Theorem dmdcan2d

Description: Cancellation law for division and multiplication. (Contributed by David Moews, 28-Feb-2017)

	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	dmdcan2d	|- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( 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	1 2 4	divcld	|- ( ph -> ( A / B ) e. CC )
7	2 3 5	divcld	|- ( ph -> ( B / C ) e. CC )
8	6 7	mulcomd	|- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( ( B / C ) x. ( A / B ) ) )
9	1 2 3 4 5	dmdcand	|- ( ph -> ( ( B / C ) x. ( A / B ) ) = ( A / C ) )
10	8 9	eqtrd	|- ( ph -> ( ( A / B ) x. ( B / C ) ) = ( A / C ) )