Theorem mvllmuld

Description: Move the left term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018)

	Ref	Expression
Hypotheses	mvllmuld.1	|- ( ph -> A e. CC )
	mvllmuld.2	|- ( ph -> B e. CC )
	mvllmuld.3	|- ( ph -> A =/= 0 )
	mvllmuld.4	|- ( ph -> ( A x. B ) = C )
Assertion	mvllmuld	|- ( ph -> B = ( C / A ) )

Proof

Step	Hyp	Ref	Expression
1		mvllmuld.1	|- ( ph -> A e. CC )
2		mvllmuld.2	|- ( ph -> B e. CC )
3		mvllmuld.3	|- ( ph -> A =/= 0 )
4		mvllmuld.4	|- ( ph -> ( A x. B ) = C )
5	2 1 3	divcan4d	|- ( ph -> ( ( B x. A ) / A ) = B )
6	1 2	mulcomd	|- ( ph -> ( A x. B ) = ( B x. A ) )
7	6 4	eqtr3d	|- ( ph -> ( B x. A ) = C )
8	7	oveq1d	|- ( ph -> ( ( B x. A ) / A ) = ( C / A ) )
9	5 8	eqtr3d	|- ( ph -> B = ( C / A ) )