Theorem submuladdd

Description: The product of a difference and a sum. Cf. addmulsub . (Contributed by Thierry Arnoux, 6-Jul-2025)

	Ref	Expression
Hypotheses	submuladdd.1	$$⊢\phi \to A\in ℂ$$
	submuladdd.2	$$⊢\phi \to B\in ℂ$$
	submuladdd.3	$$⊢\phi \to C\in ℂ$$
	submuladdd.4	$$⊢\phi \to D\in ℂ$$
Assertion	submuladdd	$$⊢\phi \to \left(A-B\right)\left(C+D\right)=AC+AD-\left(BC+BD\right)$$

Proof

Step	Hyp	Ref	Expression
1		submuladdd.1	$$⊢\phi \to A\in ℂ$$
2		submuladdd.2	$$⊢\phi \to B\in ℂ$$
3		submuladdd.3	$$⊢\phi \to C\in ℂ$$
4		submuladdd.4	$$⊢\phi \to D\in ℂ$$
5	1 2	subcld	$$⊢\phi \to A-B\in ℂ$$
6	3 4	addcld	$$⊢\phi \to C+D\in ℂ$$
7	5 6	mulcomd	$$⊢\phi \to \left(A-B\right)\left(C+D\right)=\left(C+D\right)\left(A-B\right)$$
8		addmulsub	$$⊢\left(\left(C\in ℂ\wedge D\in ℂ\right)\wedge \left(A\in ℂ\wedge B\in ℂ\right)\right)\to \left(C+D\right)\left(A-B\right)=CA+DA-\left(CB+DB\right)$$
9	3 4 1 2 8	syl22anc	$$⊢\phi \to \left(C+D\right)\left(A-B\right)=CA+DA-\left(CB+DB\right)$$
10	3 1	mulcomd	$$⊢\phi \to CA=AC$$
11	4 1	mulcomd	$$⊢\phi \to DA=AD$$
12	10 11	oveq12d	$$⊢\phi \to CA+DA=AC+AD$$
13	3 2	mulcomd	$$⊢\phi \to CB=BC$$
14	4 2	mulcomd	$$⊢\phi \to DB=BD$$
15	13 14	oveq12d	$$⊢\phi \to CB+DB=BC+BD$$
16	12 15	oveq12d	$$⊢\phi \to CA+DA-\left(CB+DB\right)=AC+AD-\left(BC+BD\right)$$
17	7 9 16	3eqtrd	$$⊢\phi \to \left(A-B\right)\left(C+D\right)=AC+AD-\left(BC+BD\right)$$