Theorem dmdcand

Description: Cancellation law for 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 )
	divdiv23d.5	|- ( ph -> C =/= 0 )
Assertion	dmdcand	|- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) )

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		divdiv23d.5	|- ( ph -> C =/= 0 )
6		dmdcan	|- ( ( ( B e. CC /\ B =/= 0 ) /\ ( C e. CC /\ C =/= 0 ) /\ A e. CC ) -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) )
7	2 4 3 5 1 6	syl221anc	|- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) )