Theorem joinlmuladdmuld

Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019)

	Ref	Expression
Hypotheses	joinlmuladdmuld.1	|- ( ph -> A e. CC )
	joinlmuladdmuld.2	|- ( ph -> B e. CC )
	joinlmuladdmuld.3	|- ( ph -> C e. CC )
	joinlmuladdmuld.4	|- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D )
Assertion	joinlmuladdmuld	|- ( ph -> ( ( A + C ) x. B ) = D )

Proof

Step	Hyp	Ref	Expression
1		joinlmuladdmuld.1	|- ( ph -> A e. CC )
2		joinlmuladdmuld.2	|- ( ph -> B e. CC )
3		joinlmuladdmuld.3	|- ( ph -> C e. CC )
4		joinlmuladdmuld.4	|- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D )
5	1 3 2	adddird	|- ( ph -> ( ( A + C ) x. B ) = ( ( A x. B ) + ( C x. B ) ) )
6	5 4	eqtrd	|- ( ph -> ( ( A + C ) x. B ) = D )