Step |
Hyp |
Ref |
Expression |
1 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
2 |
1
|
fveq2i |
|- ( theta ` 2 ) = ( theta ` ( 1 + 1 ) ) |
3 |
|
1z |
|- 1 e. ZZ |
4 |
|
2prm |
|- 2 e. Prime |
5 |
1 4
|
eqeltrri |
|- ( 1 + 1 ) e. Prime |
6 |
|
chtprm |
|- ( ( 1 e. ZZ /\ ( 1 + 1 ) e. Prime ) -> ( theta ` ( 1 + 1 ) ) = ( ( theta ` 1 ) + ( log ` ( 1 + 1 ) ) ) ) |
7 |
3 5 6
|
mp2an |
|- ( theta ` ( 1 + 1 ) ) = ( ( theta ` 1 ) + ( log ` ( 1 + 1 ) ) ) |
8 |
|
cht1 |
|- ( theta ` 1 ) = 0 |
9 |
8
|
eqcomi |
|- 0 = ( theta ` 1 ) |
10 |
1
|
fveq2i |
|- ( log ` 2 ) = ( log ` ( 1 + 1 ) ) |
11 |
9 10
|
oveq12i |
|- ( 0 + ( log ` 2 ) ) = ( ( theta ` 1 ) + ( log ` ( 1 + 1 ) ) ) |
12 |
|
2rp |
|- 2 e. RR+ |
13 |
|
relogcl |
|- ( 2 e. RR+ -> ( log ` 2 ) e. RR ) |
14 |
12 13
|
ax-mp |
|- ( log ` 2 ) e. RR |
15 |
14
|
recni |
|- ( log ` 2 ) e. CC |
16 |
15
|
addid2i |
|- ( 0 + ( log ` 2 ) ) = ( log ` 2 ) |
17 |
11 16
|
eqtr3i |
|- ( ( theta ` 1 ) + ( log ` ( 1 + 1 ) ) ) = ( log ` 2 ) |
18 |
2 7 17
|
3eqtri |
|- ( theta ` 2 ) = ( log ` 2 ) |