Theorem submuladdmuld

Description: Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023)

	Ref	Expression
Hypotheses	submuladdmuld.a	|- ( ph -> A e. CC )
	submuladdmuld.b	|- ( ph -> B e. CC )
	submuladdmuld.c	|- ( ph -> C e. CC )
	submuladdmuld.d	|- ( ph -> D e. CC )
Assertion	submuladdmuld	|- ( ph -> ( ( ( A - B ) x. C ) + ( B x. D ) ) = ( ( A x. C ) + ( B x. ( D - C ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		submuladdmuld.a	|- ( ph -> A e. CC )
2		submuladdmuld.b	|- ( ph -> B e. CC )
3		submuladdmuld.c	|- ( ph -> C e. CC )
4		submuladdmuld.d	|- ( ph -> D e. CC )
5	1 2 3	subdird	|- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )
6	5	oveq1d	|- ( ph -> ( ( ( A - B ) x. C ) + ( B x. D ) ) = ( ( ( A x. C ) - ( B x. C ) ) + ( B x. D ) ) )
7	1 3	mulcld	|- ( ph -> ( A x. C ) e. CC )
8	2 3	mulcld	|- ( ph -> ( B x. C ) e. CC )
9	2 4	mulcld	|- ( ph -> ( B x. D ) e. CC )
10	7 8 9	subadd23d	|- ( ph -> ( ( ( A x. C ) - ( B x. C ) ) + ( B x. D ) ) = ( ( A x. C ) + ( ( B x. D ) - ( B x. C ) ) ) )
11	2 4 3	subdid	|- ( ph -> ( B x. ( D - C ) ) = ( ( B x. D ) - ( B x. C ) ) )
12	11	eqcomd	|- ( ph -> ( ( B x. D ) - ( B x. C ) ) = ( B x. ( D - C ) ) )
13	12	oveq2d	|- ( ph -> ( ( A x. C ) + ( ( B x. D ) - ( B x. C ) ) ) = ( ( A x. C ) + ( B x. ( D - C ) ) ) )
14	6 10 13	3eqtrd	|- ( ph -> ( ( ( A - B ) x. C ) + ( B x. D ) ) = ( ( A x. C ) + ( B x. ( D - C ) ) ) )