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Theorem rpvmasumlem 25628
Description: Lemma for rpvmasum 25667. 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 10363 . . . . . 6 ℝ ∈ V
2 rpssre 12144 . . . . . 6 + ⊆ ℝ
31, 2ssexi 5040 . . . . 5 + ∈ V
43a1i 11 . . . 4 (𝜑 → ℝ+ ∈ V)
5 fzfid 13091 . . . . . . 7 (𝜑 → (1...(⌊‘𝑥)) ∈ Fin)
6 elfznn 12687 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
76adantl 475 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8 vmacl 25296 . . . . . . . . . 10 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
97, 8syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
109, 7nndivred 11429 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
1110recnd 10405 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
125, 11fsumcl 14871 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
1312adantr 474 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
14 relogcl 24759 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1514adantl 475 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
1615recnd 10405 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
1713, 16subcld 10734 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
18 1re 10376 . . . . . . . . 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 2778 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘𝑍)
23 rpvmasum.a . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ)
2419, 20, 21, 22, 23dchr1re 25440 . . . . . . . . . . 11 (𝜑1 :(Base‘𝑍)⟶ℝ)
2524adantr 474 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 :(Base‘𝑍)⟶ℝ)
2623nnnn0d 11702 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
27 rpvmasum.l . . . . . . . . . . . . 13 𝐿 = (ℤRHom‘𝑍)
2820, 22, 27znzrhfo 20291 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑍))
29 fof 6366 . . . . . . . . . . . 12 (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
3026, 28, 293syl 18 . . . . . . . . . . 11 (𝜑𝐿:ℤ⟶(Base‘𝑍))
31 elfzelz 12659 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
32 ffvelrn 6621 . . . . . . . . . . 11 ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑛 ∈ ℤ) → (𝐿𝑛) ∈ (Base‘𝑍))
3330, 31, 32syl2an 589 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
3425, 33ffvelrnd 6624 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
35 resubcl 10687 . . . . . . . . 9 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3618, 34, 35sylancr 581 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3736, 10remulcld 10407 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
3837recnd 10405 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
395, 38fsumcl 14871 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
4039adantr 474 . . . 4 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
41 eqidd 2779 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
42 eqidd 2779 . . . 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 7191 . . 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 10766 . . . . 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 14924 . . . . . . . 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 10371 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
4736recnd 10405 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
4846, 47, 11subdird 10832 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = ((1 · ((Λ‘𝑛) / 𝑛)) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
49 ax-1cn 10330 . . . . . . . . . . . 12 1 ∈ ℂ
5034recnd 10405 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℂ)
51 nncan 10652 . . . . . . . . . . . 12 ((1 ∈ ℂ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℂ) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5249, 50, 51sylancr 581 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5352oveq1d 6937 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5411mulid2d 10395 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 · ((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) / 𝑛))
5554oveq1d 6937 . . . . . . . . . 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 2823 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5756sumeq2dv 14841 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5845, 57eqtr3d 2816 . . . . . . 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 6937 . . . . . 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 474 . . . . 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 2814 . . . 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 4979 . . 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 2814 . 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 25623 . . 3 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
652a1i 11 . . . 4 (𝜑 → ℝ+ ⊆ ℝ)
66 1red 10377 . . . 4 (𝜑 → 1 ∈ ℝ)
67 prmdvdsfi 25285 . . . . . 6 (𝑁 ∈ ℕ → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
6823, 67syl 17 . . . . 5 (𝜑 → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
69 elrabi 3567 . . . . . 6 (𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
70 prmnn 15793 . . . . . . . . . 10 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7170adantl 475 . . . . . . . . 9 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
7271nnrpd 12179 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
7372relogcld 24806 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
74 prmuz2 15813 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
7574adantl 475 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
76 uz2m1nn 12070 . . . . . . . 8 (𝑝 ∈ (ℤ‘2) → (𝑝 − 1) ∈ ℕ)
7775, 76syl 17 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
7873, 77nndivred 11429 . . . . . 6 ((𝜑𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
7969, 78sylan2 586 . . . . 5 ((𝜑𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
8068, 79fsumrecl 14872 . . . 4 (𝜑 → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
81 fzfid 13091 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
82 simpr 479 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) = 0)
83 0re 10378 . . . . . . . . . . 11 0 ∈ ℝ
8482, 83syl6eqel 2867 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
85 eqid 2778 . . . . . . . . . . . 12 (Unit‘𝑍) = (Unit‘𝑍)
8623ad3antrrr 720 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → 𝑁 ∈ ℕ)
87 rpvmasum.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐺)
8819dchrabl 25431 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
89 ablgrp 18584 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9087, 21grpidcl 17837 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → 1𝐷)
9123, 88, 89, 904syl 19 . . . . . . . . . . . . . . 15 (𝜑1𝐷)
9291ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1𝐷)
9333adantlr 705 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
9419, 20, 87, 22, 85, 92, 93dchrn0 25427 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (( 1 ‘(𝐿𝑛)) ≠ 0 ↔ (𝐿𝑛) ∈ (Unit‘𝑍)))
9594biimpa 470 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → (𝐿𝑛) ∈ (Unit‘𝑍))
9619, 20, 21, 85, 86, 95dchr1 25434 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) = 1)
9796, 18syl6eqel 2867 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9884, 97pm2.61dane 3057 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9918, 98, 35sylancr 581 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
10010adantlr 705 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
10199, 100remulcld 10407 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
10281, 101fsumrecl 14872 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
103 0le1 10898 . . . . . . . . . . 11 0 ≤ 1
10482, 103syl6eqbr 4925 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
10518leidi 10909 . . . . . . . . . . 11 1 ≤ 1
10696, 105syl6eqbr 4925 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
107104, 106pm2.61dane 3057 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ≤ 1)
108 subge0 10888 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
10918, 98, 108sylancr 581 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
110107, 109mpbird 249 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (1 − ( 1 ‘(𝐿𝑛))))
1119adantlr 705 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
1126adantl 475 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
113 vmage0 25299 . . . . . . . . . 10 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
114112, 113syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
115112nnred 11391 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
116112nngt0d 11424 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 < 𝑛)
117 divge0 11246 . . . . . . . . 9 ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
118111, 114, 115, 116, 117syl22anc 829 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
11999, 100, 110, 118mulge0d 10952 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
12081, 101, 119fsumge0 14931 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
121102, 120absidd 14569 . . . . 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 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
123 inss2 4054 . . . . . . . . 9 ((0[,]𝑥) ∩ ℙ) ⊆ ℙ
124 rabss2 3906 . . . . . . . . 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 8468 . . . . . . . 8 (({𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin ∧ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ∈ Fin)
127122, 125, 126syl2anc 579 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ∈ Fin)
128 ssrab2 3908 . . . . . . . . . 10 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ)
129128, 123sstri 3830 . . . . . . . . 9 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ℙ
130129sseli 3817 . . . . . . . 8 (𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
13178adantlr 705 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
132130, 131sylan2 586 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
133127, 132fsumrecl 14872 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
13480adantr 474 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
135 2fveq3 6451 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → ( 1 ‘(𝐿𝑛)) = ( 1 ‘(𝐿‘(𝑝𝑘))))
136135oveq2d 6938 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → (1 − ( 1 ‘(𝐿𝑛))) = (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))))
137 fveq2 6446 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → (Λ‘𝑛) = (Λ‘(𝑝𝑘)))
138 id 22 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → 𝑛 = (𝑝𝑘))
139137, 138oveq12d 6940 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → ((Λ‘𝑛) / 𝑛) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
140136, 139oveq12d 6940 . . . . . . . . 9 (𝑛 = (𝑝𝑘) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
141 rpre 12145 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
142141ad2antrl 718 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
14338adantlr 705 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
144 simprr 763 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
145144oveq1d 6937 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = (0 / 𝑛))
1466ad2antrl 718 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
147146nncnd 11392 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℂ)
148146nnne0d 11425 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ≠ 0)
149147, 148div0d 11150 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
150145, 149eqtrd 2814 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = 0)
151150oveq2d 6938 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿𝑛))) · 0))
15247ad2ant2r 737 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
153152mul01d 10575 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · 0) = 0)
154151, 153eqtrd 2814 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = 0)
155140, 142, 143, 154fsumvma2 25391 . . . . . . . 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))))
156128a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ))
157 fzfid 13091 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
15824ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 :(Base‘𝑍)⟶ℝ)
15930ad2antrr 716 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝐿:ℤ⟶(Base‘𝑍))
16070ad2antrl 718 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
161 elfznn 12687 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
162161ad2antll 719 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
163162nnnn0d 11702 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ0)
164160, 163nnexpcld 13351 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℕ)
165164nnzd 11833 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℤ)
166159, 165ffvelrnd 6624 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝐿‘(𝑝𝑘)) ∈ (Base‘𝑍))
167158, 166ffvelrnd 6624 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ)
168 resubcl 10687 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
16918, 167, 168sylancr 581 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
170 vmacl 25296 . . . . . . . . . . . . . . . 16 ((𝑝𝑘) ∈ ℕ → (Λ‘(𝑝𝑘)) ∈ ℝ)
171164, 170syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℝ)
172171, 164nndivred 11429 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℝ)
173169, 172remulcld 10407 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
174173anassrs 461 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
175174recnd 10405 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
176157, 175fsumcl 14871 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
177130, 176sylan2 586 . . . . . . . . 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)
178 breq1 4889 . . . . . . . . . . . 12 (𝑞 = 𝑝 → (𝑞𝑁𝑝𝑁))
179178notbid 310 . . . . . . . . . . 11 (𝑞 = 𝑝 → (¬ 𝑞𝑁 ↔ ¬ 𝑝𝑁))
180 notrab 4130 . . . . . . . . . . 11 (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) = {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ ¬ 𝑞𝑁}
181179, 180elrab2 3576 . . . . . . . . . 10 (𝑝 ∈ (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) ↔ (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) ∧ ¬ 𝑝𝑁))
182123sseli 3817 . . . . . . . . . . 11 (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) → 𝑝 ∈ ℙ)
18323ad3antrrr 720 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ)
184 simplrr 768 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ¬ 𝑝𝑁)
185 simplrl 767 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℙ)
186183nnzd 11833 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℤ)
187 coprm 15827 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
188185, 186, 187syl2anc 579 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
189184, 188mpbid 224 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝 gcd 𝑁) = 1)
190 prmz 15794 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
191185, 190syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℤ)
192161adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ)
193192nnnn0d 11702 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ0)
194 rpexp1i 15837 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
195191, 186, 193, 194syl3anc 1439 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
196189, 195mpd 15 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝𝑘) gcd 𝑁) = 1)
197183nnnn0d 11702 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ0)
198165anassrs 461 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
199198adantlrr 711 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
20020, 85, 27znunit 20307 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑝𝑘) ∈ ℤ) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
201197, 199, 200syl2anc 579 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
202196, 201mpbird 249 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
20319, 20, 21, 85, 183, 202dchr1 25434 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
204203oveq2d 6938 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = (1 − 1))
205 1m1e0 11447 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
206204, 205syl6eq 2830 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = 0)
207206oveq1d 6937 . . . . . . . . . . . . . 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))))
208172recnd 10405 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
209208anassrs 461 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
210209adantlrr 711 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
211210mul02d 10574 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (0 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
212207, 211eqtrd 2814 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
213212sumeq2dv 14841 . . . . . . . . . . . 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)
214 fzfid 13091 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
215214olcd 863 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → ((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin))
216 sumz 14860 . . . . . . . . . . . . 13 (((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
217215, 216syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
218213, 217eqtrd 2814 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
219182, 218sylanr1 672 . . . . . . . . . 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)
220181, 219sylan2b 587 . . . . . . . . 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)
221 ppifi 25284 . . . . . . . . . 10 (𝑥 ∈ ℝ → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
222142, 221syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
223156, 177, 220, 222fsumss 14863 . . . . . . . 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))))
224155, 223eqtr4d 2817 . . . . . . 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))))
225157, 174fsumrecl 14872 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
226130, 225sylan2 586 . . . . . . . 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)
22773adantlr 705 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
22870adantl 475 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
229228nnrecred 11426 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ)
230228nnrpd 12179 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
231230rpreccld 12191 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ+)
232 simplrl 767 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ ℝ+)
233232relogcld 24806 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑥) ∈ ℝ)
234228nnred 11391 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
23574adantl 475 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
236 eluz2b2 12068 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
237236simprbi 492 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
238235, 237syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 < 𝑝)
239234, 238rplogcld 24812 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ+)
240233, 239rerpdivcld 12212 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑥) / (log‘𝑝)) ∈ ℝ)
241240flcld 12918 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ)
242241peano2zd 11837 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℤ)
243231, 242rpexpcld 13353 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ+)
244243rpred 12181 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ)
245229, 244resubcld 10803 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ∈ ℝ)
246235, 76syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
247246nnrpd 12179 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℝ+)
248247, 230rpdivcld 12198 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) ∈ ℝ+)
249245, 248rerpdivcld 12212 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ∈ ℝ)
250227, 249remulcld 10407 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ∈ ℝ)
251171recnd 10405 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℂ)
252164nncnd 11392 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℂ)
253164nnne0d 11425 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ≠ 0)
254251, 252, 253divrecd 11154 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))))
255 simprl 761 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
256 vmappw 25294 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
257255, 162, 256syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
258160nncnd 11392 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℂ)
259160nnne0d 11425 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ≠ 0)
260 elfzelz 12659 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℤ)
261260ad2antll 719 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℤ)
262258, 259, 261exprecd 13335 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝𝑘)))
263262eqcomd 2784 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / (𝑝𝑘)) = ((1 / 𝑝)↑𝑘))
264257, 263oveq12d 6940 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
265254, 264eqtrd 2814 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
266265, 172eqeltrrd 2860 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
267266anassrs 461 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
268 1red 10377 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 ∈ ℝ)
269 vmage0 25299 . . . . . . . . . . . . . . . . 17 ((𝑝𝑘) ∈ ℕ → 0 ≤ (Λ‘(𝑝𝑘)))
270164, 269syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ (Λ‘(𝑝𝑘)))
271164nnred 11391 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℝ)
272164nngt0d 11424 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 < (𝑝𝑘))
273 divge0 11246 . . . . . . . . . . . . . . . 16 ((((Λ‘(𝑝𝑘)) ∈ ℝ ∧ 0 ≤ (Λ‘(𝑝𝑘))) ∧ ((𝑝𝑘) ∈ ℝ ∧ 0 < (𝑝𝑘))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
274171, 270, 271, 272, 273syl22anc 829 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
27583leidi 10909 . . . . . . . . . . . . . . . . . 18 0 ≤ 0
276 simpr 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 0)
277275, 276syl5breqr 4924 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
27823ad3antrrr 720 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 𝑁 ∈ ℕ)
27991ad2antrr 716 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1𝐷)
28019, 20, 87, 22, 85, 279, 166dchrn0 25427 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0 ↔ (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍)))
281280biimpa 470 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
28219, 20, 21, 85, 278, 281dchr1 25434 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
283103, 282syl5breqr 4924 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
284277, 283pm2.61dane 3057 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
285 subge02 10891 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
28618, 167, 285sylancr 581 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
287284, 286mpbid 224 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1)
288169, 268, 172, 274, 287lemul1ad 11317 . . . . . . . . . . . . . 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))))
289208mulid2d 10395 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
290289, 265eqtrd 2814 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
291288, 290breqtrd 4912 . . . . . . . . . . . . 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)))
292291anassrs 461 . . . . . . . . . . . 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)))
293157, 174, 267, 292fsumle 14935 . . . . . . . . . . 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)))
294227recnd 10405 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℂ)
295160nnrecred 11426 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / 𝑝) ∈ ℝ)
296295, 163reexpcld 13344 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℝ)
297296recnd 10405 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
298297anassrs 461 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
299157, 294, 298fsummulc2 14920 . . . . . . . . . . . 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)))
300 fzval3 12856 . . . . . . . . . . . . . . . 16 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
301241, 300syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
302301sumeq1d 14839 . . . . . . . . . . . . . 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))
303229recnd 10405 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℂ)
304228nngt0d 11424 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 < 𝑝)
305 recgt1 11273 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
306234, 304, 305syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
307238, 306mpbid 224 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) < 1)
308229, 307ltned 10512 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ≠ 1)
309 1nn0 11660 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
310309a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℕ0)
311 log1 24769 . . . . . . . . . . . . . . . . . . . . 21 (log‘1) = 0
312 simprr 763 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
313 1rp 12141 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ+
314 simprl 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
315 logleb 24786 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
316313, 314, 315sylancr 581 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
317312, 316mpbid 224 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
318311, 317syl5eqbrr 4922 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
319318adantr 474 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ (log‘𝑥))
320233, 239, 319divge0d 12221 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑥) / (log‘𝑝)))
321 flge0nn0 12940 . . . . . . . . . . . . . . . . . 18 ((((log‘𝑥) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝑥) / (log‘𝑝))) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
322240, 320, 321syl2anc 579 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
323 nn0p1nn 11683 . . . . . . . . . . . . . . . . 17 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0 → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
324322, 323syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
325 nnuz 12029 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
326324, 325syl6eleq 2869 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ (ℤ‘1))
327303, 308, 310, 326geoserg 15002 . . . . . . . . . . . . . 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))))
328303exp1d 13322 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑1) = (1 / 𝑝))
329328oveq1d 6937 . . . . . . . . . . . . . . 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))))
330228nncnd 11392 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℂ)
331 1cnd 10371 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
332230rpcnne0d 12190 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
333 divsubdir 11069 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
334330, 331, 332, 333syl3anc 1439 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
335 divid 11062 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
336332, 335syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 / 𝑝) = 1)
337336oveq1d 6937 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
338334, 337eqtr2d 2815 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 − (1 / 𝑝)) = ((𝑝 − 1) / 𝑝))
339329, 338oveq12d 6940 . . . . . . . . . . . . . 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)))
340302, 327, 3393eqtrd 2818 . . . . . . . . . . . . 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)))
341340oveq2d 6938 . . . . . . . . . . . 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))))
342299, 341eqtr3d 2816 . . . . . . . . . . 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))))
343293, 342breqtrd 4912 . . . . . . . . . 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))))
344243rpge0d 12185 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
345229, 244subge02d 10967 . . . . . . . . . . . . . . 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)))
346344, 345mpbid 224 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (1 / 𝑝))
347247rpcnne0d 12190 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
348 dmdcan 11085 . . . . . . . . . . . . . . 15 ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ 1 ∈ ℂ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
349347, 332, 331, 348syl3anc 1439 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
350346, 349breqtrrd 4914 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))))
351246nnrecred 11426 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / (𝑝 − 1)) ∈ ℝ)
352245, 351, 248ledivmuld 12234 . . . . . . . . . . . . 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)))))
353350, 352mpbird 249 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ≤ (1 / (𝑝 − 1)))
354249, 351, 239lemul2d 12225 . . . . . . . . . . . 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)))))
355353, 354mpbid 224 . . . . . . . . . . 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))))
356246nncnd 11392 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℂ)
357246nnne0d 11425 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ≠ 0)
358294, 356, 357divrecd 11154 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) = ((log‘𝑝) · (1 / (𝑝 − 1))))
359355, 358breqtrrd 4914 . . . . . . . . . 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)))
360225, 250, 131, 343, 359letrd 10533 . . . . . . . . 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)))
361130, 360sylan2 586 . . . . . . . 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)))
362127, 226, 132, 361fsumle 14935 . . . . . . 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)))
363224, 362eqbrtrd 4908 . . . . . 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)))
36479adantlr 705 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
365239, 247rpdivcld 12198 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ+)
366365rpge0d 12185 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
36769, 366sylan2 586 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
368122, 364, 367, 125fsumless 14932 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
369102, 133, 134, 363, 368letrd 10533 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
370121, 369eqbrtrd 4908 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
37165, 40, 66, 80, 370elo1d 14675 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
372 o1sub 14754 . . 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))
37364, 371, 372sylancr 581 . 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))
37463, 373eqeltrrd 2860 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 198  wa 386  wo 836   = wceq 1601  wcel 2107  wne 2969  {crab 3094  Vcvv 3398  cdif 3789  cin 3791  wss 3792   class class class wbr 4886  cmpt 4965  wf 6131  ontowfo 6133  cfv 6135  (class class class)co 6922  𝑓 cof 7172  Fincfn 8241  cc 10270  cr 10271  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277   < clt 10411  cle 10412  cmin 10606   / cdiv 11032  cn 11374  2c2 11430  0cn0 11642  cz 11728  cuz 11992  +crp 12137  [,]cicc 12490  ...cfz 12643  ..^cfzo 12784  cfl 12910  cexp 13178  abscabs 14381  𝑂(1)co1 14625  Σcsu 14824  cdvds 15387   gcd cgcd 15622  cprime 15790  Basecbs 16255  0gc0g 16486  Grpcgrp 17809  Abelcabl 18580  Unitcui 19026  ℤRHomczrh 20244  ℤ/nczn 20247  logclog 24738  Λcvma 25270  DChrcdchr 25409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351  ax-mulf 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-tpos 7634  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-ec 8028  df-qs 8032  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-fi 8605  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-xnn0 11715  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-ioo 12491  df-ioc 12492  df-ico 12493  df-icc 12494  df-fz 12644  df-fzo 12785  df-fl 12912  df-mod 12988  df-seq 13120  df-exp 13179  df-fac 13379  df-bc 13408  df-hash 13436  df-shft 14214  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-limsup 14610  df-clim 14627  df-rlim 14628  df-o1 14629  df-lo1 14630  df-sum 14825  df-ef 15200  df-e 15201  df-sin 15202  df-cos 15203  df-pi 15205  df-dvds 15388  df-gcd 15623  df-prm 15791  df-pc 15946  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-starv 16353  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-unif 16361  df-hom 16362  df-cco 16363  df-rest 16469  df-topn 16470  df-0g 16488  df-gsum 16489  df-topgen 16490  df-pt 16491  df-prds 16494  df-xrs 16548  df-qtop 16553  df-imas 16554  df-qus 16555  df-xps 16556  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mhm 17721  df-submnd 17722  df-grp 17812  df-minusg 17813  df-sbg 17814  df-mulg 17928  df-subg 17975  df-nsg 17976  df-eqg 17977  df-ghm 18042  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-ring 18936  df-cring 18937  df-oppr 19010  df-dvdsr 19028  df-unit 19029  df-invr 19059  df-rnghom 19104  df-subrg 19170  df-lmod 19257  df-lss 19325  df-lsp 19367  df-sra 19569  df-rgmod 19570  df-lidl 19571  df-rsp 19572  df-2idl 19629  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-fbas 20139  df-fg 20140  df-cnfld 20143  df-zring 20215  df-zrh 20248  df-zn 20251  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-lp 21348  df-perf 21349  df-cn 21439  df-cnp 21440  df-haus 21527  df-cmp 21599  df-tx 21774  df-hmeo 21967  df-fil 22058  df-fm 22150  df-flim 22151  df-flf 22152  df-xms 22533  df-ms 22534  df-tms 22535  df-cncf 23089  df-limc 24067  df-dv 24068  df-log 24740  df-cxp 24741  df-cht 25275  df-vma 25276  df-chp 25277  df-ppi 25278  df-dchr 25410
This theorem is referenced by:  rpvmasum2  25653
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