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	$$⊢\phi \to A\in ℂ$$
	submuladdmuld.b	$$⊢\phi \to B\in ℂ$$
	submuladdmuld.c	$$⊢\phi \to C\in ℂ$$
	submuladdmuld.d	$$⊢\phi \to D\in ℂ$$
Assertion	submuladdmuld	$$⊢\phi \to \left(A-B\right)C+BD=AC+B\left(D-C\right)$$

Proof

Step	Hyp	Ref	Expression
1		submuladdmuld.a	$$⊢\phi \to A\in ℂ$$
2		submuladdmuld.b	$$⊢\phi \to B\in ℂ$$
3		submuladdmuld.c	$$⊢\phi \to C\in ℂ$$
4		submuladdmuld.d	$$⊢\phi \to D\in ℂ$$
5	1 2 3	subdird	$$⊢\phi \to \left(A-B\right)C=AC-BC$$
6	5	oveq1d	$$⊢\phi \to \left(A-B\right)C+BD=AC-BC+BD$$
7	1 3	mulcld	$$⊢\phi \to AC\in ℂ$$
8	2 3	mulcld	$$⊢\phi \to BC\in ℂ$$
9	2 4	mulcld	$$⊢\phi \to BD\in ℂ$$
10	7 8 9	subadd23d	$$⊢\phi \to AC-BC+BD=AC+BD-BC$$
11	2 4 3	subdid	$$⊢\phi \to B\left(D-C\right)=BD-BC$$
12	11	eqcomd	$$⊢\phi \to BD-BC=B\left(D-C\right)$$
13	12	oveq2d	$$⊢\phi \to AC+BD-BC=AC+B\left(D-C\right)$$
14	6 10 13	3eqtrd	$$⊢\phi \to \left(A-B\right)C+BD=AC+B\left(D-C\right)$$