Step |
Hyp |
Ref |
Expression |
1 |
|
peano2z |
|- ( A e. ZZ -> ( A + 1 ) e. ZZ ) |
2 |
1
|
adantr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ ) |
3 |
|
zre |
|- ( ( A + 1 ) e. ZZ -> ( A + 1 ) e. RR ) |
4 |
2 3
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. RR ) |
5 |
|
chtval |
|- ( ( A + 1 ) e. RR -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) ) |
6 |
4 5
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) ) |
7 |
|
ppisval |
|- ( ( A + 1 ) e. RR -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) ) |
8 |
4 7
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) ) |
9 |
|
flid |
|- ( ( A + 1 ) e. ZZ -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) ) |
10 |
2 9
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) ) |
11 |
10
|
oveq2d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` ( A + 1 ) ) ) = ( 2 ... ( A + 1 ) ) ) |
12 |
11
|
ineq1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) = ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) |
13 |
8 12
|
eqtrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) |
14 |
13
|
sumeq1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) ) |
15 |
6 14
|
eqtrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) ) |
16 |
|
zre |
|- ( A e. ZZ -> A e. RR ) |
17 |
16
|
adantr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. RR ) |
18 |
17
|
ltp1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A < ( A + 1 ) ) |
19 |
17 4
|
ltnled |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) ) |
20 |
18 19
|
mpbid |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) <_ A ) |
21 |
|
elinel1 |
|- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) e. ( 2 ... A ) ) |
22 |
|
elfzle2 |
|- ( ( A + 1 ) e. ( 2 ... A ) -> ( A + 1 ) <_ A ) |
23 |
21 22
|
syl |
|- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) <_ A ) |
24 |
20 23
|
nsyl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) ) |
25 |
|
disjsn |
|- ( ( ( ( 2 ... A ) i^i Prime ) i^i { ( A + 1 ) } ) = (/) <-> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) ) |
26 |
24 25
|
sylibr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( ( 2 ... A ) i^i Prime ) i^i { ( A + 1 ) } ) = (/) ) |
27 |
|
2z |
|- 2 e. ZZ |
28 |
|
zcn |
|- ( A e. ZZ -> A e. CC ) |
29 |
28
|
adantr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. CC ) |
30 |
|
ax-1cn |
|- 1 e. CC |
31 |
|
pncan |
|- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A ) |
32 |
29 30 31
|
sylancl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) = A ) |
33 |
|
prmuz2 |
|- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. ( ZZ>= ` 2 ) ) |
34 |
33
|
adantl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) ) |
35 |
|
uz2m1nn |
|- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN ) |
36 |
34 35
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) e. NN ) |
37 |
32 36
|
eqeltrrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. NN ) |
38 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
39 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
40 |
39
|
fveq2i |
|- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 ) |
41 |
38 40
|
eqtr4i |
|- NN = ( ZZ>= ` ( 2 - 1 ) ) |
42 |
37 41
|
eleqtrdi |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) ) |
43 |
|
fzsuc2 |
|- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) ) |
44 |
27 42 43
|
sylancr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) ) |
45 |
44
|
ineq1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) ) |
46 |
|
indir |
|- ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) |
47 |
45 46
|
eqtrdi |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) ) |
48 |
|
simpr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. Prime ) |
49 |
48
|
snssd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> { ( A + 1 ) } C_ Prime ) |
50 |
|
df-ss |
|- ( { ( A + 1 ) } C_ Prime <-> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } ) |
51 |
49 50
|
sylib |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } ) |
52 |
51
|
uneq2d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) |
53 |
47 52
|
eqtrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) |
54 |
|
fzfid |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) e. Fin ) |
55 |
|
inss1 |
|- ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) ) |
56 |
|
ssfi |
|- ( ( ( 2 ... ( A + 1 ) ) e. Fin /\ ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) ) ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) e. Fin ) |
57 |
54 55 56
|
sylancl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) e. Fin ) |
58 |
|
simpr |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) |
59 |
58
|
elin2d |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. Prime ) |
60 |
|
prmnn |
|- ( p e. Prime -> p e. NN ) |
61 |
59 60
|
syl |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. NN ) |
62 |
61
|
nnrpd |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. RR+ ) |
63 |
62
|
relogcld |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( log ` p ) e. RR ) |
64 |
63
|
recnd |
|- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( log ` p ) e. CC ) |
65 |
26 53 57 64
|
fsumsplit |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) = ( sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) + sum_ p e. { ( A + 1 ) } ( log ` p ) ) ) |
66 |
|
chtval |
|- ( A e. RR -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) ) |
67 |
17 66
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) ) |
68 |
|
ppisval |
|- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) |
69 |
17 68
|
syl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) |
70 |
|
flid |
|- ( A e. ZZ -> ( |_ ` A ) = A ) |
71 |
70
|
adantr |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( |_ ` A ) = A ) |
72 |
71
|
oveq2d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` A ) ) = ( 2 ... A ) ) |
73 |
72
|
ineq1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) ) |
74 |
69 73
|
eqtrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) ) |
75 |
74
|
sumeq1d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) ) |
76 |
67 75
|
eqtr2d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) = ( theta ` A ) ) |
77 |
|
prmnn |
|- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. NN ) |
78 |
77
|
adantl |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. NN ) |
79 |
78
|
nnrpd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. RR+ ) |
80 |
79
|
relogcld |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. RR ) |
81 |
80
|
recnd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. CC ) |
82 |
|
fveq2 |
|- ( p = ( A + 1 ) -> ( log ` p ) = ( log ` ( A + 1 ) ) ) |
83 |
82
|
sumsn |
|- ( ( ( A + 1 ) e. NN /\ ( log ` ( A + 1 ) ) e. CC ) -> sum_ p e. { ( A + 1 ) } ( log ` p ) = ( log ` ( A + 1 ) ) ) |
84 |
78 81 83
|
syl2anc |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. { ( A + 1 ) } ( log ` p ) = ( log ` ( A + 1 ) ) ) |
85 |
76 84
|
oveq12d |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) + sum_ p e. { ( A + 1 ) } ( log ` p ) ) = ( ( theta ` A ) + ( log ` ( A + 1 ) ) ) ) |
86 |
15 65 85
|
3eqtrd |
|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = ( ( theta ` A ) + ( log ` ( A + 1 ) ) ) ) |