Theorem mulsubfacd

Description: Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018)

	Ref	Expression
Hypotheses	muls1d.1	$$⊢\phi \to A\in ℂ$$
	muls1d.2	$$⊢\phi \to B\in ℂ$$
Assertion	mulsubfacd	$$⊢\phi \to AB-B=\left(A-1\right)B$$

Proof

Step	Hyp	Ref	Expression
1		muls1d.1	$$⊢\phi \to A\in ℂ$$
2		muls1d.2	$$⊢\phi \to B\in ℂ$$
3		1cnd	$$⊢\phi \to 1\in ℂ$$
4	1 3 2	subdird	$$⊢\phi \to \left(A-1\right)B=AB-1B$$
5	2	mullidd	$$⊢\phi \to 1B=B$$
6	5	oveq2d	$$⊢\phi \to AB-1B=AB-B$$
7	4 6	eqtr2d	$$⊢\phi \to AB-B=\left(A-1\right)B$$