Description: Imaginary part of a division. Related to remul2 . (Contributed by Mario Carneiro, 29-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | crred.1 | |- ( ph -> A e. RR ) |
|
remul2d.2 | |- ( ph -> B e. CC ) |
||
redivd.2 | |- ( ph -> A =/= 0 ) |
||
Assertion | imdivd | |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crred.1 | |- ( ph -> A e. RR ) |
|
2 | remul2d.2 | |- ( ph -> B e. CC ) |
|
3 | redivd.2 | |- ( ph -> A =/= 0 ) |
|
4 | imdiv | |- ( ( B e. CC /\ A e. RR /\ A =/= 0 ) -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) ) |
|
5 | 2 1 3 4 | syl3anc | |- ( ph -> ( Im ` ( B / A ) ) = ( ( Im ` B ) / A ) ) |