Step |
Hyp |
Ref |
Expression |
1 |
|
ancom |
|- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) <-> ( A e. CC /\ ( B e. RR /\ B =/= 0 ) ) ) |
2 |
|
3anass |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) <-> ( A e. CC /\ ( B e. RR /\ B =/= 0 ) ) ) |
3 |
1 2
|
bitr4i |
|- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) <-> ( A e. CC /\ B e. RR /\ B =/= 0 ) ) |
4 |
|
rereccl |
|- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR ) |
5 |
4
|
anim1i |
|- ( ( ( B e. RR /\ B =/= 0 ) /\ A e. CC ) -> ( ( 1 / B ) e. RR /\ A e. CC ) ) |
6 |
3 5
|
sylbir |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( ( 1 / B ) e. RR /\ A e. CC ) ) |
7 |
|
remul2 |
|- ( ( ( 1 / B ) e. RR /\ A e. CC ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) ) |
8 |
6 7
|
syl |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( ( 1 / B ) x. A ) ) = ( ( 1 / B ) x. ( Re ` A ) ) ) |
9 |
|
recn |
|- ( B e. RR -> B e. CC ) |
10 |
|
divrec2 |
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) = ( ( 1 / B ) x. A ) ) |
11 |
10
|
fveq2d |
|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) ) |
12 |
9 11
|
syl3an2 |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( Re ` ( ( 1 / B ) x. A ) ) ) |
13 |
|
recl |
|- ( A e. CC -> ( Re ` A ) e. RR ) |
14 |
13
|
recnd |
|- ( A e. CC -> ( Re ` A ) e. CC ) |
15 |
14
|
3ad2ant1 |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` A ) e. CC ) |
16 |
9
|
3ad2ant2 |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> B e. CC ) |
17 |
|
simp3 |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> B =/= 0 ) |
18 |
15 16 17
|
divrec2d |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( ( Re ` A ) / B ) = ( ( 1 / B ) x. ( Re ` A ) ) ) |
19 |
8 12 18
|
3eqtr4d |
|- ( ( A e. CC /\ B e. RR /\ B =/= 0 ) -> ( Re ` ( A / B ) ) = ( ( Re ` A ) / B ) ) |