Theorem diveq0d

Description: A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	|- ( ph -> A e. CC )
	divcld.2	|- ( ph -> B e. CC )
	divcld.3	|- ( ph -> B =/= 0 )
	diveq0d.4	|- ( ph -> ( A / B ) = 0 )
Assertion	diveq0d	|- ( ph -> 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		diveq0d.4	|- ( ph -> ( A / B ) = 0 )
5		diveq0	|- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = 0 ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( ( A / B ) = 0 <-> A = 0 ) )
7	4 6	mpbid	|- ( ph -> A = 0 )