Step |
Hyp |
Ref |
Expression |
1 |
|
rpcn |
|- ( C e. RR+ -> C e. CC ) |
2 |
1
|
adantr |
|- ( ( C e. RR+ /\ N e. ZZ ) -> C e. CC ) |
3 |
|
rpne0 |
|- ( C e. RR+ -> C =/= 0 ) |
4 |
3
|
adantr |
|- ( ( C e. RR+ /\ N e. ZZ ) -> C =/= 0 ) |
5 |
|
simpr |
|- ( ( C e. RR+ /\ N e. ZZ ) -> N e. ZZ ) |
6 |
2 4 5
|
cxpexpzd |
|- ( ( C e. RR+ /\ N e. ZZ ) -> ( C ^c N ) = ( C ^ N ) ) |
7 |
6
|
3adant1 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( C ^c N ) = ( C ^ N ) ) |
8 |
7
|
eqcomd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( C ^ N ) = ( C ^c N ) ) |
9 |
8
|
oveq2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^ N ) ) = ( B logb ( C ^c N ) ) ) |
10 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
11 |
|
relogbreexp |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. RR ) -> ( B logb ( C ^c N ) ) = ( N x. ( B logb C ) ) ) |
12 |
10 11
|
syl3an3 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^c N ) ) = ( N x. ( B logb C ) ) ) |
13 |
9 12
|
eqtrd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^ N ) ) = ( N x. ( B logb C ) ) ) |