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Theorem pntlemf 27938

Description: Lemma for pnt 27947. Add up the pieces in pntlemi 27937 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 27928	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ ℝ⁺ ∧ 𝐾 ∈ ℝ⁺ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺)))
12	11	simp3d 1162	. . . . 5 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ⁺))
13	12	simp3d 1162	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ⁺)
14	1, 2, 3, 4, 5, 6	pntlemd 27927	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ⁺ ∧ 𝐷 ∈ ℝ⁺ ∧ 𝐹 ∈ ℝ⁺))
15	14	simp1d 1160	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ⁺)
16	11	simp1d 1160	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
17		2z 12732	. . . . . . . 8 ⊢ 2 ∈ ℤ
18		rpexpcl 14227	. . . . . . . 8 ⊢ ((𝐸 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ⁺)
19	16, 17, 18	sylancl 598	. . . . . . 7 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ⁺)
20	15, 19	rpmulcld 13184	. . . . . 6 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ⁺)
21		3nn0 12628	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
22		2nn 12420	. . . . . . . . 9 ⊢ 2 ∈ ℕ
23	21, 22	decnncl 12842	. . . . . . . 8 ⊢ ;32 ∈ ℕ
24		nnrp 13136	. . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ⁺)
25	23, 24	ax-mp 5	. . . . . . 7 ⊢ ;32 ∈ ℝ⁺
26		rpmulcl 13149	. . . . . . 7 ⊢ ((;32 ∈ ℝ⁺ ∧ 𝐵 ∈ ℝ⁺) → (;32 · 𝐵) ∈ ℝ⁺)
27	25, 3, 26	sylancr 599	. . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ⁺)
28	20, 27	rpdivcld 13185	. . . . 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 27930	. . . . . . . . 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 1160	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ ℝ⁺)
36	35	rpred 13168	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ℝ)
37	34	simp2d 1161	. . . . . . . 8 ⊢ (𝜑 → (1 < 𝑍 ∧ e ≤ (√‘𝑍) ∧ (√‘𝑍) ≤ (𝑍 / 𝑌)))
38	37	simp1d 1160	. . . . . . 7 ⊢ (𝜑 → 1 < 𝑍)
39	36, 38	rplogcld 26963	. . . . . 6 ⊢ (𝜑 → (log‘𝑍) ∈ ℝ⁺)
40		rpexpcl 14227	. . . . . 6 ⊢ (((log‘𝑍) ∈ ℝ⁺ ∧ 2 ∈ ℤ) → ((log‘𝑍)↑2) ∈ ℝ⁺)
41	39, 17, 40	sylancl 598	. . . . 5 ⊢ (𝜑 → ((log‘𝑍)↑2) ∈ ℝ⁺)
42	28, 41	rpmulcld 13184	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ⁺)
43	13, 42	rpmulcld 13184	. . 3 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ⁺)
44	43	rpred 13168	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ∈ ℝ)
45	15, 16	rpmulcld 13184	. . . . . . 7 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ⁺)
46		8re 12443	. . . . . . . 8 ⊢ 8 ∈ ℝ
47		8pos 12462	. . . . . . . 8 ⊢ 0 < 8
48	46, 47	elrpii 13127	. . . . . . 7 ⊢ 8 ∈ ℝ⁺
49		rpdivcl 13151	. . . . . . 7 ⊢ (((𝐿 · 𝐸) ∈ ℝ⁺ ∧ 8 ∈ ℝ⁺) → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
50	45, 48, 49	sylancl 598	. . . . . 6 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℝ⁺)
51	50, 39	rpmulcld 13184	. . . . 5 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ⁺)
52	13, 51	rpmulcld 13184	. . . 4 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
53	52	rpred 13168	. . 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 27931	. . . . . . 7 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀) ∧ (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀)))
57	56	simp1d 1160	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
58	56	simp2d 1161	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
59		eluznn 13049	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
60	57, 58, 59	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
61	60	nnred 12354	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
62	57	nnred 12354	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
63	61, 62	resubcld 11748	. . 3 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℝ)
64	53, 63	remulcld 11343	. 2 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) ∈ ℝ)
65		fzfid 14120	. . 3 ⊢ (𝜑 → (1...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
66	7	rpred 13168	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ)
67		elfznn 13691	. . . . . 6 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℕ)
68		nndivre 12383	. . . . . 6 ⊢ ((𝑈 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑈 / 𝑛) ∈ ℝ)
69	66, 67, 68	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑈 / 𝑛) ∈ ℝ)
70	35	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑍 ∈ ℝ⁺)
71	67	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℕ)
72	71	nnrpd 13166	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ ℝ⁺)
73	70, 72	rpdivcld 13185	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (𝑍 / 𝑛) ∈ ℝ⁺)
74	1	pntrf 27896	. . . . . . . . . 10 ⊢ 𝑅:ℝ⁺⟶ℝ
75	74	ffvelcdmi 7084	. . . . . . . . 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 13199	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℝ)
78	77	recnd 11341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑅‘(𝑍 / 𝑛)) / 𝑍) ∈ ℂ)
79	78	abscld 15612	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍)) ∈ ℝ)
80	69, 79	resubcld 11748	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → ((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) ∈ ℝ)
81	72	relogcld 26957	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (log‘𝑛) ∈ ℝ)
82	80, 81	remulcld 11343	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
83	65, 82	fsumrecl 15906	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
84	45	rpcnd 13170	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℂ)
85	11	simp2d 1161	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
86	85	rpred 13168	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℝ)
87	12	simp2d 1161	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < 𝐾)
88	86, 87	rplogcld 26963	. . . . . . . . . . 11 ⊢ (𝜑 → (log‘𝐾) ∈ ℝ⁺)
89	39, 88	rpdivcld 13185	. . . . . . . . . 10 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ⁺)
90	89	rpcnd 13170	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℂ)
91		rpcnne0 13143	. . . . . . . . . 10 ⊢ (8 ∈ ℝ⁺ → (8 ∈ ℂ ∧ 8 ≠ 0))
92	48, 91	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (8 ∈ ℂ ∧ 8 ≠ 0))
93		4re 12431	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
94		4pos 12457	. . . . . . . . . . 11 ⊢ 0 < 4
95	93, 94	elrpii 13127	. . . . . . . . . 10 ⊢ 4 ∈ ℝ⁺
96		rpcnne0 13143	. . . . . . . . . 10 ⊢ (4 ∈ ℝ⁺ → (4 ∈ ℂ ∧ 4 ≠ 0))
97	95, 96	mp1i 14	. . . . . . . . 9 ⊢ (𝜑 → (4 ∈ ℂ ∧ 4 ≠ 0))
98		divmuldiv 12021	. . . . . . . . 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 852	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)))
100	10	fveq2i 6889	. . . . . . . . . . . . . 14 ⊢ (log‘𝐾) = (log‘(exp‘(𝐵 / 𝐸)))
101	3, 16	rpdivcld 13185	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ⁺)
102	101	rpred 13168	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ)
103	102	relogefd 26962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (log‘(exp‘(𝐵 / 𝐸))) = (𝐵 / 𝐸))
104	100, 103	eqtrid 2807	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐾) = (𝐵 / 𝐸))
105	104	oveq2d 7437	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = ((log‘𝑍) / (𝐵 / 𝐸)))
106	39	rpcnd 13170	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝑍) ∈ ℂ)
107	3	rpcnne0d 13177	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
108	16	rpcnne0d 13177	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
109		divdiv2 12033	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝑍) / (𝐵 / 𝐸)) = (((log‘𝑍) · 𝐸) / 𝐵))
111	105, 110	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) = (((log‘𝑍) · 𝐸) / 𝐵))
112	111	oveq2d 7437	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
113	16	rpcnd 13170	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
114	106, 113	mulcld 11333	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑍) · 𝐸) ∈ ℂ)
115		divass 11996	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = ((𝐿 · 𝐸) · (((log‘𝑍) · 𝐸) / 𝐵)))
117	15	rpcnd 13170	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
118	117, 113, 106, 113	mul4d 11526	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
119	113	sqvald 14290	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) = (𝐸 · 𝐸))
120	119	oveq2d 7437	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (log‘𝑍)) · (𝐸 · 𝐸)))
121	113	sqcld 14291	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸↑2) ∈ ℂ)
122	117, 106, 121	mul32d 11524	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 · (log‘𝑍)) · (𝐸↑2)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
123	118, 120, 122	3eqtr2d 2801	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) = ((𝐿 · (𝐸↑2)) · (log‘𝑍)))
124	123	oveq1d 7436	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) · 𝐸)) / 𝐵) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
125	112, 116, 124	3eqtr2d 2801	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵))
126		8t4e32 12940	. . . . . . . . . 10 ⊢ (8 · 4) = ;32
127	126	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (8 · 4) = ;32)
128	125, 127	oveq12d 7439	. . . . . . . 8 ⊢ (𝜑 → (((𝐿 · 𝐸) · ((log‘𝑍) / (log‘𝐾))) / (8 · 4)) = ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32))
129	20	rpcnd 13170	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℂ)
130	129, 106	mulcld 11333	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) · (log‘𝑍)) ∈ ℂ)
131		rpcnne0 13143	. . . . . . . . . . 11 ⊢ (;32 ∈ ℝ⁺ → (;32 ∈ ℂ ∧ ;32 ≠ 0))
132	25, 131	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → (;32 ∈ ℂ ∧ ;32 ≠ 0))
133		divdiv1 12032	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
135	23	nncni 12349	. . . . . . . . . . 11 ⊢ ;32 ∈ ℂ
136	3	rpcnd 13170	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℂ)
137		mulcom 11290	. . . . . . . . . . 11 ⊢ ((;32 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (;32 · 𝐵) = (𝐵 · ;32))
138	135, 136, 137	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (;32 · 𝐵) = (𝐵 · ;32))
139	138	oveq2d 7437	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (𝐵 · ;32)))
140	27	rpcnne0d 13177	. . . . . . . . . 10 ⊢ (𝜑 → ((;32 · 𝐵) ∈ ℂ ∧ (;32 · 𝐵) ≠ 0))
141		div23 11997	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) · (log‘𝑍)) / (;32 · 𝐵)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
143	134, 139, 142	3eqtr2d 2801	. . . . . . . 8 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) · (log‘𝑍)) / 𝐵) / ;32) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
144	99, 128, 143	3eqtrd 2799	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)))
145	144	oveq1d 7436	. . . . . 6 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (((log‘𝑍) / (log‘𝐾)) / 4)) · (log‘𝑍)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
146	50	rpcnd 13170	. . . . . . 7 ⊢ (𝜑 → ((𝐿 · 𝐸) / 8) ∈ ℂ)
147	89	rpred 13168	. . . . . . . . 9 ⊢ (𝜑 → ((log‘𝑍) / (log‘𝐾)) ∈ ℝ)
148		4nn 12430	. . . . . . . . 9 ⊢ 4 ∈ ℕ
149		nndivre 12383	. . . . . . . . 9 ⊢ ((((log‘𝑍) / (log‘𝐾)) ∈ ℝ ∧ 4 ∈ ℕ) → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
150	147, 148, 149	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℝ)
151	150	recnd 11341	. . . . . . 7 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ∈ ℂ)
152	146, 106, 151	mul32d 11524	. . . . . 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 14290	. . . . . . . 8 ⊢ (𝜑 → ((log‘𝑍)↑2) = ((log‘𝑍) · (log‘𝑍)))
154	153	oveq2d 7437	. . . . . . 7 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
155	28	rpcnd 13170	. . . . . . . 8 ⊢ (𝜑 → ((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) ∈ ℂ)
156	155, 106, 106	mulassd 11336	. . . . . . 7 ⊢ (𝜑 → ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍) · (log‘𝑍))))
157	154, 156	eqtr4d 2798	. . . . . 6 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) = ((((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · (log‘𝑍)) · (log‘𝑍)))
158	145, 152, 157	3eqtr4d 2805	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (((log‘𝑍) / (log‘𝐾)) / 4)) = (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)))
159	56	simp3d 1162	. . . . . 6 ⊢ (𝜑 → (((log‘𝑍) / (log‘𝐾)) / 4) ≤ (𝑁 − 𝑀))
160	150, 63, 51	lemul2d 13212	. . . . . 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 5128	. . . 4 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ≤ ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)))
163	42	rpred 13168	. . . . 5 ⊢ (𝜑 → (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2)) ∈ ℝ)
164	51	rpred 13168	. . . . . 6 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℝ)
165	164, 63	remulcld 11343	. . . . 5 ⊢ (𝜑 → ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀)) ∈ ℝ)
166	163, 165, 13	lemul2d 13212	. . . 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 13170	. . . 4 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℂ)
169	51	rpcnd 13170	. . . 4 ⊢ (𝜑 → (((𝐿 · 𝐸) / 8) · (log‘𝑍)) ∈ ℂ)
170	63	recnd 11341	. . . 4 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℂ)
171	168, 169, 170	mulassd 11336	. . 3 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)) = ((𝑈 − 𝐸) · ((((𝐿 · 𝐸) / 8) · (log‘𝑍)) · (𝑁 − 𝑀))))
172	167, 171	breqtrrd 5132	. 2 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · (𝐸↑2)) / (;32 · 𝐵)) · ((log‘𝑍)↑2))) ≤ (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
173		fzfid 14120	. . . 4 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
174	60	nnzd 12723	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
175	85, 174	rpexpcld 14395	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ⁺)
176	35, 175	rpdivcld 13185	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑁)) ∈ ℝ⁺)
177	176	rprege0d 13175	. . . . . . . . 9 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))))
178		flge0nn0 13964	. . . . . . . . 9 ⊢ (((𝑍 / (𝐾↑𝑁)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑁))) → (⌊‘(𝑍 / (𝐾↑𝑁))) ∈ ℕ₀)
179		nn0p1nn 12649	. . . . . . . . 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 13008	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
182	180, 181	eleqtrdi 2870	. . . . . . 7 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1) ∈ (ℤ_≥‘1))
183		fzss1 13701	. . . . . . 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 3908	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
186	185, 82	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
187	173, 186	fsumrecl 15906	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
188		eluzfz2 13669	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
189	58, 188	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
190		oveq1 7428	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 − 𝑀) = (𝑀 − 𝑀))
191	190	oveq2d 7437	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)))
192		oveq2 7429	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (𝐾↑𝑚) = (𝐾↑𝑀))
193	192	oveq2d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑀)))
194	193	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑀))))
195	194	oveq1d 7436	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1))
196	195	oveq1d 7436	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))))
197	196	sumeq1d 15873	. . . . . . 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 5115	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝑚 − 𝑀) = (𝑗 − 𝑀))
201	200	oveq2d 7437	. . . . . . 7 ⊢ (𝑚 = 𝑗 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)))
202		oveq2 7429	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐾↑𝑚) = (𝐾↑𝑗))
203	202	oveq2d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑗)))
204	203	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑗))))
205	204	oveq1d 7436	. . . . . . . . 9 ⊢ (𝑚 = 𝑗 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
206	205	oveq1d 7436	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))))
207	206	sumeq1d 15873	. . . . . . 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 5115	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (𝑚 − 𝑀) = ((𝑗 + 1) − 𝑀))
211	210	oveq2d 7437	. . . . . . 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 7429	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑗 + 1) → (𝐾↑𝑚) = (𝐾↑(𝑗 + 1)))
213	212	oveq2d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑗 + 1) → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑(𝑗 + 1))))
214	213	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑚 = (𝑗 + 1) → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))))
215	214	oveq1d 7436	. . . . . . . . 9 ⊢ (𝑚 = (𝑗 + 1) → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1))
216	215	oveq1d 7436	. . . . . . . 8 ⊢ (𝑚 = (𝑗 + 1) → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
217	216	sumeq1d 15873	. . . . . . 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 5115	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (𝑚 − 𝑀) = (𝑁 − 𝑀))
221	220	oveq2d 7437	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑚 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑁 − 𝑀)))
222		oveq2 7429	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝐾↑𝑚) = (𝐾↑𝑁))
223	222	oveq2d 7437	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → (𝑍 / (𝐾↑𝑚)) = (𝑍 / (𝐾↑𝑁)))
224	223	fveq2d 6890	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (⌊‘(𝑍 / (𝐾↑𝑚))) = (⌊‘(𝑍 / (𝐾↑𝑁))))
225	224	oveq1d 7436	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((⌊‘(𝑍 / (𝐾↑𝑚))) + 1) = ((⌊‘(𝑍 / (𝐾↑𝑁))) + 1))
226	225	oveq1d 7436	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (((⌊‘(𝑍 / (𝐾↑𝑚))) + 1)...(⌊‘(𝑍 / 𝑌))) = (((⌊‘(𝑍 / (𝐾↑𝑁))) + 1)...(⌊‘(𝑍 / 𝑌))))
227	226	sumeq1d 15873	. . . . . . 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 5115	. . . . . 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 12355	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℂ)
231	230	subidd 11661	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 − 𝑀) = 0)
232	231	oveq2d 7437	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0))
233	52	rpcnd 13170	. . . . . . . . 9 ⊢ (𝜑 → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
234	233	mul01d 11513	. . . . . . . 8 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 0) = 0)
235	232, 234	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑀 − 𝑀)) = 0)
236		fzfid 14120	. . . . . . . 8 ⊢ (𝜑 → (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
237	57	nnzd 12723	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℤ)
238	85, 237	rpexpcld 14395	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾↑𝑀) ∈ ℝ⁺)
239	35, 238	rpdivcld 13185	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / (𝐾↑𝑀)) ∈ ℝ⁺)
240	239	rprege0d 13175	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))))
241		flge0nn0 13964	. . . . . . . . . . . . 13 ⊢ (((𝑍 / (𝐾↑𝑀)) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑𝑀))) → (⌊‘(𝑍 / (𝐾↑𝑀))) ∈ ℕ₀)
242		nn0p1nn 12649	. . . . . . . . . . . . 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 2870	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(𝑍 / (𝐾↑𝑀))) + 1) ∈ (ℤ_≥‘1))
245		fzss1 13701	. . . . . . . . . . 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 3908	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
248	247, 82	syldan 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
249		elfzle2 13665	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
250	249	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ≤ (⌊‘(𝑍 / 𝑌)))
251	29	simpld 500	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
252	35, 251	rpdivcld 13185	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ⁺)
253	252	rpred 13168	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
254		elfzelz 13660	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))) → 𝑛 ∈ ℤ)
255		flge 13949	. . . . . . . . . . . . 13 ⊢ (((𝑍 / 𝑌) ∈ ℝ ∧ 𝑛 ∈ ℤ) → (𝑛 ≤ (𝑍 / 𝑌) ↔ 𝑛 ≤ (⌊‘(𝑍 / 𝑌))))
256	253, 254, 255	syl2an 608	. . . . . . . . . . . 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 521	. . . . . . . . . 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 27933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝑍 / 𝑌))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
261	258, 260	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
262	247, 261	syldan 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 0 ≤ (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
263	236, 248, 262	fsumge0 15968	. . . . . . 7 ⊢ (𝜑 → 0 ≤ Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑀))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)))
264	235, 263	eqbrtrd 5126	. . . . . 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 2760	. . . . . . . . . 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 27937	. . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ⁺)
270	269	rpred 13168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℝ)
271		elfzoelz 13797	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑀..^𝑁) → 𝑗 ∈ ℤ)
272	271	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℤ)
273	272	zred 12807	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℝ)
274	57	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℕ)
275	274	nnred 12354	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
276	273, 275	resubcld 11748	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℝ)
277	270, 276	remulcld 11343	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · (𝑗 − 𝑀)) ∈ ℝ)
278		fzfid 14120	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ∈ Fin)
279		ssun1 4176	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ)
281	85	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ⁺)
282	272	peano2zd 12810	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 + 1) ∈ ℤ)
283	281, 282	rpexpcld 14395	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑(𝑗 + 1)) ∈ ℝ⁺)
284	280, 283	rerpdivcld 13199	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ)
285	281, 272	rpexpcld 14395	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ⁺)
286	280, 285	rerpdivcld 13199	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑𝑗)) ∈ ℝ)
287	86	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝐾 ∈ ℝ)
288		1re 11312	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℝ
289		ltle 11402	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (1 < 𝐾 → 1 ≤ 𝐾))
290	288, 86, 289	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (1 < 𝐾 → 1 ≤ 𝐾))
291	87, 290	mpd 16	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ≤ 𝐾)
292	291	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ≤ 𝐾)
293		uzid 12984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ_≥‘𝑗))
294		peano2uz 13032	. . . . . . . . . . . . . . . . . . . . 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 14402	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ≤ (𝐾↑(𝑗 + 1)))
297	35	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑍 ∈ ℝ⁺)
298	285, 283, 297	lediv2d 13192	. . . . . . . . . . . . . . . . . . 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 13957	. . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑(𝑗 + 1))))))
302		eluzp1p1 12997	. . . . . . . . . . . . . . . . 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 13942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ∈ ℤ)
305	252	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ⁺)
306	305	rpred 13168	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / 𝑌) ∈ ℝ)
307	306	flcld 13942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ ℤ)
308	251	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ⁺)
309	308	rpred 13168	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ∈ ℝ)
310	285	rpred 13168	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝐾↑𝑗) ∈ ℝ)
311	30	simpld 500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
312	311	rpred 13168	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 ∈ ℝ)
313	312	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
314	30	simprd 501	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑌 < 𝑋)
315	314	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < 𝑋)
316		elfzofz 13814	. . . . . . . . . . . . . . . . . . . . . . 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 27932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
318	316, 317	sylan2 605	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑋 < (𝐾↑𝑗) ∧ (𝐾↑𝑗) ≤ (√‘𝑍)))
319	318	simpld 500	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑋 < (𝐾↑𝑗))
320	309, 313, 310, 315, 319	lttrd 11475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 < (𝐾↑𝑗))
321	309, 310, 320	ltled 11462	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑌 ≤ (𝐾↑𝑗))
322	308, 285, 297	lediv2d 13192	. . . . . . . . . . . . . . . . . . 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 13956	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑍 / (𝐾↑𝑗)) ∈ ℝ ∧ (𝑍 / 𝑌) ∈ ℝ ∧ (𝑍 / (𝐾↑𝑗)) ≤ (𝑍 / 𝑌)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
325	286, 306, 323, 324	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) ≤ (⌊‘(𝑍 / 𝑌)))
326		eluz2 12975	. . . . . . . . . . . . . . . . 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 1362	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / 𝑌)) ∈ (ℤ_≥‘(⌊‘(𝑍 / (𝐾↑𝑗)))))
328		fzsplit2 13687	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
331	297, 283	rpdivcld 13185	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ⁺)
332	331	rprege0d 13175	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))))
333		flge0nn0 13964	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑍 / (𝐾↑(𝑗 + 1))) ∈ ℝ ∧ 0 ≤ (𝑍 / (𝐾↑(𝑗 + 1)))) → (⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) ∈ ℕ₀)
334		nn0p1nn 12649	. . . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1) ∈ (ℤ_≥‘1))
337		fzss1 13701	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗)))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
340	339	sselda 3908	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / (𝐾↑𝑗))))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
341	82	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌)))) → (((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
342	340, 341	syldan 603	. . . . . . . . . . 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 15906	. . . . . . . . . 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 14120	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
345		ssun2 4177	. . . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))))
347	346, 338	sstrd 3918	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌))) ⊆ (1...(⌊‘(𝑍 / 𝑌))))
348	347	sselda 3908	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
349	348, 341	syldan 603	. . . . . . . . . . 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 15906	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → Σ𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑𝑗))) + 1)...(⌊‘(𝑍 / 𝑌)))(((𝑈 / 𝑛) − (abs‘((𝑅‘(𝑍 / 𝑛)) / 𝑍))) · (log‘𝑛)) ∈ ℝ)
351		le2add 11802	. . . . . . . . . 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 852	. . . . . . . . 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 708	. . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) ∈ ℂ)
355		1cnd 11306	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 1 ∈ ℂ)
356	272	zcnd 12808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑗 ∈ ℂ)
357	230	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℂ)
358	356, 357	subcld 11673	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (𝑗 − 𝑀) ∈ ℂ)
359	354, 355, 358	adddid 11337	. . . . . . . . . 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 11516	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 − 𝑀) + 1))
361	356, 355, 357	addsubd 11694	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → ((𝑗 + 1) − 𝑀) = ((𝑗 − 𝑀) + 1))
362	360, 361	eqtr4d 2798	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (1 + (𝑗 − 𝑀)) = ((𝑗 + 1) − 𝑀))
363	362	oveq2d 7437	. . . . . . . . . 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 11330	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))) · 1) = ((𝑈 − 𝐸) · (((𝐿 · 𝐸) / 8) · (log‘𝑍))))
365	364	oveq1d 7436	. . . . . . . . . 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 2803	. . . . . . . . 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 13940	. . . . . . . . . . . . 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 12251	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (⌊‘(𝑍 / (𝐾↑𝑗))) < ((⌊‘(𝑍 / (𝐾↑𝑗))) + 1))
370		fzdisj 13689	. . . . . . . . . . 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 14120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) → (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌))) ∈ Fin)
373	338	sselda 3908	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (((⌊‘(𝑍 / (𝐾↑(𝑗 + 1)))) + 1)...(⌊‘(𝑍 / 𝑌)))) → 𝑛 ∈ (1...(⌊‘(𝑍 / 𝑌))))
374	373, 341	syldan 603	. . . . . . . . . . 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 11341	. . . . . . . . . 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 15913	. . . . . . . . 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 5115	. . . . . . . 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 419	. . . . . 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 13927	. . . 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 15969	. . 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 11471	. 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 11471	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 + caddc 11207 · cmul 11209 +∞cpnf 11344 < clt 11347 ≤ cle 11348 − cmin 11545 / cdiv 11977 ℕcn 12339 2c2 12401 3c3 12402 4c4 12403 8c8 12407 ℕ₀cn0 12610 ℤcz 12697 cdc 12818 ℤ_≥cuz 12969 ℝ⁺crp 13124 (,)cioo 13480 [,)cico 13482 [,]cicc 13483 ...cfz 13643 ..^cfzo 13792 ⌊cfl 13934 ↑cexp 14208 √csqrt 15406 abscabs 15407 Σcsu 15859 expce 16233 eceu 16234 logclog 26888 ψcchp 27426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-dju 9986 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-e 16240 df-sin 16241 df-cos 16242 df-pi 16244 df-dvds 16429 df-gcd 16671 df-prm 16853 df-pc 17021 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-log 26890 df-vma 27431 df-chp 27432

This theorem is used by: pntlemo 27940

W3C validator