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Theorem rpvmasumlem 25221
Description: Lemma for rpvmasum 25260. Calculate the "trivial case" estimate Σ𝑛𝑥( 1 (𝑛)Λ(𝑛) / 𝑛) = log𝑥 + 𝑂(1), where 1 (𝑥) is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a (𝜑𝑁 ∈ ℕ)
rpvmasum.g 𝐺 = (DChr‘𝑁)
rpvmasum.d 𝐷 = (Base‘𝐺)
rpvmasum.1 1 = (0g𝐺)
Assertion
Ref Expression
rpvmasumlem (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝑛, 1   𝑛,𝑁,𝑥   𝜑,𝑛,𝑥   𝑛,𝑍,𝑥   𝐷,𝑛,𝑥   𝑛,𝐿,𝑥
Allowed substitution hints:   𝐺(𝑥,𝑛)

Proof of Theorem rpvmasumlem
Dummy variables 𝑘 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 10065 . . . . . 6 ℝ ∈ V
2 rpssre 11881 . . . . . 6 + ⊆ ℝ
31, 2ssexi 4836 . . . . 5 + ∈ V
43a1i 11 . . . 4 (𝜑 → ℝ+ ∈ V)
5 fzfid 12812 . . . . . . 7 (𝜑 → (1...(⌊‘𝑥)) ∈ Fin)
6 elfznn 12408 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
76adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8 vmacl 24889 . . . . . . . . . 10 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
97, 8syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
109, 7nndivred 11107 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
1110recnd 10106 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
125, 11fsumcl 14508 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
1312adantr 480 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
14 relogcl 24367 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1514adantl 481 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
1615recnd 10106 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
1713, 16subcld 10430 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
18 1re 10077 . . . . . . . . 9 1 ∈ ℝ
19 rpvmasum.g . . . . . . . . . . . 12 𝐺 = (DChr‘𝑁)
20 rpvmasum.z . . . . . . . . . . . 12 𝑍 = (ℤ/nℤ‘𝑁)
21 rpvmasum.1 . . . . . . . . . . . 12 1 = (0g𝐺)
22 eqid 2651 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘𝑍)
23 rpvmasum.a . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ)
2419, 20, 21, 22, 23dchr1re 25033 . . . . . . . . . . 11 (𝜑1 :(Base‘𝑍)⟶ℝ)
2524adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 :(Base‘𝑍)⟶ℝ)
2623nnnn0d 11389 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
27 rpvmasum.l . . . . . . . . . . . . 13 𝐿 = (ℤRHom‘𝑍)
2820, 22, 27znzrhfo 19944 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑍))
29 fof 6153 . . . . . . . . . . . 12 (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
3026, 28, 293syl 18 . . . . . . . . . . 11 (𝜑𝐿:ℤ⟶(Base‘𝑍))
31 elfzelz 12380 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
32 ffvelrn 6397 . . . . . . . . . . 11 ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑛 ∈ ℤ) → (𝐿𝑛) ∈ (Base‘𝑍))
3330, 31, 32syl2an 493 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
3425, 33ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
35 resubcl 10383 . . . . . . . . 9 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3618, 34, 35sylancr 696 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3736, 10remulcld 10108 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
3837recnd 10106 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
395, 38fsumcl 14508 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
4039adantr 480 . . . 4 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
41 eqidd 2652 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
42 eqidd 2652 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
434, 17, 40, 41, 42offval2 6956 . . 3 (𝜑 → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))))
4413, 16, 40sub32d 10462 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) − (log‘𝑥)))
455, 11, 38fsumsub 14564 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
46 1cnd 10094 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
4736recnd 10106 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
4846, 47, 11subdird 10525 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = ((1 · ((Λ‘𝑛) / 𝑛)) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
49 ax-1cn 10032 . . . . . . . . . . . 12 1 ∈ ℂ
5034recnd 10106 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℂ)
51 nncan 10348 . . . . . . . . . . . 12 ((1 ∈ ℂ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℂ) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5249, 50, 51sylancr 696 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5352oveq1d 6705 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5411mulid2d 10096 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 · ((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) / 𝑛))
5554oveq1d 6705 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · ((Λ‘𝑛) / 𝑛)) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
5648, 53, 553eqtr3rd 2694 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5756sumeq2dv 14477 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5845, 57eqtr3d 2687 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5958oveq1d 6705 . . . . . 6 (𝜑 → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) − (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
6059adantr 480 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) − (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
6144, 60eqtrd 2685 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
6261mpteq2dva 4777 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))))
6343, 62eqtrd 2685 . 2 (𝜑 → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))))
64 vmadivsum 25216 . . 3 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
652a1i 11 . . . 4 (𝜑 → ℝ+ ⊆ ℝ)
66 1red 10093 . . . 4 (𝜑 → 1 ∈ ℝ)
67 prmdvdsfi 24878 . . . . . 6 (𝑁 ∈ ℕ → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
6823, 67syl 17 . . . . 5 (𝜑 → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
69 elrabi 3391 . . . . . 6 (𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
70 prmnn 15435 . . . . . . . . . 10 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7170adantl 481 . . . . . . . . 9 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
7271nnrpd 11908 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
7372relogcld 24414 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
74 prmuz2 15455 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
7574adantl 481 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
76 uz2m1nn 11801 . . . . . . . 8 (𝑝 ∈ (ℤ‘2) → (𝑝 − 1) ∈ ℕ)
7775, 76syl 17 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
7873, 77nndivred 11107 . . . . . 6 ((𝜑𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
7969, 78sylan2 490 . . . . 5 ((𝜑𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
8068, 79fsumrecl 14509 . . . 4 (𝜑 → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
81 fzfid 12812 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
82 simpr 476 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) = 0)
83 0re 10078 . . . . . . . . . . 11 0 ∈ ℝ
8482, 83syl6eqel 2738 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
85 eqid 2651 . . . . . . . . . . . 12 (Unit‘𝑍) = (Unit‘𝑍)
8623ad3antrrr 766 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → 𝑁 ∈ ℕ)
87 rpvmasum.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐺)
8819dchrabl 25024 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
89 ablgrp 18244 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9087, 21grpidcl 17497 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → 1𝐷)
9123, 88, 89, 904syl 19 . . . . . . . . . . . . . . 15 (𝜑1𝐷)
9291ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1𝐷)
9333adantlr 751 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
9419, 20, 87, 22, 85, 92, 93dchrn0 25020 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (( 1 ‘(𝐿𝑛)) ≠ 0 ↔ (𝐿𝑛) ∈ (Unit‘𝑍)))
9594biimpa 500 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → (𝐿𝑛) ∈ (Unit‘𝑍))
9619, 20, 21, 85, 86, 95dchr1 25027 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) = 1)
9796, 18syl6eqel 2738 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9884, 97pm2.61dane 2910 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9918, 98, 35sylancr 696 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
10010adantlr 751 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
10199, 100remulcld 10108 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
10281, 101fsumrecl 14509 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
103 0le1 10589 . . . . . . . . . . 11 0 ≤ 1
10482, 103syl6eqbr 4724 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
10518leidi 10600 . . . . . . . . . . 11 1 ≤ 1
10696, 105syl6eqbr 4724 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
107104, 106pm2.61dane 2910 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ≤ 1)
108 subge0 10579 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
10918, 98, 108sylancr 696 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
110107, 109mpbird 247 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (1 − ( 1 ‘(𝐿𝑛))))
1119adantlr 751 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
1126adantl 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
113 vmage0 24892 . . . . . . . . . 10 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
114112, 113syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
115112nnred 11073 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
116112nngt0d 11102 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 < 𝑛)
117 divge0 10930 . . . . . . . . 9 ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
118111, 114, 115, 116, 117syl22anc 1367 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
11999, 100, 110, 118mulge0d 10642 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
12081, 101, 119fsumge0 14571 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
121102, 120absidd 14205 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
12268adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
123 inss2 3867 . . . . . . . . 9 ((0[,]𝑥) ∩ ℙ) ⊆ ℙ
124 rabss2 3718 . . . . . . . . 9 (((0[,]𝑥) ∩ ℙ) ⊆ ℙ → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ {𝑞 ∈ ℙ ∣ 𝑞𝑁})
125123, 124mp1i 13 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ {𝑞 ∈ ℙ ∣ 𝑞𝑁})
126 ssfi 8221 . . . . . . . 8 (({𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin ∧ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ∈ Fin)
127122, 125, 126syl2anc 694 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ∈ Fin)
128 ssrab2 3720 . . . . . . . . . 10 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ)
129128, 123sstri 3645 . . . . . . . . 9 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ℙ
130129sseli 3632 . . . . . . . 8 (𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
13178adantlr 751 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
132130, 131sylan2 490 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
133127, 132fsumrecl 14509 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
13480adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
135 fveq2 6229 . . . . . . . . . . . 12 (𝑛 = (𝑝𝑘) → (𝐿𝑛) = (𝐿‘(𝑝𝑘)))
136135fveq2d 6233 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → ( 1 ‘(𝐿𝑛)) = ( 1 ‘(𝐿‘(𝑝𝑘))))
137136oveq2d 6706 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → (1 − ( 1 ‘(𝐿𝑛))) = (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))))
138 fveq2 6229 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → (Λ‘𝑛) = (Λ‘(𝑝𝑘)))
139 id 22 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → 𝑛 = (𝑝𝑘))
140138, 139oveq12d 6708 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → ((Λ‘𝑛) / 𝑛) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
141137, 140oveq12d 6708 . . . . . . . . 9 (𝑛 = (𝑝𝑘) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
142 rpre 11877 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
143142ad2antrl 764 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
14438adantlr 751 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
145 simprr 811 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
146145oveq1d 6705 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = (0 / 𝑛))
1476ad2antrl 764 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
148147nncnd 11074 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℂ)
149147nnne0d 11103 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ≠ 0)
150148, 149div0d 10838 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
151146, 150eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = 0)
152151oveq2d 6706 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿𝑛))) · 0))
15347ad2ant2r 798 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
154153mul01d 10273 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · 0) = 0)
155152, 154eqtrd 2685 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = 0)
156141, 143, 144, 155fsumvma2 24984 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
157128a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ))
158 fzfid 12812 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
15924ad2antrr 762 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 :(Base‘𝑍)⟶ℝ)
16030ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝐿:ℤ⟶(Base‘𝑍))
16170ad2antrl 764 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
162 elfznn 12408 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
163162ad2antll 765 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
164163nnnn0d 11389 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ0)
165161, 164nnexpcld 13070 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℕ)
166165nnzd 11519 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℤ)
167160, 166ffvelrnd 6400 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝐿‘(𝑝𝑘)) ∈ (Base‘𝑍))
168159, 167ffvelrnd 6400 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ)
169 resubcl 10383 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
17018, 168, 169sylancr 696 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
171 vmacl 24889 . . . . . . . . . . . . . . . 16 ((𝑝𝑘) ∈ ℕ → (Λ‘(𝑝𝑘)) ∈ ℝ)
172165, 171syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℝ)
173172, 165nndivred 11107 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℝ)
174170, 173remulcld 10108 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
175174anassrs 681 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
176175recnd 10106 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
177158, 176fsumcl 14508 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
178130, 177sylan2 490 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
179 breq1 4688 . . . . . . . . . . . 12 (𝑞 = 𝑝 → (𝑞𝑁𝑝𝑁))
180179notbid 307 . . . . . . . . . . 11 (𝑞 = 𝑝 → (¬ 𝑞𝑁 ↔ ¬ 𝑝𝑁))
181 notrab 3937 . . . . . . . . . . 11 (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) = {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ ¬ 𝑞𝑁}
182180, 181elrab2 3399 . . . . . . . . . 10 (𝑝 ∈ (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) ↔ (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) ∧ ¬ 𝑝𝑁))
183123sseli 3632 . . . . . . . . . . 11 (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) → 𝑝 ∈ ℙ)
18423ad3antrrr 766 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ)
185 simplrr 818 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ¬ 𝑝𝑁)
186 simplrl 817 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℙ)
187184nnzd 11519 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℤ)
188 coprm 15470 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
189186, 187, 188syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
190185, 189mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝 gcd 𝑁) = 1)
191 prmz 15436 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
192186, 191syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℤ)
193162adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ)
194193nnnn0d 11389 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ0)
195 rpexp1i 15480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
196192, 187, 194, 195syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
197190, 196mpd 15 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝𝑘) gcd 𝑁) = 1)
198184nnnn0d 11389 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ0)
199166anassrs 681 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
200199adantlrr 757 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
20120, 85, 27znunit 19960 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑝𝑘) ∈ ℤ) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
202198, 200, 201syl2anc 694 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
203197, 202mpbird 247 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
20419, 20, 21, 85, 184, 203dchr1 25027 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
205204oveq2d 6706 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = (1 − 1))
206 1m1e0 11127 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
207205, 206syl6eq 2701 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = 0)
208207oveq1d 6705 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = (0 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
209173recnd 10106 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
210209anassrs 681 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
211210adantlrr 757 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
212211mul02d 10272 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (0 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
213208, 212eqtrd 2685 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
214213sumeq2dv 14477 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0)
215 fzfid 12812 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
216215olcd 407 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → ((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin))
217 sumz 14497 . . . . . . . . . . . . 13 (((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
218216, 217syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
219214, 218eqtrd 2685 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
220183, 219sylanr1 685 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
221182, 220sylan2b 491 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁})) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
222 ppifi 24877 . . . . . . . . . 10 (𝑥 ∈ ℝ → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
223143, 222syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
224157, 178, 221, 223fsumss 14500 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
225156, 224eqtr4d 2688 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
226158, 175fsumrecl 14509 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
227130, 226sylan2 490 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
22873adantlr 751 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
22970adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
230229nnrecred 11104 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ)
231229nnrpd 11908 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
232231rpreccld 11920 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ+)
233 simplrl 817 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ ℝ+)
234233relogcld 24414 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑥) ∈ ℝ)
235229nnred 11073 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
23674adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
237 eluz2b2 11799 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
238237simprbi 479 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
239236, 238syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 < 𝑝)
240235, 239rplogcld 24420 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ+)
241234, 240rerpdivcld 11941 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑥) / (log‘𝑝)) ∈ ℝ)
242241flcld 12639 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ)
243242peano2zd 11523 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℤ)
244232, 243rpexpcld 13072 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ+)
245244rpred 11910 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ)
246230, 245resubcld 10496 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ∈ ℝ)
247236, 76syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
248247nnrpd 11908 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℝ+)
249248, 231rpdivcld 11927 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) ∈ ℝ+)
250246, 249rerpdivcld 11941 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ∈ ℝ)
251228, 250remulcld 10108 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ∈ ℝ)
252172recnd 10106 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℂ)
253165nncnd 11074 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℂ)
254165nnne0d 11103 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ≠ 0)
255252, 253, 254divrecd 10842 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))))
256 simprl 809 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
257 vmappw 24887 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
258256, 163, 257syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
259161nncnd 11074 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℂ)
260161nnne0d 11103 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ≠ 0)
261 elfzelz 12380 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℤ)
262261ad2antll 765 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℤ)
263259, 260, 262exprecd 13056 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝𝑘)))
264263eqcomd 2657 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / (𝑝𝑘)) = ((1 / 𝑝)↑𝑘))
265258, 264oveq12d 6708 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
266255, 265eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
267266, 173eqeltrrd 2731 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
268267anassrs 681 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
269 1red 10093 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 ∈ ℝ)
270 vmage0 24892 . . . . . . . . . . . . . . . . 17 ((𝑝𝑘) ∈ ℕ → 0 ≤ (Λ‘(𝑝𝑘)))
271165, 270syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ (Λ‘(𝑝𝑘)))
272165nnred 11073 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℝ)
273165nngt0d 11102 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 < (𝑝𝑘))
274 divge0 10930 . . . . . . . . . . . . . . . 16 ((((Λ‘(𝑝𝑘)) ∈ ℝ ∧ 0 ≤ (Λ‘(𝑝𝑘))) ∧ ((𝑝𝑘) ∈ ℝ ∧ 0 < (𝑝𝑘))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
275172, 271, 272, 273, 274syl22anc 1367 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
27683leidi 10600 . . . . . . . . . . . . . . . . . 18 0 ≤ 0
277 simpr 476 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 0)
278276, 277syl5breqr 4723 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
27923ad3antrrr 766 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 𝑁 ∈ ℕ)
28091ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1𝐷)
28119, 20, 87, 22, 85, 280, 167dchrn0 25020 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0 ↔ (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍)))
282281biimpa 500 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
28319, 20, 21, 85, 279, 282dchr1 25027 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
284103, 283syl5breqr 4723 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
285278, 284pm2.61dane 2910 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
286 subge02 10582 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
28718, 168, 286sylancr 696 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
288285, 287mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1)
289170, 269, 173, 275, 288lemul1ad 11001 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
290209mulid2d 10096 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
291290, 266eqtrd 2685 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
292289, 291breqtrd 4711 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
293292anassrs 681 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
294158, 175, 268, 293fsumle 14575 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
295228recnd 10106 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℂ)
296161nnrecred 11104 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / 𝑝) ∈ ℝ)
297296, 164reexpcld 13065 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℝ)
298297recnd 10106 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
299298anassrs 681 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
300158, 295, 299fsummulc2 14560 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
301 fzval3 12576 . . . . . . . . . . . . . . . 16 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
302242, 301syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
303302sumeq1d 14475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = Σ𝑘 ∈ (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘))
304230recnd 10106 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℂ)
305229nngt0d 11102 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 < 𝑝)
306 recgt1 10957 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
307235, 305, 306syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
308239, 307mpbid 222 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) < 1)
309230, 308ltned 10211 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ≠ 1)
310 1nn0 11346 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
311310a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℕ0)
312 log1 24377 . . . . . . . . . . . . . . . . . . . . 21 (log‘1) = 0
313 simprr 811 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
314 1rp 11874 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ+
315 simprl 809 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
316 logleb 24394 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
317314, 315, 316sylancr 696 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
318313, 317mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
319312, 318syl5eqbrr 4721 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
320319adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ (log‘𝑥))
321234, 240, 320divge0d 11950 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑥) / (log‘𝑝)))
322 flge0nn0 12661 . . . . . . . . . . . . . . . . . 18 ((((log‘𝑥) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝑥) / (log‘𝑝))) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
323241, 321, 322syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
324 nn0p1nn 11370 . . . . . . . . . . . . . . . . 17 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0 → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
325323, 324syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
326 nnuz 11761 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
327325, 326syl6eleq 2740 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ (ℤ‘1))
328304, 309, 311, 327geoserg 14642 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘) = ((((1 / 𝑝)↑1) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))))
329304exp1d 13043 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑1) = (1 / 𝑝))
330329oveq1d 6705 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝)↑1) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) = ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))))
331229nncnd 11074 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℂ)
332 1cnd 10094 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
333231rpcnne0d 11919 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
334 divsubdir 10759 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
335331, 332, 333, 334syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
336 divid 10752 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
337333, 336syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 / 𝑝) = 1)
338337oveq1d 6705 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
339335, 338eqtr2d 2686 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 − (1 / 𝑝)) = ((𝑝 − 1) / 𝑝))
340330, 339oveq12d 6708 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((((1 / 𝑝)↑1) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) = (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)))
341303, 328, 3403eqtrd 2689 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)))
342341oveq2d 6706 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))))
343300, 342eqtr3d 2687 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))))
344294, 343breqtrd 4711 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))))
345244rpge0d 11914 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
346230, 245subge02d 10657 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ↔ ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (1 / 𝑝)))
347345, 346mpbid 222 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (1 / 𝑝))
348248rpcnne0d 11919 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
349 dmdcan 10773 . . . . . . . . . . . . . . 15 ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ 1 ∈ ℂ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
350348, 333, 332, 349syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
351347, 350breqtrrd 4713 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))))
352247nnrecred 11104 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / (𝑝 − 1)) ∈ ℝ)
353246, 352, 249ledivmuld 11963 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ≤ (1 / (𝑝 − 1)) ↔ ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1)))))
354351, 353mpbird 247 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ≤ (1 / (𝑝 − 1)))
355250, 352, 240lemul2d 11954 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ≤ (1 / (𝑝 − 1)) ↔ ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ≤ ((log‘𝑝) · (1 / (𝑝 − 1)))))
356354, 355mpbid 222 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ≤ ((log‘𝑝) · (1 / (𝑝 − 1))))
357247nncnd 11074 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℂ)
358247nnne0d 11103 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ≠ 0)
359295, 357, 358divrecd 10842 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) = ((log‘𝑝) · (1 / (𝑝 − 1))))
360356, 359breqtrrd 4713 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ≤ ((log‘𝑝) / (𝑝 − 1)))
361226, 251, 131, 344, 360letrd 10232 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ ((log‘𝑝) / (𝑝 − 1)))
362130, 361sylan2 490 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ ((log‘𝑝) / (𝑝 − 1)))
363127, 227, 132, 362fsumle 14575 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ≤ Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
364225, 363eqbrtrd 4707 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ≤ Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
36579adantlr 751 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
366240, 248rpdivcld 11927 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ+)
367366rpge0d 11914 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
36869, 367sylan2 490 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
369122, 365, 368, 125fsumless 14572 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
370102, 133, 134, 364, 369letrd 10232 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
371121, 370eqbrtrd 4707 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
37265, 40, 66, 80, 371elo1d 14311 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
373 o1sub 14390 . . 3 (((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))) ∈ 𝑂(1))
37464, 372, 373sylancr 696 . 2 (𝜑 → ((𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∘𝑓 − (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))) ∈ 𝑂(1))
37563, 374eqeltrrd 2731 1 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607   class class class wbr 4685  cmpt 4762  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  +crp 11870  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  cfl 12631  cexp 12900  abscabs 14018  𝑂(1)co1 14261  Σcsu 14460  cdvds 15027   gcd cgcd 15263  cprime 15432  Basecbs 15904  0gc0g 16147  Grpcgrp 17469  Abelcabl 18240  Unitcui 18685  ℤRHomczrh 19896  ℤ/nczn 19899  logclog 24346  Λcvma 24863  DChrcdchr 25002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-o1 14265  df-lo1 14266  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-qus 16216  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-nsg 17639  df-eqg 17640  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-rnghom 18763  df-subrg 18826  df-lmod 18913  df-lss 18981  df-lsp 19020  df-sra 19220  df-rgmod 19221  df-lidl 19222  df-rsp 19223  df-2idl 19280  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-zring 19867  df-zrh 19900  df-zn 19903  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-cht 24868  df-vma 24869  df-chp 24870  df-ppi 24871  df-dchr 25003
This theorem is referenced by:  rpvmasum2  25246
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