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Theorem rpvmasumlem 25221
 Description: Lemma for rpvmasum 25260. Calculate the "trivial case" estimate Σ𝑛 ≤ 𝑥( 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  –onto→wfo 5924  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937  Fincfn 7997  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   ≤ cle 10113   − cmin 10304   / cdiv 10722  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ℝ+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  ℙcprime 15432  Basecbs 15904  0gc0g 16147  Grpcgrp 17469  Abelcabl 18240  Unitcui 18685  ℤRHomczrh 19896  ℤ/nℤczn 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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