Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( A e. RR+ /\ C e. RR+ /\ E e. RR ) -> A e. RR+ ) |
2 |
|
rpcxpcl |
|- ( ( C e. RR+ /\ E e. RR ) -> ( C ^c E ) e. RR+ ) |
3 |
2
|
3adant1 |
|- ( ( A e. RR+ /\ C e. RR+ /\ E e. RR ) -> ( C ^c E ) e. RR+ ) |
4 |
1 3
|
jca |
|- ( ( A e. RR+ /\ C e. RR+ /\ E e. RR ) -> ( A e. RR+ /\ ( C ^c E ) e. RR+ ) ) |
5 |
|
relogbmul |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ ( C ^c E ) e. RR+ ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( B logb ( C ^c E ) ) ) ) |
6 |
4 5
|
sylan2 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( B logb ( C ^c E ) ) ) ) |
7 |
|
relogbreexp |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) ) |
8 |
7
|
3adant3r1 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) ) |
9 |
8
|
oveq2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( ( B logb A ) + ( B logb ( C ^c E ) ) ) = ( ( B logb A ) + ( E x. ( B logb C ) ) ) ) |
10 |
6 9
|
eqtrd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( E x. ( B logb C ) ) ) ) |