Theorem negsubi

Description: Relationship between subtraction and negative. Theorem I.3 of Apostol p. 18. (Contributed by NM, 26-Nov-1994) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Hypotheses	negidi.1	$$⊢A\in ℂ$$
	pncan3i.2	$$⊢B\in ℂ$$
Assertion	negsubi	$$⊢A+\left(-B\right)=A-B$$

Proof

Step	Hyp	Ref	Expression
1		negidi.1	$$⊢A\in ℂ$$
2		pncan3i.2	$$⊢B\in ℂ$$
3		negsub	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to A+\left(-B\right)=A-B$$
4	1 2 3	mp2an	$$⊢A+\left(-B\right)=A-B$$