Step |
Hyp |
Ref |
Expression |
1 |
|
prmorcht.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , 𝑛 , 1 ) ) |
2 |
|
nnre |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℝ ) |
3 |
|
chtval |
⊢ ( 𝐴 ∈ ℝ → ( θ ‘ 𝐴 ) = Σ 𝑘 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ∈ ℕ → ( θ ‘ 𝐴 ) = Σ 𝑘 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ) |
5 |
|
2eluzge1 |
⊢ 2 ∈ ( ℤ≥ ‘ 1 ) |
6 |
|
ppisval2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
7 |
2 5 6
|
sylancl |
⊢ ( 𝐴 ∈ ℕ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
8 |
|
nnz |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℤ ) |
9 |
|
flid |
⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) = 𝐴 ) |
10 |
8 9
|
syl |
⊢ ( 𝐴 ∈ ℕ → ( ⌊ ‘ 𝐴 ) = 𝐴 ) |
11 |
10
|
oveq2d |
⊢ ( 𝐴 ∈ ℕ → ( 1 ... ( ⌊ ‘ 𝐴 ) ) = ( 1 ... 𝐴 ) ) |
12 |
11
|
ineq1d |
⊢ ( 𝐴 ∈ ℕ → ( ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) = ( ( 1 ... 𝐴 ) ∩ ℙ ) ) |
13 |
7 12
|
eqtrd |
⊢ ( 𝐴 ∈ ℕ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 1 ... 𝐴 ) ∩ ℙ ) ) |
14 |
13
|
sumeq1d |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑘 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) = Σ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ) |
15 |
|
inss1 |
⊢ ( ( 1 ... 𝐴 ) ∩ ℙ ) ⊆ ( 1 ... 𝐴 ) |
16 |
|
elinel1 |
⊢ ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) → 𝑘 ∈ ( 1 ... 𝐴 ) ) |
17 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... 𝐴 ) → 𝑘 ∈ ℕ ) |
18 |
17
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → 𝑘 ∈ ℕ ) |
19 |
18
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → 𝑘 ∈ ℝ+ ) |
20 |
19
|
relogcld |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( log ‘ 𝑘 ) ∈ ℝ ) |
21 |
20
|
recnd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( log ‘ 𝑘 ) ∈ ℂ ) |
22 |
16 21
|
sylan2 |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ) → ( log ‘ 𝑘 ) ∈ ℂ ) |
23 |
22
|
ralrimiva |
⊢ ( 𝐴 ∈ ℕ → ∀ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ∈ ℂ ) |
24 |
|
fzfi |
⊢ ( 1 ... 𝐴 ) ∈ Fin |
25 |
24
|
olci |
⊢ ( ( 1 ... 𝐴 ) ⊆ ( ℤ≥ ‘ 1 ) ∨ ( 1 ... 𝐴 ) ∈ Fin ) |
26 |
|
sumss2 |
⊢ ( ( ( ( ( 1 ... 𝐴 ) ∩ ℙ ) ⊆ ( 1 ... 𝐴 ) ∧ ∀ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ∈ ℂ ) ∧ ( ( 1 ... 𝐴 ) ⊆ ( ℤ≥ ‘ 1 ) ∨ ( 1 ... 𝐴 ) ∈ Fin ) ) → Σ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
27 |
25 26
|
mpan2 |
⊢ ( ( ( ( 1 ... 𝐴 ) ∩ ℙ ) ⊆ ( 1 ... 𝐴 ) ∧ ∀ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) ∈ ℂ ) → Σ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
28 |
15 23 27
|
sylancr |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
29 |
14 28
|
eqtrd |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑘 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
30 |
4 29
|
eqtrd |
⊢ ( 𝐴 ∈ ℕ → ( θ ‘ 𝐴 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
31 |
|
elin |
⊢ ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ↔ ( 𝑘 ∈ ( 1 ... 𝐴 ) ∧ 𝑘 ∈ ℙ ) ) |
32 |
31
|
baibr |
⊢ ( 𝑘 ∈ ( 1 ... 𝐴 ) → ( 𝑘 ∈ ℙ ↔ 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) ) ) |
33 |
32
|
ifbid |
⊢ ( 𝑘 ∈ ( 1 ... 𝐴 ) → if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) = if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) ) |
34 |
33
|
sumeq2i |
⊢ Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ( ( 1 ... 𝐴 ) ∩ ℙ ) , ( log ‘ 𝑘 ) , 0 ) |
35 |
30 34
|
eqtr4di |
⊢ ( 𝐴 ∈ ℕ → ( θ ‘ 𝐴 ) = Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ) |
36 |
|
eleq1w |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ ) ) |
37 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( log ‘ 𝑛 ) = ( log ‘ 𝑘 ) ) |
38 |
36 37
|
ifbieq1d |
⊢ ( 𝑛 = 𝑘 → if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ) |
39 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) |
40 |
|
fvex |
⊢ ( log ‘ 𝑘 ) ∈ V |
41 |
|
0cn |
⊢ 0 ∈ ℂ |
42 |
41
|
elexi |
⊢ 0 ∈ V |
43 |
40 42
|
ifex |
⊢ if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ∈ V |
44 |
38 39 43
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ) |
45 |
18 44
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ) |
46 |
|
elnnuz |
⊢ ( 𝐴 ∈ ℕ ↔ 𝐴 ∈ ( ℤ≥ ‘ 1 ) ) |
47 |
46
|
biimpi |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ( ℤ≥ ‘ 1 ) ) |
48 |
|
ifcl |
⊢ ( ( ( log ‘ 𝑘 ) ∈ ℂ ∧ 0 ∈ ℂ ) → if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ∈ ℂ ) |
49 |
21 41 48
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ∈ ℂ ) |
50 |
45 47 49
|
fsumser |
⊢ ( 𝐴 ∈ ℕ → Σ 𝑘 ∈ ( 1 ... 𝐴 ) if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ) ‘ 𝐴 ) ) |
51 |
35 50
|
eqtrd |
⊢ ( 𝐴 ∈ ℕ → ( θ ‘ 𝐴 ) = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ) ‘ 𝐴 ) ) |
52 |
51
|
fveq2d |
⊢ ( 𝐴 ∈ ℕ → ( exp ‘ ( θ ‘ 𝐴 ) ) = ( exp ‘ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ) ‘ 𝐴 ) ) ) |
53 |
|
addcl |
⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ ) → ( 𝑘 + 𝑝 ) ∈ ℂ ) |
54 |
53
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ ) ) → ( 𝑘 + 𝑝 ) ∈ ℂ ) |
55 |
45 49
|
eqeltrd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) ∈ ℂ ) |
56 |
|
efadd |
⊢ ( ( 𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ ) → ( exp ‘ ( 𝑘 + 𝑝 ) ) = ( ( exp ‘ 𝑘 ) · ( exp ‘ 𝑝 ) ) ) |
57 |
56
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 𝑘 ∈ ℂ ∧ 𝑝 ∈ ℂ ) ) → ( exp ‘ ( 𝑘 + 𝑝 ) ) = ( ( exp ‘ 𝑘 ) · ( exp ‘ 𝑝 ) ) ) |
58 |
|
1nn |
⊢ 1 ∈ ℕ |
59 |
|
ifcl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 1 ∈ ℕ ) → if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ∈ ℕ ) |
60 |
18 58 59
|
sylancl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ∈ ℕ ) |
61 |
60
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ∈ ℝ+ ) |
62 |
61
|
reeflogd |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( exp ‘ ( log ‘ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) ) = if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) |
63 |
|
fvif |
⊢ ( log ‘ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , ( log ‘ 1 ) ) |
64 |
|
log1 |
⊢ ( log ‘ 1 ) = 0 |
65 |
|
ifeq2 |
⊢ ( ( log ‘ 1 ) = 0 → if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , ( log ‘ 1 ) ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) ) |
66 |
64 65
|
ax-mp |
⊢ if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , ( log ‘ 1 ) ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) |
67 |
63 66
|
eqtri |
⊢ ( log ‘ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) = if ( 𝑘 ∈ ℙ , ( log ‘ 𝑘 ) , 0 ) |
68 |
45 67
|
eqtr4di |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) = ( log ‘ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) ) |
69 |
68
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( exp ‘ ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) ) = ( exp ‘ ( log ‘ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) ) ) |
70 |
|
id |
⊢ ( 𝑛 = 𝑘 → 𝑛 = 𝑘 ) |
71 |
36 70
|
ifbieq1d |
⊢ ( 𝑛 = 𝑘 → if ( 𝑛 ∈ ℙ , 𝑛 , 1 ) = if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) |
72 |
|
vex |
⊢ 𝑘 ∈ V |
73 |
58
|
elexi |
⊢ 1 ∈ V |
74 |
72 73
|
ifex |
⊢ if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ∈ V |
75 |
71 1 74
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) |
76 |
18 75
|
syl |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , 𝑘 , 1 ) ) |
77 |
62 69 76
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℕ ∧ 𝑘 ∈ ( 1 ... 𝐴 ) ) → ( exp ‘ ( ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ‘ 𝑘 ) ) = ( 𝐹 ‘ 𝑘 ) ) |
78 |
54 55 47 57 77
|
seqhomo |
⊢ ( 𝐴 ∈ ℕ → ( exp ‘ ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( log ‘ 𝑛 ) , 0 ) ) ) ‘ 𝐴 ) ) = ( seq 1 ( · , 𝐹 ) ‘ 𝐴 ) ) |
79 |
52 78
|
eqtrd |
⊢ ( 𝐴 ∈ ℕ → ( exp ‘ ( θ ‘ 𝐴 ) ) = ( seq 1 ( · , 𝐹 ) ‘ 𝐴 ) ) |