Step |
Hyp |
Ref |
Expression |
1 |
|
flidm |
|- ( A e. RR -> ( |_ ` ( |_ ` A ) ) = ( |_ ` A ) ) |
2 |
1
|
oveq2d |
|- ( A e. RR -> ( 2 ... ( |_ ` ( |_ ` A ) ) ) = ( 2 ... ( |_ ` A ) ) ) |
3 |
2
|
ineq1d |
|- ( A e. RR -> ( ( 2 ... ( |_ ` ( |_ ` A ) ) ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) |
4 |
|
reflcl |
|- ( A e. RR -> ( |_ ` A ) e. RR ) |
5 |
|
ppisval |
|- ( ( |_ ` A ) e. RR -> ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( |_ ` A ) ) ) i^i Prime ) ) |
6 |
4 5
|
syl |
|- ( A e. RR -> ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( |_ ` A ) ) ) i^i Prime ) ) |
7 |
|
ppisval |
|- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) |
8 |
3 6 7
|
3eqtr4d |
|- ( A e. RR -> ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) = ( ( 0 [,] A ) i^i Prime ) ) |
9 |
8
|
sumeq1d |
|- ( A e. RR -> sum_ p e. ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) ) |
10 |
|
chtval |
|- ( ( |_ ` A ) e. RR -> ( theta ` ( |_ ` A ) ) = sum_ p e. ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) ( log ` p ) ) |
11 |
4 10
|
syl |
|- ( A e. RR -> ( theta ` ( |_ ` A ) ) = sum_ p e. ( ( 0 [,] ( |_ ` A ) ) i^i Prime ) ( log ` p ) ) |
12 |
|
chtval |
|- ( A e. RR -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) ) |
13 |
9 11 12
|
3eqtr4d |
|- ( A e. RR -> ( theta ` ( |_ ` A ) ) = ( theta ` A ) ) |