Metamath Proof Explorer


Theorem diveq0ad

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 ) )

Proof

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 ) )