Theorem div13d

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 )
	divmuld.4	|- ( ph -> B =/= 0 )
Assertion	div13d	|- ( ph -> ( ( A / B ) x. C ) = ( ( C / B ) x. 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		div13	|- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )
6	1 2 4 3 5	syl121anc	|- ( ph -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )