Theorem div12d

Description: A commutative/associative law for division. (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 )
	divassd.4	|- ( ph -> C =/= 0 )
Assertion	div12d	|- ( ph -> ( A x. ( B / C ) ) = ( B x. ( 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		divassd.4	|- ( ph -> C =/= 0 )
5		div12	|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) )
6	1 2 3 4 5	syl112anc	|- ( ph -> ( A x. ( B / C ) ) = ( B x. ( A / C ) ) )