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Theorem rpvmasumlem 27545
Description: Lemma for rpvmasum 27584. 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 11243 . . . . . 6 ℝ ∈ V
2 rpssre 13039 . . . . . 6 + ⊆ ℝ
31, 2ssexi 5327 . . . . 5 + ∈ V
43a1i 11 . . . 4 (𝜑 → ℝ+ ∈ V)
5 fzfid 14010 . . . . . . 7 (𝜑 → (1...(⌊‘𝑥)) ∈ Fin)
6 elfznn 13589 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
76adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8 vmacl 27175 . . . . . . . . . 10 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
97, 8syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
109, 7nndivred 12317 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
1110recnd 11286 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
125, 11fsumcl 15765 . . . . . 6 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
1312adantr 480 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
14 relogcl 26631 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1514adantl 481 . . . . . 6 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
1615recnd 11286 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
1713, 16subcld 11617 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
18 1re 11258 . . . . . . . . 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 2734 . . . . . . . . . . . 12 (Base‘𝑍) = (Base‘𝑍)
23 rpvmasum.a . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ)
2419, 20, 21, 22, 23dchr1re 27321 . . . . . . . . . . 11 (𝜑1 :(Base‘𝑍)⟶ℝ)
2524adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 :(Base‘𝑍)⟶ℝ)
2623nnnn0d 12584 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
27 rpvmasum.l . . . . . . . . . . . . 13 𝐿 = (ℤRHom‘𝑍)
2820, 22, 27znzrhfo 21583 . . . . . . . . . . . 12 (𝑁 ∈ ℕ0𝐿:ℤ–onto→(Base‘𝑍))
29 fof 6820 . . . . . . . . . . . 12 (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
3026, 28, 293syl 18 . . . . . . . . . . 11 (𝜑𝐿:ℤ⟶(Base‘𝑍))
31 elfzelz 13560 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
32 ffvelcdm 7100 . . . . . . . . . . 11 ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑛 ∈ ℤ) → (𝐿𝑛) ∈ (Base‘𝑍))
3330, 31, 32syl2an 596 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
3425, 33ffvelcdmd 7104 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
35 resubcl 11570 . . . . . . . . 9 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3618, 34, 35sylancr 587 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
3736, 10remulcld 11288 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
3837recnd 11286 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
395, 38fsumcl 15765 . . . . 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 2735 . . . 4 (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))))
42 eqidd 2735 . . . 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 7716 . . 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 11649 . . . . 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 15820 . . . . . . . 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 11253 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
4736recnd 11286 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
4846, 47, 11subdird 11717 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = ((1 · ((Λ‘𝑛) / 𝑛)) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))))
49 ax-1cn 11210 . . . . . . . . . . . 12 1 ∈ ℂ
5034recnd 11286 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℂ)
51 nncan 11535 . . . . . . . . . . . 12 ((1 ∈ ℂ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℂ) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5249, 50, 51sylancr 587 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 − (1 − ( 1 ‘(𝐿𝑛)))) = ( 1 ‘(𝐿𝑛)))
5352oveq1d 7445 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − (1 − ( 1 ‘(𝐿𝑛)))) · ((Λ‘𝑛) / 𝑛)) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5411mullidd 11276 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (1 · ((Λ‘𝑛) / 𝑛)) = ((Λ‘𝑛) / 𝑛))
5554oveq1d 7445 . . . . . . . . . 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 2783 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = (( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5756sumeq2dv 15734 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) − ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(( 1 ‘(𝐿𝑛)) · ((Λ‘𝑛) / 𝑛)))
5845, 57eqtr3d 2776 . . . . . . 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 7445 . . . . . 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 2774 . . . 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 5247 . . 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 2774 . 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 27540 . . 3 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
652a1i 11 . . . 4 (𝜑 → ℝ+ ⊆ ℝ)
66 1red 11259 . . . 4 (𝜑 → 1 ∈ ℝ)
67 prmdvdsfi 27164 . . . . . 6 (𝑁 ∈ ℕ → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
6823, 67syl 17 . . . . 5 (𝜑 → {𝑞 ∈ ℙ ∣ 𝑞𝑁} ∈ Fin)
69 elrabi 3689 . . . . . 6 (𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
70 prmnn 16707 . . . . . . . . . 10 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
7170adantl 481 . . . . . . . . 9 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
7271nnrpd 13072 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
7372relogcld 26679 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
74 prmuz2 16729 . . . . . . . . 9 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
7574adantl 481 . . . . . . . 8 ((𝜑𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
76 uz2m1nn 12962 . . . . . . . 8 (𝑝 ∈ (ℤ‘2) → (𝑝 − 1) ∈ ℕ)
7775, 76syl 17 . . . . . . 7 ((𝜑𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
7873, 77nndivred 12317 . . . . . 6 ((𝜑𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
7969, 78sylan2 593 . . . . 5 ((𝜑𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
8068, 79fsumrecl 15766 . . . 4 (𝜑 → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
81 fzfid 14010 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
82 simpr 484 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) = 0)
83 0re 11260 . . . . . . . . . . 11 0 ∈ ℝ
8482, 83eqeltrdi 2846 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
85 eqid 2734 . . . . . . . . . . . 12 (Unit‘𝑍) = (Unit‘𝑍)
8623ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → 𝑁 ∈ ℕ)
87 rpvmasum.d . . . . . . . . . . . . . 14 𝐷 = (Base‘𝐺)
8819dchrabl 27312 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → 𝐺 ∈ Abel)
89 ablgrp 19817 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
9087, 21grpidcl 18995 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → 1𝐷)
9123, 88, 89, 904syl 19 . . . . . . . . . . . . . . 15 (𝜑1𝐷)
9291ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1𝐷)
9333adantlr 715 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐿𝑛) ∈ (Base‘𝑍))
9419, 20, 87, 22, 85, 92, 93dchrn0 27308 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (( 1 ‘(𝐿𝑛)) ≠ 0 ↔ (𝐿𝑛) ∈ (Unit‘𝑍)))
9594biimpa 476 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → (𝐿𝑛) ∈ (Unit‘𝑍))
9619, 20, 21, 85, 86, 95dchr1 27315 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) = 1)
9796, 18eqeltrdi 2846 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9884, 97pm2.61dane 3026 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ∈ ℝ)
9918, 98, 35sylancr 587 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℝ)
10010adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
10199, 100remulcld 11288 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
10281, 101fsumrecl 15766 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℝ)
103 0le1 11783 . . . . . . . . . . 11 0 ≤ 1
10482, 103eqbrtrdi 5186 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) = 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
10518leidi 11794 . . . . . . . . . . 11 1 ≤ 1
10696, 105eqbrtrdi 5186 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ ( 1 ‘(𝐿𝑛)) ≠ 0) → ( 1 ‘(𝐿𝑛)) ≤ 1)
107104, 106pm2.61dane 3026 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ( 1 ‘(𝐿𝑛)) ≤ 1)
108 subge0 11773 . . . . . . . . . 10 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿𝑛)) ∈ ℝ) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
10918, 98, 108sylancr 587 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ (1 − ( 1 ‘(𝐿𝑛))) ↔ ( 1 ‘(𝐿𝑛)) ≤ 1))
110107, 109mpbird 257 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (1 − ( 1 ‘(𝐿𝑛))))
1119adantlr 715 . . . . . . . . 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 27178 . . . . . . . . . 10 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
114112, 113syl 17 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
115112nnred 12278 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
116112nngt0d 12312 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 < 𝑛)
117 divge0 12134 . . . . . . . . 9 ((((Λ‘𝑛) ∈ ℝ ∧ 0 ≤ (Λ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((Λ‘𝑛) / 𝑛))
118111, 114, 115, 116, 117syl22anc 839 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
11999, 100, 110, 118mulge0d 11837 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
12081, 101, 119fsumge0 15827 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)))
121102, 120absidd 15457 . . . . 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 4245 . . . . . . . . 9 ((0[,]𝑥) ∩ ℙ) ⊆ ℙ
124 rabss2 4087 . . . . . . . . 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})
126122, 125ssfid 9298 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ∈ Fin)
127 ssrab2 4089 . . . . . . . . . 10 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ)
128127, 123sstri 4004 . . . . . . . . 9 {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ℙ
129128sseli 3990 . . . . . . . 8 (𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} → 𝑝 ∈ ℙ)
13078adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
131129, 130sylan2 593 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
132126, 131fsumrecl 15766 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
13380adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
134 2fveq3 6911 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → ( 1 ‘(𝐿𝑛)) = ( 1 ‘(𝐿‘(𝑝𝑘))))
135134oveq2d 7446 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → (1 − ( 1 ‘(𝐿𝑛))) = (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))))
136 fveq2 6906 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → (Λ‘𝑛) = (Λ‘(𝑝𝑘)))
137 id 22 . . . . . . . . . . 11 (𝑛 = (𝑝𝑘) → 𝑛 = (𝑝𝑘))
138136, 137oveq12d 7448 . . . . . . . . . 10 (𝑛 = (𝑝𝑘) → ((Λ‘𝑛) / 𝑛) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
139135, 138oveq12d 7448 . . . . . . . . 9 (𝑛 = (𝑝𝑘) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))))
140 rpre 13040 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
141140ad2antrl 728 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
14238adantlr 715 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
143 simprr 773 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
144143oveq1d 7445 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = (0 / 𝑛))
1456ad2antrl 728 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
146145nncnd 12279 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℂ)
147145nnne0d 12313 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ≠ 0)
148146, 147div0d 12039 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
149144, 148eqtrd 2774 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) / 𝑛) = 0)
150149oveq2d 7446 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = ((1 − ( 1 ‘(𝐿𝑛))) · 0))
15147ad2ant2r 747 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → (1 − ( 1 ‘(𝐿𝑛))) ∈ ℂ)
152151mul01d 11457 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · 0) = 0)
153150, 152eqtrd 2774 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ (Λ‘𝑛) = 0)) → ((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) = 0)
154139, 141, 142, 153fsumvma2 27272 . . . . . . . 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))))
155127a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ⊆ ((0[,]𝑥) ∩ ℙ))
156 fzfid 14010 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
15724ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 :(Base‘𝑍)⟶ℝ)
15830ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝐿:ℤ⟶(Base‘𝑍))
15970ad2antrl 728 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
160 elfznn 13589 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
161160ad2antll 729 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
162161nnnn0d 12584 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℕ0)
163159, 162nnexpcld 14280 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℕ)
164163nnzd 12637 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℤ)
165158, 164ffvelcdmd 7104 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝐿‘(𝑝𝑘)) ∈ (Base‘𝑍))
166157, 165ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ)
167 resubcl 11570 . . . . . . . . . . . . . . 15 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
16818, 166, 167sylancr 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ∈ ℝ)
169 vmacl 27175 . . . . . . . . . . . . . . . 16 ((𝑝𝑘) ∈ ℕ → (Λ‘(𝑝𝑘)) ∈ ℝ)
170163, 169syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℝ)
171170, 163nndivred 12317 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℝ)
172168, 171remulcld 11288 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
173172anassrs 467 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
174173recnd 11286 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
175156, 174fsumcl 15765 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℂ)
176129, 175sylan2 593 . . . . . . . . 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)
177 breq1 5150 . . . . . . . . . . . 12 (𝑞 = 𝑝 → (𝑞𝑁𝑝𝑁))
178177notbid 318 . . . . . . . . . . 11 (𝑞 = 𝑝 → (¬ 𝑞𝑁 ↔ ¬ 𝑝𝑁))
179 notrab 4327 . . . . . . . . . . 11 (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) = {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ ¬ 𝑞𝑁}
180178, 179elrab2 3697 . . . . . . . . . 10 (𝑝 ∈ (((0[,]𝑥) ∩ ℙ) ∖ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁}) ↔ (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) ∧ ¬ 𝑝𝑁))
181123sseli 3990 . . . . . . . . . . 11 (𝑝 ∈ ((0[,]𝑥) ∩ ℙ) → 𝑝 ∈ ℙ)
18223ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ)
183 simplrr 778 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ¬ 𝑝𝑁)
184 simplrl 777 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℙ)
185182nnzd 12637 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℤ)
186 coprm 16744 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑝 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
187184, 185, 186syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (¬ 𝑝𝑁 ↔ (𝑝 gcd 𝑁) = 1))
188183, 187mpbid 232 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝 gcd 𝑁) = 1)
189 prmz 16708 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
190184, 189syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑝 ∈ ℤ)
191160adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ)
192191nnnn0d 12584 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑘 ∈ ℕ0)
193 rpexp1i 16756 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℕ0) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
194190, 185, 192, 193syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝 gcd 𝑁) = 1 → ((𝑝𝑘) gcd 𝑁) = 1))
195188, 194mpd 15 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝑝𝑘) gcd 𝑁) = 1)
196182nnnn0d 12584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → 𝑁 ∈ ℕ0)
197164anassrs 467 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
198197adantlrr 721 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝑝𝑘) ∈ ℤ)
19920, 85, 27znunit 21599 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑝𝑘) ∈ ℤ) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
200196, 198, 199syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍) ↔ ((𝑝𝑘) gcd 𝑁) = 1))
201195, 200mpbird 257 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
20219, 20, 21, 85, 182, 201dchr1 27315 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
203202oveq2d 7446 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = (1 − 1))
204 1m1e0 12335 . . . . . . . . . . . . . . . 16 (1 − 1) = 0
205203, 204eqtrdi 2790 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) = 0)
206205oveq1d 7445 . . . . . . . . . . . . . 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))))
207171recnd 11286 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
208207anassrs 467 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
209208adantlrr 721 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) ∈ ℂ)
210209mul02d 11456 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → (0 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
211206, 210eqtrd 2774 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
212211sumeq2dv 15734 . . . . . . . . . . . 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)
213 fzfid 14010 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin)
214213olcd 874 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → ((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin))
215 sumz 15754 . . . . . . . . . . . . 13 (((1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ⊆ (ℤ‘1) ∨ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) ∈ Fin) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
216214, 215syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))0 = 0)
217212, 216eqtrd 2774 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ ¬ 𝑝𝑁)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = 0)
218181, 217sylanr1 682 . . . . . . . . . 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)
219180, 218sylan2b 594 . . . . . . . . 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)
220 ppifi 27163 . . . . . . . . . 10 (𝑥 ∈ ℝ → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
221141, 220syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((0[,]𝑥) ∩ ℙ) ∈ Fin)
222155, 176, 219, 221fsumss 15757 . . . . . . . 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))))
223154, 222eqtr4d 2777 . . . . . . 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))))
224156, 173fsumrecl 15766 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → Σ𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))((1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) ∈ ℝ)
225129, 224sylan2 593 . . . . . . . 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)
22673adantlr 715 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ)
22770adantl 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
228227nnrecred 12314 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ)
229227nnrpd 13072 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ+)
230229rpreccld 13084 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℝ+)
231 simplrl 777 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ ℝ+)
232231relogcld 26679 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑥) ∈ ℝ)
233227nnred 12278 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
23474adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ‘2))
235 eluz2gt1 12959 . . . . . . . . . . . . . . . . . . . 20 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
236234, 235syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 < 𝑝)
237233, 236rplogcld 26685 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℝ+)
238232, 237rerpdivcld 13105 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑥) / (log‘𝑝)) ∈ ℝ)
239238flcld 13834 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ)
240239peano2zd 12722 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℤ)
241230, 240rpexpcld 14282 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ+)
242241rpred 13074 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)) ∈ ℝ)
243228, 242resubcld 11688 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ∈ ℝ)
244234, 76syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℕ)
245244nnrpd 13072 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℝ+)
246245, 229rpdivcld 13091 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) ∈ ℝ+)
247243, 246rerpdivcld 13105 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ∈ ℝ)
248226, 247remulcld 11288 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) · (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝))) ∈ ℝ)
249170recnd 11286 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℂ)
250163nncnd 12279 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℂ)
251163nnne0d 12313 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ≠ 0)
252249, 250, 251divrecd 12043 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))))
253 simprl 771 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
254 vmappw 27173 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
255253, 161, 254syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
256159nncnd 12279 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ∈ ℂ)
257159nnne0d 12313 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑝 ≠ 0)
258 elfzelz 13560 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) → 𝑘 ∈ ℤ)
259258ad2antll 729 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 𝑘 ∈ ℤ)
260256, 257, 259exprecd 14190 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝𝑘)))
261260eqcomd 2740 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / (𝑝𝑘)) = ((1 / 𝑝)↑𝑘))
262255, 261oveq12d 7448 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) · (1 / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
263252, 262eqtrd 2774 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
264263, 171eqeltrrd 2839 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
265264anassrs 467 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) ∈ ℝ)
266 1red 11259 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1 ∈ ℝ)
267 vmage0 27178 . . . . . . . . . . . . . . . . 17 ((𝑝𝑘) ∈ ℕ → 0 ≤ (Λ‘(𝑝𝑘)))
268163, 267syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ (Λ‘(𝑝𝑘)))
269163nnred 12278 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℝ)
270163nngt0d 12312 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 < (𝑝𝑘))
271 divge0 12134 . . . . . . . . . . . . . . . 16 ((((Λ‘(𝑝𝑘)) ∈ ℝ ∧ 0 ≤ (Λ‘(𝑝𝑘))) ∧ ((𝑝𝑘) ∈ ℝ ∧ 0 < (𝑝𝑘))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
272170, 268, 269, 270, 271syl22anc 839 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
27383leidi 11794 . . . . . . . . . . . . . . . . . 18 0 ≤ 0
274 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 0)
275273, 274breqtrrid 5185 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) = 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
27623ad3antrrr 730 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 𝑁 ∈ ℕ)
27791ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 1𝐷)
27819, 20, 87, 22, 85, 277, 165dchrn0 27308 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0 ↔ (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍)))
279278biimpa 476 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → (𝐿‘(𝑝𝑘)) ∈ (Unit‘𝑍))
28019, 20, 21, 85, 276, 279dchr1 27315 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → ( 1 ‘(𝐿‘(𝑝𝑘))) = 1)
281103, 280breqtrrid 5185 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ≠ 0) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
282275, 281pm2.61dane 3026 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → 0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))))
283 subge02 11776 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℝ ∧ ( 1 ‘(𝐿‘(𝑝𝑘))) ∈ ℝ) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
28418, 166, 283sylancr 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (0 ≤ ( 1 ‘(𝐿‘(𝑝𝑘))) ↔ (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1))
285282, 284mpbid 232 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 − ( 1 ‘(𝐿‘(𝑝𝑘)))) ≤ 1)
286168, 266, 171, 272, 285lemul1ad 12204 . . . . . . . . . . . . . 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))))
287207mullidd 11276 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((Λ‘(𝑝𝑘)) / (𝑝𝑘)))
288287, 263eqtrd 2774 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 · ((Λ‘(𝑝𝑘)) / (𝑝𝑘))) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
289286, 288breqtrd 5173 . . . . . . . . . . . . 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)))
290289anassrs 467 . . . . . . . . . . . 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)))
291156, 173, 265, 290fsumle 15831 . . . . . . . . . . 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)))
292226recnd 11286 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (log‘𝑝) ∈ ℂ)
293159nnrecred 12314 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → (1 / 𝑝) ∈ ℝ)
294293, 162reexpcld 14199 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℝ)
295294recnd 11286 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝)))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
296295anassrs 467 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝑥) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
297156, 292, 296fsummulc2 15816 . . . . . . . . . . . 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)))
298 fzval3 13769 . . . . . . . . . . . . . . . 16 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℤ → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
299239, 298syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1...(⌊‘((log‘𝑥) / (log‘𝑝)))) = (1..^((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
300299sumeq1d 15732 . . . . . . . . . . . . . 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))
301228recnd 11286 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ∈ ℂ)
302227nngt0d 12312 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 < 𝑝)
303 recgt1 12161 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
304233, 302, 303syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
305236, 304mpbid 232 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) < 1)
306228, 305ltned 11394 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / 𝑝) ≠ 1)
307 1nn0 12539 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
308307a1i 11 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℕ0)
309 log1 26641 . . . . . . . . . . . . . . . . . . . . 21 (log‘1) = 0
310 simprr 773 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
311 1rp 13035 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℝ+
312 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
313 logleb 26659 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℝ+𝑥 ∈ ℝ+) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
314311, 312, 313sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (log‘1) ≤ (log‘𝑥)))
315310, 314mpbid 232 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘1) ≤ (log‘𝑥))
316309, 315eqbrtrrid 5183 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ (log‘𝑥))
317316adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ (log‘𝑥))
318232, 237, 317divge0d 13114 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑥) / (log‘𝑝)))
319 flge0nn0 13856 . . . . . . . . . . . . . . . . . 18 ((((log‘𝑥) / (log‘𝑝)) ∈ ℝ ∧ 0 ≤ ((log‘𝑥) / (log‘𝑝))) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
320238, 318, 319syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0)
321 nn0p1nn 12562 . . . . . . . . . . . . . . . . 17 ((⌊‘((log‘𝑥) / (log‘𝑝))) ∈ ℕ0 → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
322320, 321syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ ℕ)
323 nnuz 12918 . . . . . . . . . . . . . . . 16 ℕ = (ℤ‘1)
324322, 323eleqtrdi 2848 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((⌊‘((log‘𝑥) / (log‘𝑝))) + 1) ∈ (ℤ‘1))
325301, 306, 308, 324geoserg 15898 . . . . . . . . . . . . . 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))))
326301exp1d 14177 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝)↑1) = (1 / 𝑝))
327326oveq1d 7445 . . . . . . . . . . . . . . 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))))
328227nncnd 12279 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℂ)
329 1cnd 11253 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
330229rpcnne0d 13083 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
331 divsubdir 11958 . . . . . . . . . . . . . . . . 17 ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
332328, 329, 330, 331syl3anc 1370 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
333 divid 11950 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
334330, 333syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 / 𝑝) = 1)
335334oveq1d 7445 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
336332, 335eqtr2d 2775 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 − (1 / 𝑝)) = ((𝑝 − 1) / 𝑝))
337327, 336oveq12d 7448 . . . . . . . . . . . . . 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)))
338300, 325, 3373eqtrd 2778 . . . . . . . . . . . . 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)))
339338oveq2d 7446 . . . . . . . . . . . 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))))
340297, 339eqtr3d 2776 . . . . . . . . . . 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))))
341291, 340breqtrd 5173 . . . . . . . . . 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))))
342241rpge0d 13078 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1)))
343228, 242subge02d 11852 . . . . . . . . . . . . . . 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)))
344342, 343mpbid 232 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (1 / 𝑝))
345245rpcnne0d 13083 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
346 dmdcan 11974 . . . . . . . . . . . . . . 15 ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ 1 ∈ ℂ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
347345, 330, 329, 346syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))) = (1 / 𝑝))
348344, 347breqtrrd 5175 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) ≤ (((𝑝 − 1) / 𝑝) · (1 / (𝑝 − 1))))
349244nnrecred 12314 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (1 / (𝑝 − 1)) ∈ ℝ)
350243, 349, 246ledivmuld 13127 . . . . . . . . . . . . 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)))))
351348, 350mpbird 257 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (((1 / 𝑝) − ((1 / 𝑝)↑((⌊‘((log‘𝑥) / (log‘𝑝))) + 1))) / ((𝑝 − 1) / 𝑝)) ≤ (1 / (𝑝 − 1)))
352247, 349, 237lemul2d 13118 . . . . . . . . . . . 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)))))
353351, 352mpbid 232 . . . . . . . . . . 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))))
354244nncnd 12279 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈ ℂ)
355244nnne0d 12313 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ≠ 0)
356292, 354, 355divrecd 12043 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) = ((log‘𝑝) · (1 / (𝑝 − 1))))
357353, 356breqtrrd 5175 . . . . . . . . . 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)))
358224, 248, 130, 341, 357letrd 11415 . . . . . . . . 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)))
359129, 358sylan2 593 . . . . . . . 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)))
360126, 225, 131, 359fsumle 15831 . . . . . . 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)))
361223, 360eqbrtrd 5169 . . . . . 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)))
36279adantlr 715 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ)
363237, 245rpdivcld 13091 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → ((log‘𝑝) / (𝑝 − 1)) ∈ ℝ+)
364363rpge0d 13078 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ ℙ) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
36569, 364sylan2 593 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁}) → 0 ≤ ((log‘𝑝) / (𝑝 − 1)))
366122, 362, 365, 125fsumless 15828 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑝 ∈ {𝑞 ∈ ((0[,]𝑥) ∩ ℙ) ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
367102, 132, 133, 361, 366letrd 11415 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛)) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
368121, 367eqbrtrd 5169 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ≤ Σ𝑝 ∈ {𝑞 ∈ ℙ ∣ 𝑞𝑁} ((log‘𝑝) / (𝑝 − 1)))
36965, 40, 66, 80, 368elo1d 15568 . . 3 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((1 − ( 1 ‘(𝐿𝑛))) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
370 o1sub 15648 . . 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))
37164, 369, 370sylancr 587 . 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))
37263, 371eqeltrrd 2839 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 206  wa 395  wo 847   = wceq 1536  wcel 2105  wne 2937  {crab 3432  Vcvv 3477  cdif 3959  cin 3961  wss 3962   class class class wbr 5147  cmpt 5230  wf 6558  ontowfo 6560  cfv 6562  (class class class)co 7430  f cof 7694  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  +crp 13031  [,]cicc 13386  ...cfz 13543  ..^cfzo 13690  cfl 13826  cexp 14098  abscabs 15269  𝑂(1)co1 15518  Σcsu 15718  cdvds 16286   gcd cgcd 16527  cprime 16704  Basecbs 17244  0gc0g 17485  Grpcgrp 18963  Abelcabl 19813  Unitcui 20371  ℤRHomczrh 21527  ℤ/nczn 21530  logclog 26610  Λcvma 27149  DChrcdchr 27290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-o1 15522  df-lo1 15523  df-sum 15719  df-ef 16099  df-e 16100  df-sin 16101  df-cos 16102  df-pi 16104  df-dvds 16287  df-gcd 16528  df-prm 16705  df-pc 16870  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-qus 17555  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-nsg 19154  df-eqg 19155  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-lsp 20987  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-rsp 21236  df-2idl 21277  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-zn 21534  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-log 26612  df-cxp 26613  df-cht 27154  df-vma 27155  df-chp 27156  df-ppi 27157  df-dchr 27291
This theorem is referenced by:  rpvmasum2  27570
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