Step |
Hyp |
Ref |
Expression |
1 |
|
flid |
⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) = 𝐴 ) |
2 |
1
|
oveq2d |
⊢ ( 𝐴 ∈ ℤ → ( 1 ... ( ⌊ ‘ 𝐴 ) ) = ( 1 ... 𝐴 ) ) |
3 |
2
|
sumeq1d |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = Σ 𝑛 ∈ ( 1 ... 𝐴 ) ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) ) |
4 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → ( Λ ‘ 𝑛 ) = ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) ) |
5 |
|
eleq1 |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → ( 𝑛 ∈ ℙ ↔ ( 𝑝 ↑ 𝑘 ) ∈ ℙ ) ) |
6 |
|
fveq2 |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → ( log ‘ 𝑛 ) = ( log ‘ ( 𝑝 ↑ 𝑘 ) ) ) |
7 |
5 6
|
ifbieq1d |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) = if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) |
8 |
4 7
|
oveq12d |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) = ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) ) |
9 |
|
id |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → 𝑛 = ( 𝑝 ↑ 𝑘 ) ) |
10 |
8 9
|
oveq12d |
⊢ ( 𝑛 = ( 𝑝 ↑ 𝑘 ) → ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ) |
11 |
|
zre |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℝ ) |
12 |
|
elfznn |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑛 ∈ ℕ ) |
13 |
12
|
adantl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℕ ) |
14 |
|
vmacl |
⊢ ( 𝑛 ∈ ℕ → ( Λ ‘ 𝑛 ) ∈ ℝ ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( Λ ‘ 𝑛 ) ∈ ℝ ) |
16 |
13
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℝ+ ) |
17 |
16
|
relogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( log ‘ 𝑛 ) ∈ ℝ ) |
18 |
|
0re |
⊢ 0 ∈ ℝ |
19 |
|
ifcl |
⊢ ( ( ( log ‘ 𝑛 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ∈ ℝ ) |
20 |
17 18 19
|
sylancl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ∈ ℝ ) |
21 |
15 20
|
resubcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ∈ ℝ ) |
22 |
21 13
|
nndivred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) ∈ ℝ ) |
23 |
22
|
recnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) ∈ ℂ ) |
24 |
|
simprr |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( Λ ‘ 𝑛 ) = 0 ) |
25 |
|
vmaprm |
⊢ ( 𝑛 ∈ ℙ → ( Λ ‘ 𝑛 ) = ( log ‘ 𝑛 ) ) |
26 |
|
prmnn |
⊢ ( 𝑛 ∈ ℙ → 𝑛 ∈ ℕ ) |
27 |
26
|
nnred |
⊢ ( 𝑛 ∈ ℙ → 𝑛 ∈ ℝ ) |
28 |
|
prmgt1 |
⊢ ( 𝑛 ∈ ℙ → 1 < 𝑛 ) |
29 |
27 28
|
rplogcld |
⊢ ( 𝑛 ∈ ℙ → ( log ‘ 𝑛 ) ∈ ℝ+ ) |
30 |
25 29
|
eqeltrd |
⊢ ( 𝑛 ∈ ℙ → ( Λ ‘ 𝑛 ) ∈ ℝ+ ) |
31 |
30
|
rpne0d |
⊢ ( 𝑛 ∈ ℙ → ( Λ ‘ 𝑛 ) ≠ 0 ) |
32 |
31
|
necon2bi |
⊢ ( ( Λ ‘ 𝑛 ) = 0 → ¬ 𝑛 ∈ ℙ ) |
33 |
32
|
ad2antll |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ¬ 𝑛 ∈ ℙ ) |
34 |
33
|
iffalsed |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) = 0 ) |
35 |
24 34
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) = ( 0 − 0 ) ) |
36 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
37 |
35 36
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) = 0 ) |
38 |
37
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = ( 0 / 𝑛 ) ) |
39 |
12
|
ad2antrl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → 𝑛 ∈ ℕ ) |
40 |
39
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → 𝑛 ∈ ℝ+ ) |
41 |
40
|
rpcnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) ) |
42 |
|
div0 |
⊢ ( ( 𝑛 ∈ ℂ ∧ 𝑛 ≠ 0 ) → ( 0 / 𝑛 ) = 0 ) |
43 |
41 42
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( 0 / 𝑛 ) = 0 ) |
44 |
38 43
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ ( Λ ‘ 𝑛 ) = 0 ) ) → ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = 0 ) |
45 |
10 11 23 44
|
fsumvma2 |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ) |
46 |
3 45
|
eqtr3d |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑛 ∈ ( 1 ... 𝐴 ) ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) = Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ) |
47 |
|
fzfid |
⊢ ( 𝐴 ∈ ℤ → ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ∈ Fin ) |
48 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) |
49 |
48
|
elin2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ℙ ) |
50 |
|
prmnn |
⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ℕ ) |
51 |
49 50
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ℕ ) |
52 |
51
|
nnred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ℝ ) |
53 |
11
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝐴 ∈ ℝ ) |
54 |
|
zcn |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ ) |
55 |
54
|
abscld |
⊢ ( 𝐴 ∈ ℤ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
56 |
|
peano2re |
⊢ ( ( abs ‘ 𝐴 ) ∈ ℝ → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
57 |
55 56
|
syl |
⊢ ( 𝐴 ∈ ℤ → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
58 |
57
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℝ ) |
59 |
|
elinel1 |
⊢ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) → 𝑝 ∈ ( 0 [,] 𝐴 ) ) |
60 |
|
elicc2 |
⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝑝 ∈ ( 0 [,] 𝐴 ) ↔ ( 𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴 ) ) ) |
61 |
18 11 60
|
sylancr |
⊢ ( 𝐴 ∈ ℤ → ( 𝑝 ∈ ( 0 [,] 𝐴 ) ↔ ( 𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴 ) ) ) |
62 |
59 61
|
syl5ib |
⊢ ( 𝐴 ∈ ℤ → ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) → ( 𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴 ) ) ) |
63 |
62
|
imp |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴 ) ) |
64 |
63
|
simp3d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ≤ 𝐴 ) |
65 |
54
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝐴 ∈ ℂ ) |
66 |
65
|
abscld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( abs ‘ 𝐴 ) ∈ ℝ ) |
67 |
53
|
leabsd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝐴 ≤ ( abs ‘ 𝐴 ) ) |
68 |
66
|
lep1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( abs ‘ 𝐴 ) ≤ ( ( abs ‘ 𝐴 ) + 1 ) ) |
69 |
53 66 58 67 68
|
letrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝐴 ≤ ( ( abs ‘ 𝐴 ) + 1 ) ) |
70 |
52 53 58 64 69
|
letrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ≤ ( ( abs ‘ 𝐴 ) + 1 ) ) |
71 |
|
prmuz2 |
⊢ ( 𝑝 ∈ ℙ → 𝑝 ∈ ( ℤ≥ ‘ 2 ) ) |
72 |
49 71
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ( ℤ≥ ‘ 2 ) ) |
73 |
|
nn0abscl |
⊢ ( 𝐴 ∈ ℤ → ( abs ‘ 𝐴 ) ∈ ℕ0 ) |
74 |
|
nn0p1nn |
⊢ ( ( abs ‘ 𝐴 ) ∈ ℕ0 → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℕ ) |
75 |
73 74
|
syl |
⊢ ( 𝐴 ∈ ℤ → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℕ ) |
76 |
75
|
nnzd |
⊢ ( 𝐴 ∈ ℤ → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℤ ) |
77 |
76
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℤ ) |
78 |
|
elfz5 |
⊢ ( ( 𝑝 ∈ ( ℤ≥ ‘ 2 ) ∧ ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℤ ) → ( 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ↔ 𝑝 ≤ ( ( abs ‘ 𝐴 ) + 1 ) ) ) |
79 |
72 77 78
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ↔ 𝑝 ≤ ( ( abs ‘ 𝐴 ) + 1 ) ) ) |
80 |
70 79
|
mpbird |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) |
81 |
80
|
ex |
⊢ ( 𝐴 ∈ ℤ → ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) → 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) ) |
82 |
81
|
ssrdv |
⊢ ( 𝐴 ∈ ℤ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) ⊆ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) |
83 |
47 82
|
ssfid |
⊢ ( 𝐴 ∈ ℤ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∈ Fin ) |
84 |
|
fzfid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ∈ Fin ) |
85 |
|
simprl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) |
86 |
85
|
elin2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑝 ∈ ℙ ) |
87 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ℕ ) |
88 |
87
|
ad2antll |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑘 ∈ ℕ ) |
89 |
|
vmappw |
⊢ ( ( 𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ ) → ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) = ( log ‘ 𝑝 ) ) |
90 |
86 88 89
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) = ( log ‘ 𝑝 ) ) |
91 |
51
|
adantrr |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑝 ∈ ℕ ) |
92 |
91
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑝 ∈ ℝ+ ) |
93 |
92
|
relogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( log ‘ 𝑝 ) ∈ ℝ ) |
94 |
90 93
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
95 |
88
|
nnnn0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → 𝑘 ∈ ℕ0 ) |
96 |
|
nnexpcl |
⊢ ( ( 𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑝 ↑ 𝑘 ) ∈ ℕ ) |
97 |
91 95 96
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( 𝑝 ↑ 𝑘 ) ∈ ℕ ) |
98 |
97
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( 𝑝 ↑ 𝑘 ) ∈ ℝ+ ) |
99 |
98
|
relogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( log ‘ ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
100 |
|
ifcl |
⊢ ( ( ( log ‘ ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ∈ ℝ ) |
101 |
99 18 100
|
sylancl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ∈ ℝ ) |
102 |
94 101
|
resubcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) ∈ ℝ ) |
103 |
102 97
|
nndivred |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
104 |
103
|
anassrs |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
105 |
84 104
|
fsumrecl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
106 |
83 105
|
fsumrecl |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
107 |
51
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ℝ+ ) |
108 |
107
|
relogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝑝 ) ∈ ℝ ) |
109 |
|
uz2m1nn |
⊢ ( 𝑝 ∈ ( ℤ≥ ‘ 2 ) → ( 𝑝 − 1 ) ∈ ℕ ) |
110 |
72 109
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 − 1 ) ∈ ℕ ) |
111 |
51 110
|
nnmulcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 · ( 𝑝 − 1 ) ) ∈ ℕ ) |
112 |
108 111
|
nndivred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ ) |
113 |
83 112
|
fsumrecl |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ ) |
114 |
|
2re |
⊢ 2 ∈ ℝ |
115 |
114
|
a1i |
⊢ ( 𝐴 ∈ ℤ → 2 ∈ ℝ ) |
116 |
18
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 0 ∈ ℝ ) |
117 |
51
|
nngt0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 0 < 𝑝 ) |
118 |
116 52 53 117 64
|
ltletrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 0 < 𝐴 ) |
119 |
53 118
|
elrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝐴 ∈ ℝ+ ) |
120 |
119
|
relogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝐴 ) ∈ ℝ ) |
121 |
|
prmgt1 |
⊢ ( 𝑝 ∈ ℙ → 1 < 𝑝 ) |
122 |
49 121
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 1 < 𝑝 ) |
123 |
52 122
|
rplogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝑝 ) ∈ ℝ+ ) |
124 |
120 123
|
rerpdivcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ∈ ℝ ) |
125 |
123
|
rpcnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝑝 ) ∈ ℂ ) |
126 |
125
|
mulid2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 · ( log ‘ 𝑝 ) ) = ( log ‘ 𝑝 ) ) |
127 |
107 119
|
logled |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ≤ 𝐴 ↔ ( log ‘ 𝑝 ) ≤ ( log ‘ 𝐴 ) ) ) |
128 |
64 127
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝑝 ) ≤ ( log ‘ 𝐴 ) ) |
129 |
126 128
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 · ( log ‘ 𝑝 ) ) ≤ ( log ‘ 𝐴 ) ) |
130 |
|
1re |
⊢ 1 ∈ ℝ |
131 |
130
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 1 ∈ ℝ ) |
132 |
131 120 123
|
lemuldivd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 · ( log ‘ 𝑝 ) ) ≤ ( log ‘ 𝐴 ) ↔ 1 ≤ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) |
133 |
129 132
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 1 ≤ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) |
134 |
|
flge1nn |
⊢ ( ( ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ∈ ℝ ∧ 1 ≤ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ℕ ) |
135 |
124 133 134
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ℕ ) |
136 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
137 |
135 136
|
eleqtrdi |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ( ℤ≥ ‘ 1 ) ) |
138 |
103
|
recnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℂ ) |
139 |
138
|
anassrs |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℂ ) |
140 |
|
oveq2 |
⊢ ( 𝑘 = 1 → ( 𝑝 ↑ 𝑘 ) = ( 𝑝 ↑ 1 ) ) |
141 |
140
|
fveq2d |
⊢ ( 𝑘 = 1 → ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) = ( Λ ‘ ( 𝑝 ↑ 1 ) ) ) |
142 |
140
|
eleq1d |
⊢ ( 𝑘 = 1 → ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ ↔ ( 𝑝 ↑ 1 ) ∈ ℙ ) ) |
143 |
140
|
fveq2d |
⊢ ( 𝑘 = 1 → ( log ‘ ( 𝑝 ↑ 𝑘 ) ) = ( log ‘ ( 𝑝 ↑ 1 ) ) ) |
144 |
142 143
|
ifbieq1d |
⊢ ( 𝑘 = 1 → if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) = if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) |
145 |
141 144
|
oveq12d |
⊢ ( 𝑘 = 1 → ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) = ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) ) |
146 |
145 140
|
oveq12d |
⊢ ( 𝑘 = 1 → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) / ( 𝑝 ↑ 1 ) ) ) |
147 |
137 139 146
|
fsum1p |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) / ( 𝑝 ↑ 1 ) ) + Σ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ) ) |
148 |
51
|
nncnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ∈ ℂ ) |
149 |
148
|
exp1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ↑ 1 ) = 𝑝 ) |
150 |
149
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( Λ ‘ ( 𝑝 ↑ 1 ) ) = ( Λ ‘ 𝑝 ) ) |
151 |
|
vmaprm |
⊢ ( 𝑝 ∈ ℙ → ( Λ ‘ 𝑝 ) = ( log ‘ 𝑝 ) ) |
152 |
49 151
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( Λ ‘ 𝑝 ) = ( log ‘ 𝑝 ) ) |
153 |
150 152
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( Λ ‘ ( 𝑝 ↑ 1 ) ) = ( log ‘ 𝑝 ) ) |
154 |
149 49
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ↑ 1 ) ∈ ℙ ) |
155 |
154
|
iftrued |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) = ( log ‘ ( 𝑝 ↑ 1 ) ) ) |
156 |
149
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( log ‘ ( 𝑝 ↑ 1 ) ) = ( log ‘ 𝑝 ) ) |
157 |
155 156
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) = ( log ‘ 𝑝 ) ) |
158 |
153 157
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) = ( ( log ‘ 𝑝 ) − ( log ‘ 𝑝 ) ) ) |
159 |
125
|
subidd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) − ( log ‘ 𝑝 ) ) = 0 ) |
160 |
158 159
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) = 0 ) |
161 |
160 149
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) / ( 𝑝 ↑ 1 ) ) = ( 0 / 𝑝 ) ) |
162 |
107
|
rpcnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) ) |
163 |
|
div0 |
⊢ ( ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) → ( 0 / 𝑝 ) = 0 ) |
164 |
162 163
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 0 / 𝑝 ) = 0 ) |
165 |
161 164
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) / ( 𝑝 ↑ 1 ) ) = 0 ) |
166 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
167 |
166
|
oveq1i |
⊢ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) = ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) |
168 |
167
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) = ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) |
169 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 2 ) ) |
170 |
|
eluz2nn |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 2 ) → 𝑘 ∈ ℕ ) |
171 |
169 170
|
syl |
⊢ ( 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ℕ ) |
172 |
171 167
|
eleq2s |
⊢ ( 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ℕ ) |
173 |
49 172 89
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) = ( log ‘ 𝑝 ) ) |
174 |
51
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → 𝑝 ∈ ℕ ) |
175 |
|
nnq |
⊢ ( 𝑝 ∈ ℕ → 𝑝 ∈ ℚ ) |
176 |
174 175
|
syl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → 𝑝 ∈ ℚ ) |
177 |
169 167
|
eleq2s |
⊢ ( 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 2 ) ) |
178 |
177
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 2 ) ) |
179 |
|
expnprm |
⊢ ( ( 𝑝 ∈ ℚ ∧ 𝑘 ∈ ( ℤ≥ ‘ 2 ) ) → ¬ ( 𝑝 ↑ 𝑘 ) ∈ ℙ ) |
180 |
176 178 179
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ¬ ( 𝑝 ↑ 𝑘 ) ∈ ℙ ) |
181 |
180
|
iffalsed |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) = 0 ) |
182 |
173 181
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) = ( ( log ‘ 𝑝 ) − 0 ) ) |
183 |
125
|
subid1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) − 0 ) = ( log ‘ 𝑝 ) ) |
184 |
183
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( log ‘ 𝑝 ) − 0 ) = ( log ‘ 𝑝 ) ) |
185 |
182 184
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) = ( log ‘ 𝑝 ) ) |
186 |
185
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) |
187 |
168 186
|
sumeq12dv |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) |
188 |
165 187
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( ( Λ ‘ ( 𝑝 ↑ 1 ) ) − if ( ( 𝑝 ↑ 1 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 1 ) ) , 0 ) ) / ( 𝑝 ↑ 1 ) ) + Σ 𝑘 ∈ ( ( 1 + 1 ) ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ) = ( 0 + Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) ) |
189 |
|
fzfid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ∈ Fin ) |
190 |
108
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( log ‘ 𝑝 ) ∈ ℝ ) |
191 |
|
nnnn0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℕ0 ) |
192 |
51 191 96
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( 𝑝 ↑ 𝑘 ) ∈ ℕ ) |
193 |
190 192
|
nndivred |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
194 |
171 193
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
195 |
189 194
|
fsumrecl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℝ ) |
196 |
195
|
recnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ∈ ℂ ) |
197 |
196
|
addid2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 0 + Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) |
198 |
147 188 197
|
3eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ) |
199 |
107
|
rpreccld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / 𝑝 ) ∈ ℝ+ ) |
200 |
124
|
flcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ℤ ) |
201 |
200
|
peano2zd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ℤ ) |
202 |
199 201
|
rpexpcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ∈ ℝ+ ) |
203 |
202
|
rpge0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 0 ≤ ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) |
204 |
51
|
nnrecred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / 𝑝 ) ∈ ℝ ) |
205 |
204
|
resqcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) ↑ 2 ) ∈ ℝ ) |
206 |
135
|
peano2nnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ℕ ) |
207 |
206
|
nnnn0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ℕ0 ) |
208 |
204 207
|
reexpcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ∈ ℝ ) |
209 |
205 208
|
subge02d |
⊢ ( ( 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 ) ) ) |
210 |
203 209
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) ≤ ( ( 1 / 𝑝 ) ↑ 2 ) ) |
211 |
110
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 − 1 ) ∈ ℝ+ ) |
212 |
211
|
rpcnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 𝑝 − 1 ) ∈ ℂ ∧ ( 𝑝 − 1 ) ≠ 0 ) ) |
213 |
199
|
rpcnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / 𝑝 ) ∈ ℂ ) |
214 |
|
dmdcan |
⊢ ( ( ( ( 𝑝 − 1 ) ∈ ℂ ∧ ( 𝑝 − 1 ) ≠ 0 ) ∧ ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) ∧ ( 1 / 𝑝 ) ∈ ℂ ) → ( ( ( 𝑝 − 1 ) / 𝑝 ) · ( ( 1 / 𝑝 ) / ( 𝑝 − 1 ) ) ) = ( ( 1 / 𝑝 ) / 𝑝 ) ) |
215 |
212 162 213 214
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( 𝑝 − 1 ) / 𝑝 ) · ( ( 1 / 𝑝 ) / ( 𝑝 − 1 ) ) ) = ( ( 1 / 𝑝 ) / 𝑝 ) ) |
216 |
131
|
recnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 1 ∈ ℂ ) |
217 |
|
divsubdir |
⊢ ( ( 𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) ) → ( ( 𝑝 − 1 ) / 𝑝 ) = ( ( 𝑝 / 𝑝 ) − ( 1 / 𝑝 ) ) ) |
218 |
148 216 162 217
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 𝑝 − 1 ) / 𝑝 ) = ( ( 𝑝 / 𝑝 ) − ( 1 / 𝑝 ) ) ) |
219 |
|
divid |
⊢ ( ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) → ( 𝑝 / 𝑝 ) = 1 ) |
220 |
162 219
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 / 𝑝 ) = 1 ) |
221 |
220
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 𝑝 / 𝑝 ) − ( 1 / 𝑝 ) ) = ( 1 − ( 1 / 𝑝 ) ) ) |
222 |
218 221
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 𝑝 − 1 ) / 𝑝 ) = ( 1 − ( 1 / 𝑝 ) ) ) |
223 |
|
divdiv1 |
⊢ ( ( 1 ∈ ℂ ∧ ( 𝑝 ∈ ℂ ∧ 𝑝 ≠ 0 ) ∧ ( ( 𝑝 − 1 ) ∈ ℂ ∧ ( 𝑝 − 1 ) ≠ 0 ) ) → ( ( 1 / 𝑝 ) / ( 𝑝 − 1 ) ) = ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
224 |
216 162 212 223
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) / ( 𝑝 − 1 ) ) = ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
225 |
222 224
|
oveq12d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( 𝑝 − 1 ) / 𝑝 ) · ( ( 1 / 𝑝 ) / ( 𝑝 − 1 ) ) ) = ( ( 1 − ( 1 / 𝑝 ) ) · ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) ) |
226 |
51
|
nnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 𝑝 ≠ 0 ) |
227 |
213 148 226
|
divrecd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) / 𝑝 ) = ( ( 1 / 𝑝 ) · ( 1 / 𝑝 ) ) ) |
228 |
213
|
sqvald |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) ↑ 2 ) = ( ( 1 / 𝑝 ) · ( 1 / 𝑝 ) ) ) |
229 |
227 228
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) / 𝑝 ) = ( ( 1 / 𝑝 ) ↑ 2 ) ) |
230 |
215 225 229
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 − ( 1 / 𝑝 ) ) · ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) = ( ( 1 / 𝑝 ) ↑ 2 ) ) |
231 |
210 230
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) ≤ ( ( 1 − ( 1 / 𝑝 ) ) · ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) ) |
232 |
205 208
|
resubcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) ∈ ℝ ) |
233 |
111
|
nnrecred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ ) |
234 |
|
resubcl |
⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝑝 ) ∈ ℝ ) → ( 1 − ( 1 / 𝑝 ) ) ∈ ℝ ) |
235 |
130 204 234
|
sylancr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 − ( 1 / 𝑝 ) ) ∈ ℝ ) |
236 |
|
recgt1 |
⊢ ( ( 𝑝 ∈ ℝ ∧ 0 < 𝑝 ) → ( 1 < 𝑝 ↔ ( 1 / 𝑝 ) < 1 ) ) |
237 |
52 117 236
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 < 𝑝 ↔ ( 1 / 𝑝 ) < 1 ) ) |
238 |
122 237
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / 𝑝 ) < 1 ) |
239 |
|
posdif |
⊢ ( ( ( 1 / 𝑝 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 𝑝 ) < 1 ↔ 0 < ( 1 − ( 1 / 𝑝 ) ) ) ) |
240 |
204 130 239
|
sylancl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( 1 / 𝑝 ) < 1 ↔ 0 < ( 1 − ( 1 / 𝑝 ) ) ) ) |
241 |
238 240
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 0 < ( 1 − ( 1 / 𝑝 ) ) ) |
242 |
|
ledivmul |
⊢ ( ( ( ( ( 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 ) ) ) ) ) ) |
243 |
232 233 235 241 242
|
syl112anc |
⊢ ( ( 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 ) ) ) ) ) ) |
244 |
231 243
|
mpbird |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ≤ ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
245 |
235 241
|
elrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 − ( 1 / 𝑝 ) ) ∈ ℝ+ ) |
246 |
232 245
|
rerpdivcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ∈ ℝ ) |
247 |
246 233 123
|
lemul2d |
⊢ ( ( 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 ) ) ) ) ) ) |
248 |
244 247
|
mpbid |
⊢ ( ( 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 ) ) ) ) ) |
249 |
125
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( log ‘ 𝑝 ) ∈ ℂ ) |
250 |
192
|
nncnd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( 𝑝 ↑ 𝑘 ) ∈ ℂ ) |
251 |
192
|
nnne0d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( 𝑝 ↑ 𝑘 ) ≠ 0 ) |
252 |
249 250 251
|
divrecd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( 1 / ( 𝑝 ↑ 𝑘 ) ) ) ) |
253 |
148
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → 𝑝 ∈ ℂ ) |
254 |
51
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → 𝑝 ∈ ℕ ) |
255 |
254
|
nnne0d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → 𝑝 ≠ 0 ) |
256 |
|
nnz |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℤ ) |
257 |
256
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℤ ) |
258 |
253 255 257
|
exprecd |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( 1 / 𝑝 ) ↑ 𝑘 ) = ( 1 / ( 𝑝 ↑ 𝑘 ) ) ) |
259 |
258
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( log ‘ 𝑝 ) · ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( 1 / ( 𝑝 ↑ 𝑘 ) ) ) ) |
260 |
252 259
|
eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) ) |
261 |
171 260
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) ) |
262 |
261
|
sumeq2dv |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) · ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) ) |
263 |
171
|
nnnn0d |
⊢ ( 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) → 𝑘 ∈ ℕ0 ) |
264 |
|
expcl |
⊢ ( ( ( 1 / 𝑝 ) ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( ( 1 / 𝑝 ) ↑ 𝑘 ) ∈ ℂ ) |
265 |
213 263 264
|
syl2an |
⊢ ( ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) ∧ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ) → ( ( 1 / 𝑝 ) ↑ 𝑘 ) ∈ ℂ ) |
266 |
189 125 265
|
fsummulc2 |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) · Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) · ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) ) |
267 |
|
fzval3 |
⊢ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ℤ → ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) = ( 2 ..^ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) |
268 |
200 267
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) = ( 2 ..^ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) |
269 |
268
|
sumeq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) = Σ 𝑘 ∈ ( 2 ..^ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) |
270 |
204 238
|
ltned |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 1 / 𝑝 ) ≠ 1 ) |
271 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
272 |
271
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → 2 ∈ ℕ0 ) |
273 |
|
eluzp1p1 |
⊢ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ∈ ( ℤ≥ ‘ 1 ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) |
274 |
137 273
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) |
275 |
|
df-2 |
⊢ 2 = ( 1 + 1 ) |
276 |
275
|
fveq2i |
⊢ ( ℤ≥ ‘ 2 ) = ( ℤ≥ ‘ ( 1 + 1 ) ) |
277 |
274 276
|
eleqtrrdi |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ∈ ( ℤ≥ ‘ 2 ) ) |
278 |
213 270 272 277
|
geoserg |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ..^ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) = ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ) |
279 |
269 278
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) = ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ) |
280 |
279
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) · Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( 1 / 𝑝 ) ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ) ) |
281 |
262 266 280
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) = ( ( log ‘ 𝑝 ) · ( ( ( ( 1 / 𝑝 ) ↑ 2 ) − ( ( 1 / 𝑝 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) + 1 ) ) ) / ( 1 − ( 1 / 𝑝 ) ) ) ) ) |
282 |
111
|
nncnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 · ( 𝑝 − 1 ) ) ∈ ℂ ) |
283 |
111
|
nnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( 𝑝 · ( 𝑝 − 1 ) ) ≠ 0 ) |
284 |
125 282 283
|
divrecd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) = ( ( log ‘ 𝑝 ) · ( 1 / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) ) |
285 |
248 281 284
|
3brtr4d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 2 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 ↑ 𝑘 ) ) ≤ ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
286 |
198 285
|
eqbrtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ≤ ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
287 |
83 105 112 286
|
fsumle |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ≤ Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
288 |
|
elfzuz |
⊢ ( 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) → 𝑝 ∈ ( ℤ≥ ‘ 2 ) ) |
289 |
|
eluz2nn |
⊢ ( 𝑝 ∈ ( ℤ≥ ‘ 2 ) → 𝑝 ∈ ℕ ) |
290 |
288 289
|
syl |
⊢ ( 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) → 𝑝 ∈ ℕ ) |
291 |
290
|
adantl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → 𝑝 ∈ ℕ ) |
292 |
291
|
nnred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → 𝑝 ∈ ℝ ) |
293 |
288
|
adantl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → 𝑝 ∈ ( ℤ≥ ‘ 2 ) ) |
294 |
|
eluz2gt1 |
⊢ ( 𝑝 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝑝 ) |
295 |
293 294
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → 1 < 𝑝 ) |
296 |
292 295
|
rplogcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( log ‘ 𝑝 ) ∈ ℝ+ ) |
297 |
293 109
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( 𝑝 − 1 ) ∈ ℕ ) |
298 |
291 297
|
nnmulcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( 𝑝 · ( 𝑝 − 1 ) ) ∈ ℕ ) |
299 |
298
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( 𝑝 · ( 𝑝 − 1 ) ) ∈ ℝ+ ) |
300 |
296 299
|
rpdivcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ+ ) |
301 |
300
|
rpred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ ) |
302 |
47 301
|
fsumrecl |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ∈ ℝ ) |
303 |
300
|
rpge0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ) → 0 ≤ ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
304 |
47 301 303 82
|
fsumless |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ≤ Σ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ) |
305 |
|
rplogsumlem1 |
⊢ ( ( ( abs ‘ 𝐴 ) + 1 ) ∈ ℕ → Σ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ≤ 2 ) |
306 |
75 305
|
syl |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( 2 ... ( ( abs ‘ 𝐴 ) + 1 ) ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ≤ 2 ) |
307 |
113 302 115 304 306
|
letrd |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( ( log ‘ 𝑝 ) / ( 𝑝 · ( 𝑝 − 1 ) ) ) ≤ 2 ) |
308 |
106 113 115 287 307
|
letrd |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ 𝑝 ) ) ) ) ( ( ( Λ ‘ ( 𝑝 ↑ 𝑘 ) ) − if ( ( 𝑝 ↑ 𝑘 ) ∈ ℙ , ( log ‘ ( 𝑝 ↑ 𝑘 ) ) , 0 ) ) / ( 𝑝 ↑ 𝑘 ) ) ≤ 2 ) |
309 |
46 308
|
eqbrtrd |
⊢ ( 𝐴 ∈ ℤ → Σ 𝑛 ∈ ( 1 ... 𝐴 ) ( ( ( Λ ‘ 𝑛 ) − if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) / 𝑛 ) ≤ 2 ) |