Step |
Hyp |
Ref |
Expression |
1 |
|
neg1rr |
|- -u 1 e. RR |
2 |
|
relogbmulexp |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ /\ -u 1 e. RR ) ) -> ( B logb ( A x. ( C ^c -u 1 ) ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) ) |
3 |
1 2
|
mp3anr3 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. ( C ^c -u 1 ) ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) ) |
4 |
|
rpcn |
|- ( A e. RR+ -> A e. CC ) |
5 |
4
|
adantr |
|- ( ( A e. RR+ /\ C e. RR+ ) -> A e. CC ) |
6 |
|
rpcn |
|- ( C e. RR+ -> C e. CC ) |
7 |
6
|
adantl |
|- ( ( A e. RR+ /\ C e. RR+ ) -> C e. CC ) |
8 |
|
rpne0 |
|- ( C e. RR+ -> C =/= 0 ) |
9 |
8
|
adantl |
|- ( ( A e. RR+ /\ C e. RR+ ) -> C =/= 0 ) |
10 |
5 7 9
|
divrecd |
|- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( 1 / C ) ) ) |
11 |
|
1cnd |
|- ( C e. RR+ -> 1 e. CC ) |
12 |
6 8 11
|
cxpnegd |
|- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / ( C ^c 1 ) ) ) |
13 |
6
|
cxp1d |
|- ( C e. RR+ -> ( C ^c 1 ) = C ) |
14 |
13
|
oveq2d |
|- ( C e. RR+ -> ( 1 / ( C ^c 1 ) ) = ( 1 / C ) ) |
15 |
12 14
|
eqtrd |
|- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / C ) ) |
16 |
15
|
adantl |
|- ( ( A e. RR+ /\ C e. RR+ ) -> ( C ^c -u 1 ) = ( 1 / C ) ) |
17 |
16
|
oveq2d |
|- ( ( A e. RR+ /\ C e. RR+ ) -> ( A x. ( C ^c -u 1 ) ) = ( A x. ( 1 / C ) ) ) |
18 |
10 17
|
eqtr4d |
|- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) ) |
19 |
18
|
adantl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) ) |
20 |
19
|
oveq2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( B logb ( A x. ( C ^c -u 1 ) ) ) ) |
21 |
|
rpcndif0 |
|- ( C e. RR+ -> C e. ( CC \ { 0 } ) ) |
22 |
21
|
adantl |
|- ( ( A e. RR+ /\ C e. RR+ ) -> C e. ( CC \ { 0 } ) ) |
23 |
|
logbcl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ C e. ( CC \ { 0 } ) ) -> ( B logb C ) e. CC ) |
24 |
22 23
|
sylan2 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) e. CC ) |
25 |
|
mulm1 |
|- ( ( B logb C ) e. CC -> ( -u 1 x. ( B logb C ) ) = -u ( B logb C ) ) |
26 |
25
|
oveq2d |
|- ( ( B logb C ) e. CC -> ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) = ( ( B logb A ) + -u ( B logb C ) ) ) |
27 |
24 26
|
syl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) = ( ( B logb A ) + -u ( B logb C ) ) ) |
28 |
|
rpcndif0 |
|- ( A e. RR+ -> A e. ( CC \ { 0 } ) ) |
29 |
28
|
adantr |
|- ( ( A e. RR+ /\ C e. RR+ ) -> A e. ( CC \ { 0 } ) ) |
30 |
|
logbcl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ A e. ( CC \ { 0 } ) ) -> ( B logb A ) e. CC ) |
31 |
29 30
|
sylan2 |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) e. CC ) |
32 |
31 24
|
negsubd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) + -u ( B logb C ) ) = ( ( B logb A ) - ( B logb C ) ) ) |
33 |
27 32
|
eqtr2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) - ( B logb C ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) ) |
34 |
3 20 33
|
3eqtr4d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) ) |