Theorem addneg1mul

Description: Addition with product with minus one is a subtraction. (Contributed by AV, 18-Oct-2021)

	Ref	Expression
Assertion	addneg1mul	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+-1B=A-B$$

Proof

Step	Hyp	Ref	Expression
1		mulm1	$$⊢B\in ℂ\to -1B=-B$$
2	1	adantl	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to -1B=-B$$
3	2	oveq2d	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+-1B=A+\left(-B\right)$$
4		negsub	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+\left(-B\right)=A-B$$
5	3 4	eqtrd	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+-1B=A-B$$