𝑀..^𝑁) → ((𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))) → (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · ((𝑗 + 1) − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))))
381	199, 209, 219, 229, 265, 380	fzind2 13158	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛))))
382	189, 381	mpcom 38	. . 3 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
383	65, 82, 261, 184	fsumless 15153	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
384	64, 187, 83, 382, 383	letrd 10799	. 2 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
385	44, 64, 83, 172, 384	letrd 10799	1 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 +∞cpnf 10674 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 3c3 11696 4c4 11697 8c8 11701 ℕ₀cn0 11900 ℤcz 11984 cdc 12101 ℤ_≥cuz 12246 ℝ⁺crp 12392 (,)cioo 12741 [,)cico 12743 [,]cicc 12744 ...cfz 12895 ..^cfzo 13036 ⌊cfl 13163 ↑cexp 13432 √csqrt 14594 abscabs 14595 Σcsu 15044 expce 15417 eceu 15418 logclog 25140 ψcchp 25672

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-e 15424 df-sin 15425 df-cos 15426 df-pi 15428 df-dvds 15610 df-gcd 15846 df-prm 16018 df-pc 16176 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-vma 25677 df-chp 25678

This theorem is referenced by: pntlemo 26185

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