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 ) )