Theorem muls1d

Description: Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017)

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

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 2 3	subdid	$$⊢\phi \to A\left(B-1\right)=AB-A\cdot 1$$
5	1	mulridd	$$⊢\phi \to A\cdot 1=A$$
6	5	oveq2d	$$⊢\phi \to AB-A\cdot 1=AB-A$$
7	4 6	eqtrd	$$⊢\phi \to A\left(B-1\right)=AB-A$$