Step |
Hyp |
Ref |
Expression |
1 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
2 |
1
|
fveq2i |
|- ( theta ` 3 ) = ( theta ` ( 2 + 1 ) ) |
3 |
|
2z |
|- 2 e. ZZ |
4 |
|
3prm |
|- 3 e. Prime |
5 |
1 4
|
eqeltrri |
|- ( 2 + 1 ) e. Prime |
6 |
|
chtprm |
|- ( ( 2 e. ZZ /\ ( 2 + 1 ) e. Prime ) -> ( theta ` ( 2 + 1 ) ) = ( ( theta ` 2 ) + ( log ` ( 2 + 1 ) ) ) ) |
7 |
3 5 6
|
mp2an |
|- ( theta ` ( 2 + 1 ) ) = ( ( theta ` 2 ) + ( log ` ( 2 + 1 ) ) ) |
8 |
|
2rp |
|- 2 e. RR+ |
9 |
|
3rp |
|- 3 e. RR+ |
10 |
|
relogmul |
|- ( ( 2 e. RR+ /\ 3 e. RR+ ) -> ( log ` ( 2 x. 3 ) ) = ( ( log ` 2 ) + ( log ` 3 ) ) ) |
11 |
8 9 10
|
mp2an |
|- ( log ` ( 2 x. 3 ) ) = ( ( log ` 2 ) + ( log ` 3 ) ) |
12 |
|
3cn |
|- 3 e. CC |
13 |
|
2cn |
|- 2 e. CC |
14 |
|
3t2e6 |
|- ( 3 x. 2 ) = 6 |
15 |
12 13 14
|
mulcomli |
|- ( 2 x. 3 ) = 6 |
16 |
15
|
fveq2i |
|- ( log ` ( 2 x. 3 ) ) = ( log ` 6 ) |
17 |
|
cht2 |
|- ( theta ` 2 ) = ( log ` 2 ) |
18 |
17
|
eqcomi |
|- ( log ` 2 ) = ( theta ` 2 ) |
19 |
1
|
fveq2i |
|- ( log ` 3 ) = ( log ` ( 2 + 1 ) ) |
20 |
18 19
|
oveq12i |
|- ( ( log ` 2 ) + ( log ` 3 ) ) = ( ( theta ` 2 ) + ( log ` ( 2 + 1 ) ) ) |
21 |
11 16 20
|
3eqtr3ri |
|- ( ( theta ` 2 ) + ( log ` ( 2 + 1 ) ) ) = ( log ` 6 ) |
22 |
2 7 21
|
3eqtri |
|- ( theta ` 3 ) = ( log ` 6 ) |