Theorem negneg

Description: A number is equal to the negative of its negative. Theorem I.4 of Apostol p. 18. (Contributed by NM, 12-Jan-2002) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	negneg	$$⊢A\in ℂ\to -\left(-A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		df-neg	$$⊢-\left(-A\right)=0-\left(-A\right)$$
2		0cn	$$⊢0\in ℂ$$
3		subneg	$$⊢\left(0\in ℂ\wedge A\in ℂ\right)\to 0-\left(-A\right)=0+A$$
4	2 3	mpan	$$⊢A\in ℂ\to 0-\left(-A\right)=0+A$$
5	1 4	eqtrid	$$⊢A\in ℂ\to -\left(-A\right)=0+A$$
6		addid2	$$⊢A\in ℂ\to 0+A=A$$
7	5 6	eqtrd	$$⊢A\in ℂ\to -\left(-A\right)=A$$