Theorem divmuld

Description: Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	|- ( ph -> A e. CC )
	divcld.2	|- ( ph -> B e. CC )
	divmuld.3	|- ( ph -> C e. CC )
	divmuld.4	|- ( ph -> B =/= 0 )
Assertion	divmuld	|- ( ph -> ( ( A / B ) = C <-> ( B x. C ) = A ) )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	|- ( ph -> A e. CC )
2		divcld.2	|- ( ph -> B e. CC )
3		divmuld.3	|- ( ph -> C e. CC )
4		divmuld.4	|- ( ph -> B =/= 0 )
5		divmul	|- ( ( A e. CC /\ C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A / B ) = C <-> ( B x. C ) = A ) )
6	1 3 2 4 5	syl112anc	|- ( ph -> ( ( A / B ) = C <-> ( B x. C ) = A ) )