Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
|- 1 e. RR |
2 |
|
chtval |
|- ( 1 e. RR -> ( theta ` 1 ) = sum_ p e. ( ( 0 [,] 1 ) i^i Prime ) ( log ` p ) ) |
3 |
1 2
|
ax-mp |
|- ( theta ` 1 ) = sum_ p e. ( ( 0 [,] 1 ) i^i Prime ) ( log ` p ) |
4 |
|
ppisval |
|- ( 1 e. RR -> ( ( 0 [,] 1 ) i^i Prime ) = ( ( 2 ... ( |_ ` 1 ) ) i^i Prime ) ) |
5 |
1 4
|
ax-mp |
|- ( ( 0 [,] 1 ) i^i Prime ) = ( ( 2 ... ( |_ ` 1 ) ) i^i Prime ) |
6 |
|
1z |
|- 1 e. ZZ |
7 |
|
flid |
|- ( 1 e. ZZ -> ( |_ ` 1 ) = 1 ) |
8 |
6 7
|
ax-mp |
|- ( |_ ` 1 ) = 1 |
9 |
8
|
oveq2i |
|- ( 2 ... ( |_ ` 1 ) ) = ( 2 ... 1 ) |
10 |
|
1lt2 |
|- 1 < 2 |
11 |
|
2z |
|- 2 e. ZZ |
12 |
|
fzn |
|- ( ( 2 e. ZZ /\ 1 e. ZZ ) -> ( 1 < 2 <-> ( 2 ... 1 ) = (/) ) ) |
13 |
11 6 12
|
mp2an |
|- ( 1 < 2 <-> ( 2 ... 1 ) = (/) ) |
14 |
10 13
|
mpbi |
|- ( 2 ... 1 ) = (/) |
15 |
9 14
|
eqtri |
|- ( 2 ... ( |_ ` 1 ) ) = (/) |
16 |
15
|
ineq1i |
|- ( ( 2 ... ( |_ ` 1 ) ) i^i Prime ) = ( (/) i^i Prime ) |
17 |
|
0in |
|- ( (/) i^i Prime ) = (/) |
18 |
5 16 17
|
3eqtri |
|- ( ( 0 [,] 1 ) i^i Prime ) = (/) |
19 |
18
|
sumeq1i |
|- sum_ p e. ( ( 0 [,] 1 ) i^i Prime ) ( log ` p ) = sum_ p e. (/) ( log ` p ) |
20 |
|
sum0 |
|- sum_ p e. (/) ( log ` p ) = 0 |
21 |
3 19 20
|
3eqtri |
|- ( theta ` 1 ) = 0 |