Step |
Hyp |
Ref |
Expression |
1 |
|
logbval |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) ) |
2 |
1
|
oveq2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B ^c ( B logb X ) ) = ( B ^c ( ( log ` X ) / ( log ` B ) ) ) ) |
3 |
|
eldifi |
|- ( B e. ( CC \ { 0 , 1 } ) -> B e. CC ) |
4 |
3
|
adantr |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> B e. CC ) |
5 |
|
eldif |
|- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ -. B e. { 0 , 1 } ) ) |
6 |
|
c0ex |
|- 0 e. _V |
7 |
6
|
prid1 |
|- 0 e. { 0 , 1 } |
8 |
|
eleq1 |
|- ( B = 0 -> ( B e. { 0 , 1 } <-> 0 e. { 0 , 1 } ) ) |
9 |
7 8
|
mpbiri |
|- ( B = 0 -> B e. { 0 , 1 } ) |
10 |
9
|
necon3bi |
|- ( -. B e. { 0 , 1 } -> B =/= 0 ) |
11 |
10
|
adantl |
|- ( ( B e. CC /\ -. B e. { 0 , 1 } ) -> B =/= 0 ) |
12 |
5 11
|
sylbi |
|- ( B e. ( CC \ { 0 , 1 } ) -> B =/= 0 ) |
13 |
12
|
adantr |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> B =/= 0 ) |
14 |
|
eldif |
|- ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ -. X e. { 0 } ) ) |
15 |
6
|
snid |
|- 0 e. { 0 } |
16 |
|
eleq1 |
|- ( X = 0 -> ( X e. { 0 } <-> 0 e. { 0 } ) ) |
17 |
15 16
|
mpbiri |
|- ( X = 0 -> X e. { 0 } ) |
18 |
17
|
necon3bi |
|- ( -. X e. { 0 } -> X =/= 0 ) |
19 |
18
|
anim2i |
|- ( ( X e. CC /\ -. X e. { 0 } ) -> ( X e. CC /\ X =/= 0 ) ) |
20 |
14 19
|
sylbi |
|- ( X e. ( CC \ { 0 } ) -> ( X e. CC /\ X =/= 0 ) ) |
21 |
|
logcl |
|- ( ( X e. CC /\ X =/= 0 ) -> ( log ` X ) e. CC ) |
22 |
20 21
|
syl |
|- ( X e. ( CC \ { 0 } ) -> ( log ` X ) e. CC ) |
23 |
22
|
adantl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( log ` X ) e. CC ) |
24 |
10
|
anim2i |
|- ( ( B e. CC /\ -. B e. { 0 , 1 } ) -> ( B e. CC /\ B =/= 0 ) ) |
25 |
5 24
|
sylbi |
|- ( B e. ( CC \ { 0 , 1 } ) -> ( B e. CC /\ B =/= 0 ) ) |
26 |
|
logcl |
|- ( ( B e. CC /\ B =/= 0 ) -> ( log ` B ) e. CC ) |
27 |
25 26
|
syl |
|- ( B e. ( CC \ { 0 , 1 } ) -> ( log ` B ) e. CC ) |
28 |
27
|
adantr |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( log ` B ) e. CC ) |
29 |
|
eldifpr |
|- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) ) |
30 |
29
|
biimpi |
|- ( B e. ( CC \ { 0 , 1 } ) -> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) ) |
31 |
30
|
adantr |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) ) |
32 |
|
logccne0 |
|- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> ( log ` B ) =/= 0 ) |
33 |
31 32
|
syl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( log ` B ) =/= 0 ) |
34 |
23 28 33
|
divcld |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( ( log ` X ) / ( log ` B ) ) e. CC ) |
35 |
4 13 34
|
cxpefd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B ^c ( ( log ` X ) / ( log ` B ) ) ) = ( exp ` ( ( ( log ` X ) / ( log ` B ) ) x. ( log ` B ) ) ) ) |
36 |
|
eldifsn |
|- ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) ) |
37 |
36 21
|
sylbi |
|- ( X e. ( CC \ { 0 } ) -> ( log ` X ) e. CC ) |
38 |
37
|
adantl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( log ` X ) e. CC ) |
39 |
29 32
|
sylbi |
|- ( B e. ( CC \ { 0 , 1 } ) -> ( log ` B ) =/= 0 ) |
40 |
39
|
adantr |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( log ` B ) =/= 0 ) |
41 |
38 28 40
|
divcan1d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( ( ( log ` X ) / ( log ` B ) ) x. ( log ` B ) ) = ( log ` X ) ) |
42 |
41
|
fveq2d |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( exp ` ( ( ( log ` X ) / ( log ` B ) ) x. ( log ` B ) ) ) = ( exp ` ( log ` X ) ) ) |
43 |
|
eflog |
|- ( ( X e. CC /\ X =/= 0 ) -> ( exp ` ( log ` X ) ) = X ) |
44 |
36 43
|
sylbi |
|- ( X e. ( CC \ { 0 } ) -> ( exp ` ( log ` X ) ) = X ) |
45 |
44
|
adantl |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( exp ` ( log ` X ) ) = X ) |
46 |
42 45
|
eqtrd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( exp ` ( ( ( log ` X ) / ( log ` B ) ) x. ( log ` B ) ) ) = X ) |
47 |
2 35 46
|
3eqtrd |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B ^c ( B logb X ) ) = X ) |