Theorem rediveq0d

Description: A ratio is zero iff the numerator is zero. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivcan2d.a	|- ( ph -> A e. RR )
	redivcan2d.b	|- ( ph -> B e. RR )
	redivcan2d.z	|- ( ph -> B =/= 0 )
Assertion	rediveq0d	|- ( ph -> ( ( A /R B ) = 0 <-> A = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		redivcan2d.a	|- ( ph -> A e. RR )
2		redivcan2d.b	|- ( ph -> B e. RR )
3		redivcan2d.z	|- ( ph -> B =/= 0 )
4		0red	|- ( ph -> 0 e. RR )
5	1 4 2 3	redivmul2d	|- ( ph -> ( ( A /R B ) = 0 <-> A = ( B x. 0 ) ) )
6		remul01	|- ( B e. RR -> ( B x. 0 ) = 0 )
7	2 6	syl	|- ( ph -> ( B x. 0 ) = 0 )
8	7	eqeq2d	|- ( ph -> ( A = ( B x. 0 ) <-> A = 0 ) )
9	5 8	bitrd	|- ( ph -> ( ( A /R B ) = 0 <-> A = 0 ) )