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Theorem pntlemf 27731

Description: Lemma for pnt 27740. Add up the pieces in pntlemi 27730 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 27721	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
12	11	simp3d 1160	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
13	12	simp3d 1160	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
14	1, 2, 3, 4, 5, 6	pntlemd 27720	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
15	14	simp1d 1158	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
16	11	simp1d 1158	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17		2z 12622	. . . . . . . 8 ⊢ 2 ∈ ℤ
18		rpexpcl 14112	. . . . . . . 8 ⊢ ((𝐸 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ⁺)
19	16, 17, 18	sylancl 597	. . . . . . 7 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ⁺)
20	15, 19	rpmulcld 13072	. . . . . 6 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ⁺)
21		3nn0 12518	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
22		2nn 12310	. . . . . . . . 9 ⊢ 2 ∈ ℕ
23	21, 22	decnncl 12731	. . . . . . . 8 ⊢ ;32 ∈ ℕ
24		nnrp 13024	. . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ⁺)
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ ;32 ∈ ℝ⁺
26		rpmulcl 13037	. . . . . . 7 ⊢ ((;32 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (;32 · 𝐵) ∈ ℝ⁺)
27	25, 3, 26	sylancr 598	. . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ⁺)
28	20, 27	rpdivcld 13073	. . . . 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 27723	. . . . . . . . 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 1158	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
36	35	rpred 13056	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	34	simp2d 1159	. . . . . . . 8 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
38	37	simp1d 1158	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑍)
39	36, 38	rplogcld 26756	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
40		rpexpcl 14112	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ⁺)
41	39, 17, 40	sylancl 597	. . . . 5 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ⁺)
42	28, 41	rpmulcld 13072	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ⁺)
43	13, 42	rpmulcld 13072	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ⁺)
44	43	rpred 13056	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
45	15, 16	rpmulcld 13072	. . . . . . 7 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
46		8re 12333	. . . . . . . 8 ⊢ 8 ∈ ℝ
47		8pos 12352	. . . . . . . 8 ⊢ 0 < 8
48	46, 47	elrpii 13015	. . . . . . 7 ⊢ 8 ∈ ℝ⁺
49		rpdivcl 13039	. . . . . . 7 ⊢ (((𝐿 · 𝐸) ∈ ℝ⁺ ∧ 8 ∈ ℝ⁺) → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
50	45, 48, 49	sylancl 597	. . . . . 6 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
51	50, 39	rpmulcld 13072	. . . . 5 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ⁺)
52	13, 51	rpmulcld 13072	. . . 4 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
53	52	rpred 13056	. . 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 27724	. . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
57	56	simp1d 1158	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
58	56	simp2d 1159	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
59		eluznn 12938	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
60	57, 58, 59	syl2anc 595	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
61	60	nnred 12244	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
62	57	nnred 12244	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	61, 62	resubcld 11638	. . 3 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
64	53, 63	remulcld 11235	. 2 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ∈ ℝ)
65		fzfid 14005	. . 3 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
66	7	rpred 13056	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
67		elfznn 13577	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68		nndivre 12273	. . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
69	66, 67, 68	syl2an 607	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
70	35	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ⁺)
71	67	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
72	71	nnrpd 13054	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
73	70, 72	rpdivcld 13073	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ⁺)
74	1	pntrf 27689	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
75	74	ffvelcdmi 7076	. . . . . . . . 9 ⊢ ((𝑍 / 𝑛) ∈ ℝ⁺ → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
76	73, 75	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑅‘(𝑍 / 𝑛)) ∈ ℝ)
77	76, 70	rerpdivcld 13087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
78	77	recnd 11233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
79	78	abscld 15486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
80	69, 79	resubcld 11638	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
81	72	relogcld 26750	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
82	80, 81	remulcld 11235	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
83	65, 82	fsumrecl 15781	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
84	45	rpcnd 13058	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
85	11	simp2d 1159	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
86	85	rpred 13056	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
87	12	simp2d 1159	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝐾)
88	86, 87	rplogcld 26756	. . . . . . . . . . 11 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
89	39, 88	rpdivcld 13073	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
90	89	rpcnd 13058	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91		rpcnne0 13031	. . . . . . . . . 10 ⊢ (8 ∈ ℝ⁺ → (8 ∈ ℂ ∧ 8 ≠ 0))
92	48, 91	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93		4re 12321	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
94		4pos 12347	. . . . . . . . . . 11 ⊢ 0 < 4
95	93, 94	elrpii 13015	. . . . . . . . . 10 ⊢ 4 ∈ ℝ⁺
96		rpcnne0 13031	. . . . . . . . . 10 ⊢ (4 ∈ ℝ⁺ → (4 ∈ ℂ ∧ 4 ≠ 0))
97	95, 96	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98		divmuldiv 11911	. . . . . . . . 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 851	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
100	10	fveq2i 6882	. . . . . . . . . . . . . 14 ⊢ (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
101	3, 16	rpdivcld 13073	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
102	101	rpred 13056	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103	102	relogefd 26755	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104	100, 103	eqtrid 2816	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105	104	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
106	39	rpcnd 13058	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
107	3	rpcnne0d 13065	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
108	16	rpcnne0d 13065	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109		divdiv2 11923	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111	105, 110	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112	111	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
113	16	rpcnd 13058	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
114	106, 113	mulcld 11225	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115		divass 11886	. . . . . . . . . . 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 1396	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
117	15	rpcnd 13058	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
118	117, 113, 106, 113	mul4d 11418	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119	113	sqvald 14175	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120	119	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121	113	sqcld 14176	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
122	117, 106, 121	mul32d 11416	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123	118, 120, 122	3eqtr2d 2810	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124	123	oveq1d 7423	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125	112, 116, 124	3eqtr2d 2810	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126		8t4e32 12829	. . . . . . . . . 10 ⊢ (8 · 4) = ;32
127	126	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (8 · 4) = ;32)
128	125, 127	oveq12d 7426	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32))
129	20	rpcnd 13058	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130	129, 106	mulcld 11225	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131		rpcnne0 13031	. . . . . . . . . . 11 ⊢ (;32 ∈ ℝ⁺ → (;32 ∈ ℂ ∧ ;32 ≠ 0))
132	25, 131	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → (;32 ∈ ℂ ∧ ;32 ≠ 0))
133		divdiv1 11922	. . . . . . . . . 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 1396	. . . . . . . . 9 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
135	23	nncni 12239	. . . . . . . . . . 11 ⊢ ;32 ∈ ℂ
136	3	rpcnd 13058	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
137		mulcom 11182	. . . . . . . . . . 11 ⊢ ((;32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;32 · 𝐵) = (𝐵 · ;32))
138	135, 136, 137	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (;32 · 𝐵) = (𝐵 · ;32))
139	138	oveq2d 7424	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
140	27	rpcnne0d 13065	. . . . . . . . . 10 ⊢ (𝜑 → ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0))
141		div23 11887	. . . . . . . . . 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 1396	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
143	134, 139, 142	3eqtr2d 2810	. . . . . . . 8 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
144	99, 128, 143	3eqtrd 2808	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
145	144	oveq1d 7423	. . . . . 6 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
146	50	rpcnd 13058	. . . . . . 7 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
147	89	rpred 13056	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148		4nn 12320	. . . . . . . . 9 ⊢ 4 ∈ ℕ
149		nndivre 12273	. . . . . . . . 9 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150	147, 148, 149	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151	150	recnd 11233	. . . . . . 7 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152	146, 106, 151	mul32d 11416	. . . . . 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 14175	. . . . . . . 8 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154	153	oveq2d 7424	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
155	28	rpcnd 13058	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℂ)
156	155, 106, 106	mulassd 11228	. . . . . . 7 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157	154, 156	eqtr4d 2807	. . . . . 6 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158	145, 152, 157	3eqtr4d 2814	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))
159	56	simp3d 1160	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
160	150, 63, 51	lemul2d 13100	. . . . . 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 235	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))
162	158, 161	eqbrtrrd 5136	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))
163	42	rpred 13056	. . . . 5 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
164	51	rpred 13056	. . . . . 6 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165	164, 63	remulcld 11235	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)) ∈ ℝ)
166	163, 165, 13	lemul2d 13100	. . . 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 235	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
168	13	rpcnd 13058	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℂ)
169	51	rpcnd 13058	. . . 4 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
170	63	recnd 11233	. . . 4 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
171	168, 169, 170	mulassd 11228	. . 3 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) = ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
172	167, 171	breqtrrd 5140	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
173		fzfid 14005	. . . 4 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
174	60	nnzd 12613	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
175	85, 174	rpexpcld 14279	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ⁺)
176	35, 175	rpdivcld 13073	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑁)) ∈ ℝ⁺)
177	176	rprege0d 13063	. . . . . . . . 9 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))))
178		flge0nn0 13849	. . . . . . . . 9 ⊢ (((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))) → (⌊‘(𝑍 / (𝐾↑𝑁))) ∈ ℕ₀)
179		nn0p1nn 12539	. . . . . . . . 9 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑁))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ ℕ)
180	177, 178, 179	3syl 19	. . . . . . . 8 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ ℕ)
181		nnuz 12897	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
182	180, 181	eleqtrdi 2879	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ (ℤ_≥‘1))
183		fzss1 13587	. . . . . . 7 ⊢ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
184	182, 183	syl 18	. . . . . 6 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
185	184	sselda 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186	185, 82	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187	173, 186	fsumrecl 15781	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188		eluzfz2 13556	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
189	58, 188	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
190		oveq1 7415	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 − 𝑀) = (𝑀 − 𝑀))
191	190	oveq2d 7424	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)))
192		oveq2 7416	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝐾↑𝑚) = (𝐾↑𝑀))
193	192	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑀)))
194	193	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑀))))
195	194	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1))
196	195	oveq1d 7423	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197	196	sumeq1d 15747	. . . . . . 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 5123	. . . . . 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 7415	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝑚 − 𝑀) = (𝑗 − 𝑀))
201	200	oveq2d 7424	. . . . . . 7 ⊢ (𝑚 = 𝑗 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)))
202		oveq2 7416	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐾↑𝑚) = (𝐾↑𝑗))
203	202	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑗)))
204	203	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑗))))
205	204	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
206	205	oveq1d 7423	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207	206	sumeq1d 15747	. . . . . . 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 5123	. . . . . 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 7415	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (𝑚 − 𝑀) = ((𝑗 + 1) − 𝑀))
211	210	oveq2d 7424	. . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑗 + 1) → (𝐾↑𝑚) = (𝐾↑(𝑗 + 1)))
213	212	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214	213	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215	214	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216	215	oveq1d 7423	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217	216	sumeq1d 15747	. . . . . . 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 5123	. . . . . 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 7415	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑀) = (𝑁 − 𝑀))
221	220	oveq2d 7424	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
222		oveq2 7416	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝐾↑𝑚) = (𝐾↑𝑁))
223	222	oveq2d 7424	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑁)))
224	223	fveq2d 6883	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑁))))
225	224	oveq1d 7423	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1))
226	225	oveq1d 7423	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227	226	sumeq1d 15747	. . . . . . 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 5123	. . . . . 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 12245	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
231	230	subidd 11553	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
232	231	oveq2d 7424	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
233	52	rpcnd 13058	. . . . . . . . 9 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234	233	mul01d 11405	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235	232, 234	eqtrd 2804	. . . . . . 7 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = 0)
236		fzfid 14005	. . . . . . . 8 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
237	57	nnzd 12613	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
238	85, 237	rpexpcld 14279	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾↑𝑀) ∈ ℝ⁺)
239	35, 238	rpdivcld 13073	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑀)) ∈ ℝ⁺)
240	239	rprege0d 13063	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))))
241		flge0nn0 13849	. . . . . . . . . . . . 13 ⊢ (((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))) → (⌊‘(𝑍 / (𝐾↑𝑀))) ∈ ℕ₀)
242		nn0p1nn 12539	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑀))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ ℕ)
243	240, 241, 242	3syl 19	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ ℕ)
244	243, 181	eleqtrdi 2879	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ (ℤ_≥‘1))
245		fzss1 13587	. . . . . . . . . . 11 ⊢ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
246	244, 245	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
247	246	sselda 3945	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248	247, 82	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249		elfzle2 13552	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250	249	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
251	29	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
252	35, 251	rpdivcld 13073	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ⁺)
253	252	rpred 13056	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254		elfzelz 13548	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255		flge 13834	. . . . . . . . . . . . 13 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256	253, 254, 255	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
257	250, 256	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (𝑍 / 𝑌))
258	71, 257	jca 520	. . . . . . . . . 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 27726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261	258, 260	syldan 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262	247, 261	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263	236, 248, 262	fsumge0 15843	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264	235, 263	eqbrtrd 5134	. . . . . 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 2769	. . . . . . . . . 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 27730	. . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
270	269	rpred 13056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271		elfzoelz 13683	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272	271	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273	272	zred 12696	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
274	57	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275	274	nnred 12244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276	273, 275	resubcld 11638	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℝ)
277	270, 276	remulcld 11235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ∈ ℝ)
278		fzfid 14005	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∈ Fin)
279		ssun1 4139	. . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
281	85	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ⁺)
282	272	peano2zd 12699	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283	281, 282	rpexpcld 14279	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ⁺)
284	280, 283	rerpdivcld 13087	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285	281, 272	rpexpcld 14279	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ⁺)
286	280, 285	rerpdivcld 13087	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑𝑗)) ∈ ℝ)
287	86	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288		1re 11204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
289		ltle 11294	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290	288, 86, 289	sylancr 598	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
291	87, 290	mpd 16	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ≤ 𝐾)
292	291	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293		uzid 12873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
294		peano2uz 12921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (ℤ_≥‘𝑗) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
295	272, 293, 294	3syl 19	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ (ℤ_≥‘𝑗))
296	287, 292, 295	leexp2ad 14286	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ≤ (𝐾↑(𝑗 + 1)))
297	35	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ⁺)
298	285, 283, 297	lediv2d 13080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝐾↑𝑗) ≤ (𝐾↑(𝑗 + 1)) ↔ (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾↑𝑗))))
299	296, 298	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ≤ (𝑍 / (𝐾↑𝑗)))
300		flword2 13842	. . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302		eluzp1p1 12886	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))) → ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
303	301, 302	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1) ∈ (ℤ_≥‘((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)))
304	286	flcld 13827	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℤ)
305	252	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ⁺)
306	305	rpred 13056	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307	306	flcld 13827	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308	251	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ⁺)
309	308	rpred 13056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310	285	rpred 13056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ)
311	30	simpld 499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
312	311	rpred 13056	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 ∈ ℝ)
313	312	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
314	30	simprd 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 < 𝑋)
315	314	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316		elfzofz 13700	. . . . . . . . . . . . . . . . . . . . . . 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 27725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
318	316, 317	sylan2 604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
319	318	simpld 499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾↑𝑗))
320	309, 313, 310, 315, 319	lttrd 11367	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾↑𝑗))
321	309, 310, 320	ltled 11354	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾↑𝑗))
322	308, 285, 297	lediv2d 13080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑌 ≤ (𝐾↑𝑗) ↔ (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌)))
323	321, 322	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌))
324		flwordi 13841	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑍 / (𝐾↑𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325	286, 306, 323, 324	syl3anc 1396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326		eluz2 12864	. . . . . . . . . . . . . . . . 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 1360	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑𝑗)))))
328		fzsplit2 13573	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331	297, 283	rpdivcld 13073	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ⁺)
332	331	rprege0d 13063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333		flge0nn0 13849	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ₀)
334		nn0p1nn 12539	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ₀ → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
335	332, 333, 334	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ ℕ)
336	335, 181	eleqtrdi 2879	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ_≥‘1))
337		fzss1 13587	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ_≥‘1) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
338	336, 337	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
339	330, 338	sstrd 3955	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340	339	sselda 3945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
341	82	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342	340, 341	syldan 602	. . . . . . . . . . 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 15781	. . . . . . . . . 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 14005	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345		ssun2 4140	. . . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347	346, 338	sstrd 3955	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348	347	sselda 3945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349	348, 341	syldan 602	. . . . . . . . . . 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 15781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351		le2add 11692	. . . . . . . . . 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 851	. . . . . . . . 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 707	. . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355		1cnd 11198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356	272	zcnd 12697	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357	230	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358	356, 357	subcld 11565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℂ)
359	354, 355, 358	adddid 11229	. . . . . . . . . 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 11408	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 − 𝑀) + 1))
361	356, 355, 357	addsubd 11586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗 − 𝑀) + 1))
362	360, 361	eqtr4d 2807	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 + 1) − 𝑀))
363	362	oveq2d 7424	. . . . . . . . . 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	mulridd 11222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365	364	oveq1d 7423	. . . . . . . . . 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 2812	. . . . . . . . 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 13825	. . . . . . . . . . . . 13 ⊢ ((𝑍 / (𝐾↑𝑗)) ∈ ℝ → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℝ)
368	286, 367	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℝ)
369	368	ltp1d 12141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) < ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
370		fzdisj 13575	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∩ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) = ∅)
372		fzfid 14005	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373	338	sselda 3945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374	373, 341	syldan 602	. . . . . . . . . . 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 11233	. . . . . . . . . 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 15788	. . . . . . . . 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 5123	. . . . . . . 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 262	. . . . . . 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 418	. . . . . 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 30	. . . . 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 13813	. . . 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 39	. . 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 15844	. . 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 11363	. 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 11363	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ∪ cun 3911 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6533 (class class class)co 7408 ℂcc 11094 ℝcr 11095 0cc0 11096 1c1 11097 + caddc 11099 · cmul 11101 +∞cpnf 11236 < clt 11239 ≤ cle 11240 − cmin 11437 / cdiv 11867 ℕcn 12229 2c2 12291 3c3 12292 4c4 12293 8c8 12297 ℕ₀cn0 12500 ℤcz 12587 cdc 12707 ℤ_≥cuz 12858 ℝ⁺crp 13012 (,)cioo 13368 [,)cico 13370 [,]cicc 13371 ...cfz 13531 ..^cfzo 13678 ⌊cfl 13819 ↑cexp 14093 √csqrt 15280 abscabs 15281 Σcsu 15733 expce 16111 eceu 16112 logclog 26681 ψcchp 27219

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-inf2 9606 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-dju 9883 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-ioc 13373 df-ico 13374 df-icc 13375 df-fz 13532 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16117 df-e 16118 df-sin 16119 df-cos 16120 df-pi 16122 df-dvds 16307 df-gcd 16549 df-prm 16726 df-pc 16893 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-fbas 21484 df-fg 21485 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-ntr 23142 df-cls 23143 df-nei 23220 df-lp 23258 df-perf 23259 df-cn 23349 df-cnp 23350 df-haus 23437 df-tx 23684 df-hmeo 23877 df-fil 23968 df-fm 24060 df-flim 24061 df-flf 24062 df-xms 24442 df-ms 24443 df-tms 24444 df-cncf 25002 df-limc 25990 df-dv 25991 df-log 26683 df-vma 27224 df-chp 27225

This theorem is referenced by: pntlemo 27733

W3C validator