Theorem int-mulassocd

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

	Ref	Expression
Hypotheses	int-mulassocd.1	|- ( ph -> B e. RR )
	int-mulassocd.2	|- ( ph -> C e. RR )
	int-mulassocd.3	|- ( ph -> D e. RR )
	int-mulassocd.4	|- ( ph -> A = B )
Assertion	int-mulassocd	|- ( ph -> ( B x. ( C x. D ) ) = ( ( A x. C ) x. D ) )

Proof

Step	Hyp	Ref	Expression
1		int-mulassocd.1	|- ( ph -> B e. RR )
2		int-mulassocd.2	|- ( ph -> C e. RR )
3		int-mulassocd.3	|- ( ph -> D e. RR )
4		int-mulassocd.4	|- ( ph -> A = B )
5	1	recnd	|- ( ph -> B e. CC )
6	2	recnd	|- ( ph -> C e. CC )
7	3	recnd	|- ( ph -> D e. CC )
8	5 6 7	mulassd	|- ( ph -> ( ( B x. C ) x. D ) = ( B x. ( C x. D ) ) )
9	4	eqcomd	|- ( ph -> B = A )
10	9	oveq1d	|- ( ph -> ( B x. C ) = ( A x. C ) )
11	10	oveq1d	|- ( ph -> ( ( B x. C ) x. D ) = ( ( A x. C ) x. D ) )
12	8 11	eqtr3d	|- ( ph -> ( B x. ( C x. D ) ) = ( ( A x. C ) x. D ) )