Step |
Hyp |
Ref |
Expression |
1 |
|
efcl |
|- ( A e. CC -> ( exp ` A ) e. CC ) |
2 |
|
efcl |
|- ( B e. CC -> ( exp ` B ) e. CC ) |
3 |
|
efne0 |
|- ( B e. CC -> ( exp ` B ) =/= 0 ) |
4 |
|
divrec |
|- ( ( ( exp ` A ) e. CC /\ ( exp ` B ) e. CC /\ ( exp ` B ) =/= 0 ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) ) |
5 |
1 2 3 4
|
syl3an |
|- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) ) |
6 |
5
|
3anidm23 |
|- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) ) |
7 |
|
efcan |
|- ( B e. CC -> ( ( exp ` B ) x. ( exp ` -u B ) ) = 1 ) |
8 |
7
|
eqcomd |
|- ( B e. CC -> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) ) |
9 |
|
negcl |
|- ( B e. CC -> -u B e. CC ) |
10 |
|
efcl |
|- ( -u B e. CC -> ( exp ` -u B ) e. CC ) |
11 |
9 10
|
syl |
|- ( B e. CC -> ( exp ` -u B ) e. CC ) |
12 |
|
ax-1cn |
|- 1 e. CC |
13 |
|
divmul2 |
|- ( ( 1 e. CC /\ ( exp ` -u B ) e. CC /\ ( ( exp ` B ) e. CC /\ ( exp ` B ) =/= 0 ) ) -> ( ( 1 / ( exp ` B ) ) = ( exp ` -u B ) <-> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) ) ) |
14 |
12 13
|
mp3an1 |
|- ( ( ( exp ` -u B ) e. CC /\ ( ( exp ` B ) e. CC /\ ( exp ` B ) =/= 0 ) ) -> ( ( 1 / ( exp ` B ) ) = ( exp ` -u B ) <-> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) ) ) |
15 |
11 2 3 14
|
syl12anc |
|- ( B e. CC -> ( ( 1 / ( exp ` B ) ) = ( exp ` -u B ) <-> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) ) ) |
16 |
8 15
|
mpbird |
|- ( B e. CC -> ( 1 / ( exp ` B ) ) = ( exp ` -u B ) ) |
17 |
16
|
oveq2d |
|- ( B e. CC -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) ) |
18 |
17
|
adantl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) ) |
19 |
|
efadd |
|- ( ( A e. CC /\ -u B e. CC ) -> ( exp ` ( A + -u B ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) ) |
20 |
9 19
|
sylan2 |
|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A + -u B ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) ) |
21 |
18 20
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( exp ` ( A + -u B ) ) ) |
22 |
|
negsub |
|- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) ) |
23 |
22
|
fveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A + -u B ) ) = ( exp ` ( A - B ) ) ) |
24 |
6 21 23
|
3eqtrrd |
|- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) ) |