Theorem negsubd

Description: Relationship between subtraction and negative. Theorem I.3 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	$$⊢\phi \to A\in ℂ$$
	pncand.2	$$⊢\phi \to B\in ℂ$$
Assertion	negsubd	$$⊢\phi \to A+\left(-B\right)=A-B$$

Proof

Step	Hyp	Ref	Expression
1		negidd.1	$$⊢\phi \to A\in ℂ$$
2		pncand.2	$$⊢\phi \to B\in ℂ$$
3		negsub	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+\left(-B\right)=A-B$$
4	1 2 3	syl2anc	$$⊢\phi \to A+\left(-B\right)=A-B$$