MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rplogsumlem2 Structured version   Visualization version   GIF version

Theorem rplogsumlem2 24918
Description: Lemma for rplogsum 24960. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Assertion
Ref Expression
rplogsumlem2 (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)
Distinct variable group:   𝐴,𝑛

Proof of Theorem rplogsumlem2
Dummy variables 𝑘 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flid 12428 . . . . 5 (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
21oveq2d 6542 . . . 4 (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
32sumeq1d 14227 . . 3 (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛))
4 fveq2 6087 . . . . . 6 (𝑛 = (𝑝𝑘) → (Λ‘𝑛) = (Λ‘(𝑝𝑘)))
5 eleq1 2675 . . . . . . 7 (𝑛 = (𝑝𝑘) → (𝑛 ∈ ℙ ↔ (𝑝𝑘) ∈ ℙ))
6 fveq2 6087 . . . . . . 7 (𝑛 = (𝑝𝑘) → (log‘𝑛) = (log‘(𝑝𝑘)))
75, 6ifbieq1d 4058 . . . . . 6 (𝑛 = (𝑝𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0))
84, 7oveq12d 6544 . . . . 5 (𝑛 = (𝑝𝑘) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)))
9 id 22 . . . . 5 (𝑛 = (𝑝𝑘) → 𝑛 = (𝑝𝑘))
108, 9oveq12d 6544 . . . 4 (𝑛 = (𝑝𝑘) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)))
11 zre 11216 . . . 4 (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12 elfznn 12198 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
1312adantl 480 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14 vmacl 24588 . . . . . . . 8 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
1513, 14syl 17 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
1613nnrpd 11704 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
1716relogcld 24117 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18 0re 9896 . . . . . . . 8 0 ∈ ℝ
19 ifcl 4079 . . . . . . . 8 (((log‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
2017, 18, 19sylancl 692 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
2115, 20resubcld 10309 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
2221, 13nndivred 10918 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
2322recnd 9924 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℂ)
24 simprr 791 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
25 vmaprm 24587 . . . . . . . . . . . . 13 (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26 prmnn 15174 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
2726nnred 10884 . . . . . . . . . . . . . 14 (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28 prmgt1 15195 . . . . . . . . . . . . . 14 (𝑛 ∈ ℙ → 1 < 𝑛)
2927, 28rplogcld 24123 . . . . . . . . . . . . 13 (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ+)
3025, 29eqeltrd 2687 . . . . . . . . . . . 12 (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ+)
3130rpne0d 11711 . . . . . . . . . . 11 (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
3231necon2bi 2811 . . . . . . . . . 10 ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
3332ad2antll 760 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
3433iffalsed 4046 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
3524, 34oveq12d 6544 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36 0m0e0 10979 . . . . . . 7 (0 − 0) = 0
3735, 36syl6eq 2659 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
3837oveq1d 6541 . . . . 5 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (0 / 𝑛))
3912ad2antrl 759 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
4039nnrpd 11704 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ+)
4140rpcnne0d 11715 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42 div0 10566 . . . . . 6 ((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) → (0 / 𝑛) = 0)
4341, 42syl 17 . . . . 5 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
4438, 43eqtrd 2643 . . . 4 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = 0)
4510, 11, 23, 44fsumvma2 24683 . . 3 (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)))
463, 45eqtr3d 2645 . 2 (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)))
47 fzfid 12591 . . . . 5 (𝐴 ∈ ℤ → (2...((abs‘𝐴) + 1)) ∈ Fin)
48 inss2 3795 . . . . . . . . . . . 12 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
49 simpr 475 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
5048, 49sseldi 3565 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
51 prmnn 15174 . . . . . . . . . . 11 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5250, 51syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
5352nnred 10884 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
5411adantr 479 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
55 zcn 11217 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
5655abscld 13971 . . . . . . . . . . 11 (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
57 peano2re 10060 . . . . . . . . . . 11 ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
5856, 57syl 17 . . . . . . . . . 10 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℝ)
5958adantr 479 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℝ)
60 inss1 3794 . . . . . . . . . . . . 13 ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴)
6160sseli 3563 . . . . . . . . . . . 12 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
62 elicc2 12067 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6318, 11, 62sylancr 693 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6461, 63syl5ib 232 . . . . . . . . . . 11 (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6564imp 443 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
6665simp3d 1067 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
6755adantr 479 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℂ)
6867abscld 13971 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
6954leabsd 13949 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
7068lep1d 10806 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
7154, 68, 59, 69, 70letrd 10045 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
7253, 54, 59, 66, 71letrd 10045 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
73 prmuz2 15194 . . . . . . . . . 10 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
7450, 73syl 17 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
75 nn0abscl 13848 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0)
76 nn0p1nn 11181 . . . . . . . . . . . 12 ((abs‘𝐴) ∈ ℕ0 → ((abs‘𝐴) + 1) ∈ ℕ)
7775, 76syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
7877nnzd 11315 . . . . . . . . . 10 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
7978adantr 479 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
80 elfz5 12162 . . . . . . . . 9 ((𝑝 ∈ (ℤ‘2) ∧ ((abs‘𝐴) + 1) ∈ ℤ) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
8174, 79, 80syl2anc 690 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
8272, 81mpbird 245 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (2...((abs‘𝐴) + 1)))
8382ex 448 . . . . . 6 (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (2...((abs‘𝐴) + 1))))
8483ssrdv 3573 . . . . 5 (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
85 ssfi 8042 . . . . 5 (((2...((abs‘𝐴) + 1)) ∈ Fin ∧ ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1))) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8647, 84, 85syl2anc 690 . . . 4 (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
87 fzfid 12591 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
88 simprl 789 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
8948, 88sseldi 3565 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
90 elfznn 12198 . . . . . . . . . . 11 (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
9190ad2antll 760 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
92 vmappw 24586 . . . . . . . . . 10 ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
9389, 91, 92syl2anc 690 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
9452adantrr 748 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
9594nnrpd 11704 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ+)
9695relogcld 24117 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
9793, 96eqeltrd 2687 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℝ)
9891nnnn0d 11200 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ0)
99 nnexpcl 12692 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑘) ∈ ℕ)
10094, 98, 99syl2anc 690 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℕ)
101100nnrpd 11704 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℝ+)
102101relogcld 24117 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝𝑘)) ∈ ℝ)
103 ifcl 4079 . . . . . . . . 9 (((log‘(𝑝𝑘)) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) ∈ ℝ)
104102, 18, 103sylancl 692 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) ∈ ℝ)
10597, 104resubcld 10309 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) ∈ ℝ)
106105, 100nndivred 10918 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
107106anassrs 677 . . . . 5 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
10887, 107fsumrecl 14260 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
10986, 108fsumrecl 14260 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
11052nnrpd 11704 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
111110relogcld 24117 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
112 uz2m1nn 11597 . . . . . . 7 (𝑝 ∈ (ℤ‘2) → (𝑝 − 1) ∈ ℕ)
11374, 112syl 17 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℕ)
11452, 113nnmulcld 10917 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
115111, 114nndivred 10918 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
11686, 115fsumrecl 14260 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
117 2re 10939 . . . 4 2 ∈ ℝ
118117a1i 11 . . 3 (𝐴 ∈ ℤ → 2 ∈ ℝ)
11918a1i 11 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
12052nngt0d 10913 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
121119, 53, 54, 120, 66ltletrd 10048 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
12254, 121elrpd 11703 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ+)
123122relogcld 24117 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
124 prmgt1 15195 . . . . . . . . . . . 12 (𝑝 ∈ ℙ → 1 < 𝑝)
12550, 124syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
12653, 125rplogcld 24123 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
127123, 126rerpdivcld 11737 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
128126rpcnd 11708 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
129128mulid2d 9914 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
130110, 122logled 24121 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
13166, 130mpbid 220 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
132129, 131eqbrtrd 4599 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
133 1re 9895 . . . . . . . . . . . 12 1 ∈ ℝ
134133a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
135134, 123, 126lemuldivd 11755 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
136132, 135mpbid 220 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
137 flge1nn 12441 . . . . . . . . 9 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
138127, 136, 137syl2anc 690 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
139 nnuz 11557 . . . . . . . 8 ℕ = (ℤ‘1)
140138, 139syl6eleq 2697 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ‘1))
141106recnd 9924 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℂ)
142141anassrs 677 . . . . . . 7 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℂ)
143 oveq2 6534 . . . . . . . . . 10 (𝑘 = 1 → (𝑝𝑘) = (𝑝↑1))
144143fveq2d 6091 . . . . . . . . 9 (𝑘 = 1 → (Λ‘(𝑝𝑘)) = (Λ‘(𝑝↑1)))
145143eleq1d 2671 . . . . . . . . . 10 (𝑘 = 1 → ((𝑝𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
146143fveq2d 6091 . . . . . . . . . 10 (𝑘 = 1 → (log‘(𝑝𝑘)) = (log‘(𝑝↑1)))
147145, 146ifbieq1d 4058 . . . . . . . . 9 (𝑘 = 1 → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
148144, 147oveq12d 6544 . . . . . . . 8 (𝑘 = 1 → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) = ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)))
149148, 143oveq12d 6544 . . . . . . 7 (𝑘 = 1 → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) = (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)))
150140, 142, 149fsum1p 14274 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) = ((((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) + Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘))))
15152nncnd 10885 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
152151exp1d 12822 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
153152fveq2d 6091 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
154 vmaprm 24587 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → (Λ‘𝑝) = (log‘𝑝))
15550, 154syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑝) = (log‘𝑝))
156153, 155eqtrd 2643 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
157152, 50eqeltrd 2687 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
158157iftrued 4043 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
159152fveq2d 6091 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
160158, 159eqtrd 2643 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
161156, 160oveq12d 6544 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = ((log‘𝑝) − (log‘𝑝)))
162128subidd 10231 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
163161, 162eqtrd 2643 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
164163, 152oveq12d 6544 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = (0 / 𝑝))
165110rpcnne0d 11715 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
166 div0 10566 . . . . . . . . 9 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (0 / 𝑝) = 0)
167165, 166syl 17 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 / 𝑝) = 0)
168164, 167eqtrd 2643 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
169 1p1e2 10983 . . . . . . . . . 10 (1 + 1) = 2
170169oveq1i 6536 . . . . . . . . 9 ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝))))
171170a1i 11 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝)))))
172 elfzuz 12166 . . . . . . . . . . . . . 14 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ‘2))
173 eluz2nn 11560 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
174172, 173syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
175174, 170eleq2s 2705 . . . . . . . . . . . 12 (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
17650, 175, 92syl2an 492 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
17752adantr 479 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℕ)
178 nnq 11635 . . . . . . . . . . . . . 14 (𝑝 ∈ ℕ → 𝑝 ∈ ℚ)
179177, 178syl 17 . . . . . . . . . . . . 13 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℚ)
180172, 170eleq2s 2705 . . . . . . . . . . . . . 14 (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ‘2))
181180adantl 480 . . . . . . . . . . . . 13 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑘 ∈ (ℤ‘2))
182 expnprm 15392 . . . . . . . . . . . . 13 ((𝑝 ∈ ℚ ∧ 𝑘 ∈ (ℤ‘2)) → ¬ (𝑝𝑘) ∈ ℙ)
183179, 181, 182syl2anc 690 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ¬ (𝑝𝑘) ∈ ℙ)
184183iffalsed 4046 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) = 0)
185176, 184oveq12d 6544 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) = ((log‘𝑝) − 0))
186128subid1d 10232 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − 0) = (log‘𝑝))
187186adantr 479 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) − 0) = (log‘𝑝))
188185, 187eqtrd 2643 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) = (log‘𝑝))
189188oveq1d 6541 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) = ((log‘𝑝) / (𝑝𝑘)))
190171, 189sumeq12dv 14232 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)))
191168, 190oveq12d 6544 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) + Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘))) = (0 + Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘))))
192 fzfid 12591 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
193111adantr 479 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ)
194 nnnn0 11148 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
19552, 194, 99syl2an 492 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ∈ ℕ)
196193, 195nndivred 10918 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
197174, 196sylan2 489 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
198192, 197fsumrecl 14260 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
199198recnd 9924 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ∈ ℂ)
200199addid2d 10088 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 + Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘))) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)))
201150, 191, 2003eqtrd 2647 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)))
202110rpreccld 11716 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ+)
203127flcld 12418 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
204203peano2zd 11319 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
205202, 204rpexpcld 12851 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ+)
206205rpge0d 11710 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
20752nnrecred 10915 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
208207resqcld 12854 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
209138peano2nnd 10886 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
210209nnnn0d 11200 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ0)
211207, 210reexpcld 12844 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
212208, 211subge02d 10470 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2)))
213206, 212mpbid 220 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2))
214113nnrpd 11704 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ+)
215214rpcnne0d 11715 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
216202rpcnd 11708 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
217 dmdcan 10586 . . . . . . . . . . 11 ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ (1 / 𝑝) ∈ ℂ) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
218215, 165, 216, 217syl3anc 1317 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
219134recnd 9924 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
220 divsubdir 10572 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
221151, 219, 165, 220syl3anc 1317 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
222 divid 10565 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
223165, 222syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 / 𝑝) = 1)
224223oveq1d 6541 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
225221, 224eqtrd 2643 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
226 divdiv1 10587 . . . . . . . . . . . 12 ((1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
227219, 165, 215, 226syl3anc 1317 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
228225, 227oveq12d 6544 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
22952nnne0d 10914 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
230216, 151, 229divrecd 10655 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
231216sqvald 12824 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
232230, 231eqtr4d 2646 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
233218, 228, 2323eqtr3d 2651 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
234213, 233breqtrrd 4605 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
235208, 211resubcld 10309 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
236114nnrecred 10915 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
237 resubcl 10196 . . . . . . . . . 10 ((1 ∈ ℝ ∧ (1 / 𝑝) ∈ ℝ) → (1 − (1 / 𝑝)) ∈ ℝ)
238133, 207, 237sylancr 693 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ)
239 recgt1 10770 . . . . . . . . . . . 12 ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
24053, 120, 239syl2anc 690 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
241125, 240mpbid 220 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) < 1)
242 posdif 10372 . . . . . . . . . . 11 (((1 / 𝑝) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
243207, 133, 242sylancl 692 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
244241, 243mpbid 220 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < (1 − (1 / 𝑝)))
245 ledivmul 10750 . . . . . . . . 9 (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ ∧ (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ ∧ ((1 − (1 / 𝑝)) ∈ ℝ ∧ 0 < (1 − (1 / 𝑝)))) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1))))))
246235, 236, 238, 244, 245syl112anc 1321 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1))))))
247234, 246mpbird 245 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))))
248238, 244elrpd 11703 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ+)
249235, 248rerpdivcld 11737 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ∈ ℝ)
250249, 236, 126lemul2d 11750 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))) ≤ ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1))))))
251247, 250mpbid 220 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))) ≤ ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
252128adantr 479 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℂ)
253195nncnd 10885 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ∈ ℂ)
254195nnne0d 10914 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ≠ 0)
255252, 253, 254divrecd 10655 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · (1 / (𝑝𝑘))))
256151adantr 479 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ)
25752adantr 479 . . . . . . . . . . . . 13 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ)
258257nnne0d 10914 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
259 nnz 11234 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
260259adantl 480 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
261256, 258, 260exprecd 12835 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝𝑘)))
262261oveq2d 6542 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝𝑘))))
263255, 262eqtr4d 2646 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
264174, 263sylan2 489 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
265264sumeq2dv 14229 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
266174nnnn0d 11200 . . . . . . . . 9 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ0)
267 expcl 12697 . . . . . . . . 9 (((1 / 𝑝) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
268216, 266, 267syl2an 492 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
269192, 128, 268fsummulc2 14306 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
270 fzval3 12361 . . . . . . . . . . 11 ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
271203, 270syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
272271sumeq1d 14227 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = Σ𝑘 ∈ (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘))
273207, 241ltned 10024 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
274 2nn0 11158 . . . . . . . . . . 11 2 ∈ ℕ0
275274a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ0)
276 eluzp1p1 11547 . . . . . . . . . . . 12 ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ‘1) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ‘(1 + 1)))
277140, 276syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ‘(1 + 1)))
278 df-2 10928 . . . . . . . . . . . 12 2 = (1 + 1)
279278fveq2i 6090 . . . . . . . . . . 11 (ℤ‘2) = (ℤ‘(1 + 1))
280277, 279syl6eleqr 2698 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ‘2))
281216, 273, 275, 280geoserg 14385 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘) = ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))))
282272, 281eqtrd 2643 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))))
283282oveq2d 6542 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))))
284265, 269, 2833eqtr2d 2649 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))))
285114nncnd 10885 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
286114nnne0d 10914 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
287128, 285, 286divrecd 10655 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
288251, 284, 2873brtr4d 4609 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
289201, 288eqbrtrd 4599 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
29086, 108, 115, 289fsumle 14320 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))))
291 elfzuz 12166 . . . . . . . . . . 11 (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ‘2))
292 eluz2nn 11560 . . . . . . . . . . 11 (𝑝 ∈ (ℤ‘2) → 𝑝 ∈ ℕ)
293291, 292syl 17 . . . . . . . . . 10 (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ ℕ)
294293adantl 480 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℕ)
295294nnred 10884 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℝ)
296291adantl 480 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ (ℤ‘2))
297 eluz2b2 11595 . . . . . . . . . 10 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
298297simprbi 478 . . . . . . . . 9 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
299296, 298syl 17 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 1 < 𝑝)
300295, 299rplogcld 24123 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (log‘𝑝) ∈ ℝ+)
301296, 112syl 17 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 − 1) ∈ ℕ)
302294, 301nnmulcld 10917 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
303302nnrpd 11704 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ+)
304300, 303rpdivcld 11723 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ+)
305304rpred 11706 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
30647, 305fsumrecl 14260 . . . 4 (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
307304rpge0d 11710 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
30847, 305, 307, 84fsumless 14317 . . . 4 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
309 rplogsumlem1 24917 . . . . 5 (((abs‘𝐴) + 1) ∈ ℕ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
31077, 309syl 17 . . . 4 (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
311116, 306, 118, 308, 310letrd 10045 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
312109, 116, 118, 290, 311letrd 10045 . 2 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ≤ 2)
31346, 312eqbrtrd 4599 1 (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  cin 3538  wss 3539  ifcif 4035   class class class wbr 4577  cfv 5789  (class class class)co 6526  Fincfn 7818  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797   < clt 9930  cle 9931  cmin 10117   / cdiv 10535  cn 10869  2c2 10919  0cn0 11141  cz 11212  cuz 11521  cq 11622  +crp 11666  [,]cicc 12007  ...cfz 12154  ..^cfzo 12291  cfl 12410  cexp 12679  abscabs 13770  Σcsu 14212  cprime 15171  logclog 24049  Λcvma 24562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10536  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-q 11623  df-rp 11667  df-xneg 11780  df-xadd 11781  df-xmul 11782  df-ioo 12008  df-ioc 12009  df-ico 12010  df-icc 12011  df-fz 12155  df-fzo 12292  df-fl 12412  df-mod 12488  df-seq 12621  df-exp 12680  df-fac 12880  df-bc 12909  df-hash 12937  df-shft 13603  df-cj 13635  df-re 13636  df-im 13637  df-sqrt 13771  df-abs 13772  df-limsup 13998  df-clim 14015  df-rlim 14016  df-sum 14213  df-ef 14585  df-sin 14587  df-cos 14588  df-tan 14589  df-pi 14590  df-dvds 14770  df-gcd 15003  df-prm 15172  df-pc 15328  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-ress 15650  df-plusg 15729  df-mulr 15730  df-starv 15731  df-sca 15732  df-vsca 15733  df-ip 15734  df-tset 15735  df-ple 15736  df-ds 15739  df-unif 15740  df-hom 15741  df-cco 15742  df-rest 15854  df-topn 15855  df-0g 15873  df-gsum 15874  df-topgen 15875  df-pt 15876  df-prds 15879  df-xrs 15933  df-qtop 15938  df-imas 15939  df-xps 15941  df-mre 16017  df-mrc 16018  df-acs 16020  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-submnd 17107  df-mulg 17312  df-cntz 17521  df-cmn 17966  df-psmet 19507  df-xmet 19508  df-met 19509  df-bl 19510  df-mopn 19511  df-fbas 19512  df-fg 19513  df-cnfld 19516  df-top 20468  df-bases 20469  df-topon 20470  df-topsp 20471  df-cld 20580  df-ntr 20581  df-cls 20582  df-nei 20659  df-lp 20697  df-perf 20698  df-cn 20788  df-cnp 20789  df-haus 20876  df-cmp 20947  df-tx 21122  df-hmeo 21315  df-fil 21407  df-fm 21499  df-flim 21500  df-flf 21501  df-xms 21882  df-ms 21883  df-tms 21884  df-cncf 22436  df-limc 23380  df-dv 23381  df-log 24051  df-cxp 24052  df-vma 24568
This theorem is referenced by:  rplogsum  24960
  Copyright terms: Public domain W3C validator