Step |
Hyp |
Ref |
Expression |
1 |
|
0red |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 e. RR ) |
2 |
|
0le0 |
|- 0 <_ 0 |
3 |
2
|
a1i |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 <_ 0 ) |
4 |
|
simplr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> B e. RR ) |
5 |
|
recxpcl |
|- ( ( 0 e. RR /\ 0 <_ 0 /\ B e. RR ) -> ( 0 ^c B ) e. RR ) |
6 |
1 3 4 5
|
syl3anc |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( 0 ^c B ) e. RR ) |
7 |
|
cxpge0 |
|- ( ( 0 e. RR /\ 0 <_ 0 /\ B e. RR ) -> 0 <_ ( 0 ^c B ) ) |
8 |
1 3 4 7
|
syl3anc |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> 0 <_ ( 0 ^c B ) ) |
9 |
6 8
|
absidd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( 0 ^c B ) ) = ( 0 ^c B ) ) |
10 |
|
simpr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> A = 0 ) |
11 |
10
|
oveq1d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( A ^c B ) = ( 0 ^c B ) ) |
12 |
11
|
fveq2d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( A ^c B ) ) = ( abs ` ( 0 ^c B ) ) ) |
13 |
10
|
abs00bd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` A ) = 0 ) |
14 |
13
|
oveq1d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( ( abs ` A ) ^c B ) = ( 0 ^c B ) ) |
15 |
9 12 14
|
3eqtr4d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A = 0 ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) ) |
16 |
|
simplr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> B e. RR ) |
17 |
16
|
recnd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> B e. CC ) |
18 |
|
logcl |
|- ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC ) |
19 |
18
|
adantlr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( log ` A ) e. CC ) |
20 |
17 19
|
mulcld |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( B x. ( log ` A ) ) e. CC ) |
21 |
|
absef |
|- ( ( B x. ( log ` A ) ) e. CC -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) ) |
22 |
20 21
|
syl |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) ) |
23 |
16 19
|
remul2d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( B x. ( log ` A ) ) ) = ( B x. ( Re ` ( log ` A ) ) ) ) |
24 |
|
relog |
|- ( ( A e. CC /\ A =/= 0 ) -> ( Re ` ( log ` A ) ) = ( log ` ( abs ` A ) ) ) |
25 |
24
|
adantlr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( log ` A ) ) = ( log ` ( abs ` A ) ) ) |
26 |
25
|
oveq2d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( B x. ( Re ` ( log ` A ) ) ) = ( B x. ( log ` ( abs ` A ) ) ) ) |
27 |
23 26
|
eqtrd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( Re ` ( B x. ( log ` A ) ) ) = ( B x. ( log ` ( abs ` A ) ) ) ) |
28 |
27
|
fveq2d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( exp ` ( Re ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) ) |
29 |
22 28
|
eqtrd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) ) |
30 |
|
simpll |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> A e. CC ) |
31 |
|
simpr |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> A =/= 0 ) |
32 |
|
cxpef |
|- ( ( A e. CC /\ A =/= 0 /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) ) |
33 |
30 31 17 32
|
syl3anc |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) ) |
34 |
33
|
fveq2d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( A ^c B ) ) = ( abs ` ( exp ` ( B x. ( log ` A ) ) ) ) ) |
35 |
30
|
abscld |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) e. RR ) |
36 |
35
|
recnd |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) e. CC ) |
37 |
|
abs00 |
|- ( A e. CC -> ( ( abs ` A ) = 0 <-> A = 0 ) ) |
38 |
37
|
adantr |
|- ( ( A e. CC /\ B e. RR ) -> ( ( abs ` A ) = 0 <-> A = 0 ) ) |
39 |
38
|
necon3bid |
|- ( ( A e. CC /\ B e. RR ) -> ( ( abs ` A ) =/= 0 <-> A =/= 0 ) ) |
40 |
39
|
biimpar |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` A ) =/= 0 ) |
41 |
|
cxpef |
|- ( ( ( abs ` A ) e. CC /\ ( abs ` A ) =/= 0 /\ B e. CC ) -> ( ( abs ` A ) ^c B ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) ) |
42 |
36 40 17 41
|
syl3anc |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( ( abs ` A ) ^c B ) = ( exp ` ( B x. ( log ` ( abs ` A ) ) ) ) ) |
43 |
29 34 42
|
3eqtr4d |
|- ( ( ( A e. CC /\ B e. RR ) /\ A =/= 0 ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) ) |
44 |
15 43
|
pm2.61dane |
|- ( ( A e. CC /\ B e. RR ) -> ( abs ` ( A ^c B ) ) = ( ( abs ` A ) ^c B ) ) |