Theorem int-mulcomd

Description: MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-mulcomd.1	|- ( ph -> B e. RR )
	int-mulcomd.2	|- ( ph -> C e. RR )
	int-mulcomd.3	|- ( ph -> A = B )
Assertion	int-mulcomd	|- ( ph -> ( B x. C ) = ( C x. A ) )

Proof

Step	Hyp	Ref	Expression
1		int-mulcomd.1	|- ( ph -> B e. RR )
2		int-mulcomd.2	|- ( ph -> C e. RR )
3		int-mulcomd.3	|- ( ph -> A = B )
4	1	recnd	|- ( ph -> B e. CC )
5	2	recnd	|- ( ph -> C e. CC )
6	4 5	mulcomd	|- ( ph -> ( B x. C ) = ( C x. B ) )
7	3	eqcomd	|- ( ph -> B = A )
8	7	oveq2d	|- ( ph -> ( C x. B ) = ( C x. A ) )
9	6 8	eqtrd	|- ( ph -> ( B x. C ) = ( C x. A ) )