Theorem divcan5rd

Description: Cancellation of common factor in a ratio. (Contributed by Mario Carneiro, 1-Jan-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	divcan5rd	|- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( A / B ) )

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 3	mulcomd	|- ( ph -> ( A x. C ) = ( C x. A ) )
7	2 3	mulcomd	|- ( ph -> ( B x. C ) = ( C x. B ) )
8	6 7	oveq12d	|- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( ( C x. A ) / ( C x. B ) ) )
9	1 2 3 4 5	divcan5d	|- ( ph -> ( ( C x. A ) / ( C x. B ) ) = ( A / B ) )
10	8 9	eqtrd	|- ( ph -> ( ( A x. C ) / ( B x. C ) ) = ( A / B ) )