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Theorem rplogsumlem2 25219
 Description: Lemma for rplogsum 25261. 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 12649 . . . . 5 (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
21oveq2d 6706 . . . 4 (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
32sumeq1d 14475 . . 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 6229 . . . . . 6 (𝑛 = (𝑝𝑘) → (Λ‘𝑛) = (Λ‘(𝑝𝑘)))
5 eleq1 2718 . . . . . . 7 (𝑛 = (𝑝𝑘) → (𝑛 ∈ ℙ ↔ (𝑝𝑘) ∈ ℙ))
6 fveq2 6229 . . . . . . 7 (𝑛 = (𝑝𝑘) → (log‘𝑛) = (log‘(𝑝𝑘)))
75, 6ifbieq1d 4142 . . . . . 6 (𝑛 = (𝑝𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0))
84, 7oveq12d 6708 . . . . 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 6708 . . . 4 (𝑛 = (𝑝𝑘) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)))
11 zre 11419 . . . 4 (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12 elfznn 12408 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
1312adantl 481 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14 vmacl 24889 . . . . . . . 8 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
1513, 14syl 17 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
1613nnrpd 11908 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
1716relogcld 24414 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18 0re 10078 . . . . . . . 8 0 ∈ ℝ
19 ifcl 4163 . . . . . . . 8 (((log‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
2017, 18, 19sylancl 695 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
2115, 20resubcld 10496 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
2221, 13nndivred 11107 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
2322recnd 10106 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℂ)
24 simprr 811 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
25 vmaprm 24888 . . . . . . . . . . . . 13 (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26 prmnn 15435 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
2726nnred 11073 . . . . . . . . . . . . . 14 (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28 prmgt1 15456 . . . . . . . . . . . . . 14 (𝑛 ∈ ℙ → 1 < 𝑛)
2927, 28rplogcld 24420 . . . . . . . . . . . . 13 (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ+)
3025, 29eqeltrd 2730 . . . . . . . . . . . 12 (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ+)
3130rpne0d 11915 . . . . . . . . . . 11 (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
3231necon2bi 2853 . . . . . . . . . 10 ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
3332ad2antll 765 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
3433iffalsed 4130 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
3524, 34oveq12d 6708 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36 0m0e0 11168 . . . . . . 7 (0 − 0) = 0
3735, 36syl6eq 2701 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
3837oveq1d 6705 . . . . 5 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (0 / 𝑛))
3912ad2antrl 764 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
4039nnrpd 11908 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ+)
4140rpcnne0d 11919 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42 div0 10753 . . . . . 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 2685 . . . 4 ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = 0)
4510, 11, 23, 44fsumvma2 24984 . . 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 2687 . 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 12812 . . . . 5 (𝐴 ∈ ℤ → (2...((abs‘𝐴) + 1)) ∈ Fin)
48 inss2 3867 . . . . . . . . . . . 12 ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
49 simpr 476 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
5048, 49sseldi 3634 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
51 prmnn 15435 . . . . . . . . . . 11 (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
5250, 51syl 17 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
5352nnred 11073 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
5411adantr 480 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
55 zcn 11420 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
5655abscld 14219 . . . . . . . . . . 11 (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
57 peano2re 10247 . . . . . . . . . . 11 ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
5856, 57syl 17 . . . . . . . . . 10 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℝ)
5958adantr 480 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℝ)
60 inss1 3866 . . . . . . . . . . . . 13 ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴)
6160sseli 3632 . . . . . . . . . . . 12 (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
62 elicc2 12276 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6318, 11, 62sylancr 696 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6461, 63syl5ib 234 . . . . . . . . . . 11 (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴)))
6564imp 444 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝𝑝𝐴))
6665simp3d 1095 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝𝐴)
6755adantr 480 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℂ)
6867abscld 14219 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
6954leabsd 14197 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
7068lep1d 10993 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
7154, 68, 59, 69, 70letrd 10232 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
7253, 54, 59, 66, 71letrd 10232 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
73 prmuz2 15455 . . . . . . . . . 10 (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ‘2))
7450, 73syl 17 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ‘2))
75 nn0abscl 14096 . . . . . . . . . . . 12 (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ0)
76 nn0p1nn 11370 . . . . . . . . . . . 12 ((abs‘𝐴) ∈ ℕ0 → ((abs‘𝐴) + 1) ∈ ℕ)
7775, 76syl 17 . . . . . . . . . . 11 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
7877nnzd 11519 . . . . . . . . . 10 (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
7978adantr 480 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
80 elfz5 12372 . . . . . . . . 9 ((𝑝 ∈ (ℤ‘2) ∧ ((abs‘𝐴) + 1) ∈ ℤ) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
8174, 79, 80syl2anc 694 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
8272, 81mpbird 247 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (2...((abs‘𝐴) + 1)))
8382ex 449 . . . . . 6 (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (2...((abs‘𝐴) + 1))))
8483ssrdv 3642 . . . . 5 (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
85 ssfi 8221 . . . . 5 (((2...((abs‘𝐴) + 1)) ∈ Fin ∧ ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1))) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8647, 84, 85syl2anc 694 . . . 4 (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
87 fzfid 12812 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
88 simprl 809 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
8948, 88sseldi 3634 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
90 elfznn 12408 . . . . . . . . . . 11 (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
9190ad2antll 765 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
92 vmappw 24887 . . . . . . . . . 10 ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
9389, 91, 92syl2anc 694 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
9452adantrr 753 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
9594nnrpd 11908 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ+)
9695relogcld 24414 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
9793, 96eqeltrd 2730 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝𝑘)) ∈ ℝ)
9891nnnn0d 11389 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ0)
99 nnexpcl 12913 . . . . . . . . . . . 12 ((𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (𝑝𝑘) ∈ ℕ)
10094, 98, 99syl2anc 694 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℕ)
101100nnrpd 11908 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝𝑘) ∈ ℝ+)
102101relogcld 24414 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝𝑘)) ∈ ℝ)
103 ifcl 4163 . . . . . . . . 9 (((log‘(𝑝𝑘)) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) ∈ ℝ)
104102, 18, 103sylancl 695 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) ∈ ℝ)
10597, 104resubcld 10496 . . . . . . 7 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) ∈ ℝ)
106105, 100nndivred 11107 . . . . . 6 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
107106anassrs 681 . . . . 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 14509 . . . 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 14509 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℝ)
11052nnrpd 11908 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+)
111110relogcld 24414 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
112 uz2m1nn 11801 . . . . . . 7 (𝑝 ∈ (ℤ‘2) → (𝑝 − 1) ∈ ℕ)
11374, 112syl 17 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℕ)
11452, 113nnmulcld 11106 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
115111, 114nndivred 11107 . . . 4 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
11686, 115fsumrecl 14509 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
117 2re 11128 . . . 4 2 ∈ ℝ
118117a1i 11 . . 3 (𝐴 ∈ ℤ → 2 ∈ ℝ)
11918a1i 11 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
12052nngt0d 11102 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
121119, 53, 54, 120, 66ltletrd 10235 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
12254, 121elrpd 11907 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ+)
123122relogcld 24414 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
124 prmgt1 15456 . . . . . . . . . . . 12 (𝑝 ∈ ℙ → 1 < 𝑝)
12550, 124syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
12653, 125rplogcld 24420 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ+)
127123, 126rerpdivcld 11941 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
128126rpcnd 11912 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
129128mulid2d 10096 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
130110, 122logled 24418 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
13166, 130mpbid 222 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
132129, 131eqbrtrd 4707 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
133 1re 10077 . . . . . . . . . . . 12 1 ∈ ℝ
134133a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
135134, 123, 126lemuldivd 11959 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
136132, 135mpbid 222 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
137 flge1nn 12662 . . . . . . . . 9 ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
138127, 136, 137syl2anc 694 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
139 nnuz 11761 . . . . . . . 8 ℕ = (ℤ‘1)
140138, 139syl6eleq 2740 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ‘1))
141106recnd 10106 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) / (𝑝𝑘)) ∈ ℂ)
142141anassrs 681 . . . . . . 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 6698 . . . . . . . . . 10 (𝑘 = 1 → (𝑝𝑘) = (𝑝↑1))
144143fveq2d 6233 . . . . . . . . 9 (𝑘 = 1 → (Λ‘(𝑝𝑘)) = (Λ‘(𝑝↑1)))
145143eleq1d 2715 . . . . . . . . . 10 (𝑘 = 1 → ((𝑝𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
146143fveq2d 6233 . . . . . . . . . 10 (𝑘 = 1 → (log‘(𝑝𝑘)) = (log‘(𝑝↑1)))
147145, 146ifbieq1d 4142 . . . . . . . . 9 (𝑘 = 1 → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
148144, 147oveq12d 6708 . . . . . . . 8 (𝑘 = 1 → ((Λ‘(𝑝𝑘)) − if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0)) = ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)))
149148, 143oveq12d 6708 . . . . . . 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 14526 . . . . . 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 11074 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
152151exp1d 13043 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
153152fveq2d 6233 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
154 vmaprm 24888 . . . . . . . . . . . . 13 (𝑝 ∈ ℙ → (Λ‘𝑝) = (log‘𝑝))
15550, 154syl 17 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑝) = (log‘𝑝))
156153, 155eqtrd 2685 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
157152, 50eqeltrd 2730 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
158157iftrued 4127 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
159152fveq2d 6233 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
160158, 159eqtrd 2685 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
161156, 160oveq12d 6708 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = ((log‘𝑝) − (log‘𝑝)))
162128subidd 10418 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
163161, 162eqtrd 2685 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
164163, 152oveq12d 6708 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = (0 / 𝑝))
165110rpcnne0d 11919 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
166 div0 10753 . . . . . . . . 9 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (0 / 𝑝) = 0)
167165, 166syl 17 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 / 𝑝) = 0)
168164, 167eqtrd 2685 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
169 1p1e2 11172 . . . . . . . . . 10 (1 + 1) = 2
170169oveq1i 6700 . . . . . . . . 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 12376 . . . . . . . . . . . . . 14 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ‘2))
173 eluz2nn 11764 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘2) → 𝑘 ∈ ℕ)
174172, 173syl 17 . . . . . . . . . . . . 13 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
175174, 170eleq2s 2748 . . . . . . . . . . . 12 (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
17650, 175, 92syl2an 493 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (Λ‘(𝑝𝑘)) = (log‘𝑝))
17752adantr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℕ)
178 nnq 11839 . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . 14 (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ‘2))
181180adantl 481 . . . . . . . . . . . . 13 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑘 ∈ (ℤ‘2))
182 expnprm 15653 . . . . . . . . . . . . 13 ((𝑝 ∈ ℚ ∧ 𝑘 ∈ (ℤ‘2)) → ¬ (𝑝𝑘) ∈ ℙ)
183179, 181, 182syl2anc 694 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ¬ (𝑝𝑘) ∈ ℙ)
184183iffalsed 4130 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → if((𝑝𝑘) ∈ ℙ, (log‘(𝑝𝑘)), 0) = 0)
185176, 184oveq12d 6708 . . . . . . . . . 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 10419 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − 0) = (log‘𝑝))
187186adantr 480 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) − 0) = (log‘𝑝))
188185, 187eqtrd 2685 . . . . . . . . 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 6705 . . . . . . . 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 14481 . . . . . . 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 6708 . . . . . 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 12812 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
193111adantr 480 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ)
194 nnnn0 11337 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
19552, 194, 99syl2an 493 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ∈ ℕ)
196193, 195nndivred 11107 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
197174, 196sylan2 490 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
198192, 197fsumrecl 14509 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ∈ ℝ)
199198recnd 10106 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ∈ ℂ)
200199addid2d 10275 . . . . . 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 2689 . . . . 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 11920 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ+)
203127flcld 12639 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
204203peano2zd 11523 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
205202, 204rpexpcld 13072 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ+)
206205rpge0d 11914 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
20752nnrecred 11104 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
208207resqcld 13075 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
209138peano2nnd 11075 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
210209nnnn0d 11389 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ0)
211207, 210reexpcld 13065 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
212208, 211subge02d 10657 . . . . . . . . . 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 222 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2))
214113nnrpd 11908 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ+)
215214rpcnne0d 11919 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
216202rpcnd 11912 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
217 dmdcan 10773 . . . . . . . . . . 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 1366 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
219134recnd 10106 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
220 divsubdir 10759 . . . . . . . . . . . . 13 ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
221151, 219, 165, 220syl3anc 1366 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
222 divid 10752 . . . . . . . . . . . . . 14 ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
223165, 222syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 / 𝑝) = 1)
224223oveq1d 6705 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
225221, 224eqtrd 2685 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
226 divdiv1 10774 . . . . . . . . . . . 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 1366 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
228225, 227oveq12d 6708 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
22952nnne0d 11103 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
230216, 151, 229divrecd 10842 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
231216sqvald 13045 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
232230, 231eqtr4d 2688 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
233218, 228, 2323eqtr3d 2693 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
234213, 233breqtrrd 4713 . . . . . . . 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 10496 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
236114nnrecred 11104 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
237 resubcl 10383 . . . . . . . . . 10 ((1 ∈ ℝ ∧ (1 / 𝑝) ∈ ℝ) → (1 − (1 / 𝑝)) ∈ ℝ)
238133, 207, 237sylancr 696 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ)
239 recgt1 10957 . . . . . . . . . . . 12 ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
24053, 120, 239syl2anc 694 . . . . . . . . . . 11 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
241125, 240mpbid 222 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) < 1)
242 posdif 10559 . . . . . . . . . . 11 (((1 / 𝑝) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
243207, 133, 242sylancl 695 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
244241, 243mpbid 222 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < (1 − (1 / 𝑝)))
245 ledivmul 10937 . . . . . . . . 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 1370 . . . . . . . 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 247 . . . . . . 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 11907 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ+)
249235, 248rerpdivcld 11941 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ∈ ℝ)
250249, 236, 126lemul2d 11954 . . . . . . 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 222 . . . . . 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 480 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℂ)
253195nncnd 11074 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ∈ ℂ)
254195nnne0d 11103 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝𝑘) ≠ 0)
255252, 253, 254divrecd 10842 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · (1 / (𝑝𝑘))))
256151adantr 480 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ)
25752adantr 480 . . . . . . . . . . . . 13 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ)
258257nnne0d 11103 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
259 nnz 11437 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
260259adantl 481 . . . . . . . . . . . 12 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
261256, 258, 260exprecd 13056 . . . . . . . . . . 11 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝𝑘)))
262261oveq2d 6706 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝𝑘))))
263255, 262eqtr4d 2688 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
264174, 263sylan2 490 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
265264sumeq2dv 14477 . . . . . . 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 11389 . . . . . . . . 9 (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ0)
267 expcl 12918 . . . . . . . . 9 (((1 / 𝑝) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
268216, 266, 267syl2an 493 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
269192, 128, 268fsummulc2 14560 . . . . . . 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 12576 . . . . . . . . . . 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 14475 . . . . . . . . 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 10211 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
274 2nn0 11347 . . . . . . . . . . 11 2 ∈ ℕ0
275274a1i 11 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ0)
276 eluzp1p1 11751 . . . . . . . . . . . 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 11117 . . . . . . . . . . . 12 2 = (1 + 1)
279278fveq2i 6232 . . . . . . . . . . 11 (ℤ‘2) = (ℤ‘(1 + 1))
280277, 279syl6eleqr 2741 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ‘2))
281216, 273, 275, 280geoserg 14642 . . . . . . . . 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 2685 . . . . . . . 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 6706 . . . . . . 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 2691 . . . . . 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 11074 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
286114nnne0d 11103 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
287128, 285, 286divrecd 10842 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
288251, 284, 2873brtr4d 4717 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
289201, 288eqbrtrd 4707 . . . 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 14575 . . 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 12376 . . . . . . . . . . 11 (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ‘2))
292 eluz2nn 11764 . . . . . . . . . . 11 (𝑝 ∈ (ℤ‘2) → 𝑝 ∈ ℕ)
293291, 292syl 17 . . . . . . . . . 10 (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ ℕ)
294293adantl 481 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℕ)
295294nnred 11073 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℝ)
296291adantl 481 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ (ℤ‘2))
297 eluz2b2 11799 . . . . . . . . . 10 (𝑝 ∈ (ℤ‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
298297simprbi 479 . . . . . . . . 9 (𝑝 ∈ (ℤ‘2) → 1 < 𝑝)
299296, 298syl 17 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 1 < 𝑝)
300295, 299rplogcld 24420 . . . . . . 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 11106 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
303302nnrpd 11908 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ+)
304300, 303rpdivcld 11927 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ+)
305304rpred 11910 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
30647, 305fsumrecl 14509 . . . 4 (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
307304rpge0d 11914 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
30847, 305, 307, 84fsumless 14572 . . . 4 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
309 rplogsumlem1 25218 . . . . 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 10232 . . 3 (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
312109, 116, 118, 290, 311letrd 10232 . 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 4707 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ∩ cin 3606   ⊆ wss 3607  ifcif 4119   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112   ≤ cle 10113   − cmin 10304   / cdiv 10722  ℕcn 11058  2c2 11108  ℕ0cn0 11330  ℤcz 11415  ℤ≥cuz 11725  ℚcq 11826  ℝ+crp 11870  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  ⌊cfl 12631  ↑cexp 12900  abscabs 14018  Σcsu 14460  ℙcprime 15432  logclog 24346  Λcvma 24863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-ef 14842  df-sin 14844  df-cos 14845  df-tan 14846  df-pi 14847  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-vma 24869 This theorem is referenced by:  rplogsum  25261
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