Description: A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 . (Contributed by David Moews, 28-Feb-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | div1d.1 | |- ( ph -> A e. CC ) |
|
divcld.2 | |- ( ph -> B e. CC ) |
||
divcld.3 | |- ( ph -> B =/= 0 ) |
||
Assertion | diveq0ad | |- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | |- ( ph -> A e. CC ) |
|
2 | divcld.2 | |- ( ph -> B e. CC ) |
|
3 | divcld.3 | |- ( ph -> B =/= 0 ) |
|
4 | diveq0 | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = 0 ) ) |
|
5 | 1 2 3 4 | syl3anc | |- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) ) |