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 e. CC -> -u -u A = A )

Proof

Step	Hyp	Ref	Expression
1		df-neg	|- -u -u A = ( 0 - -u A )
2		0cn	|- 0 e. CC
3		subneg	|- ( ( 0 e. CC /\ A e. CC ) -> ( 0 - -u A ) = ( 0 + A ) )
4	2 3	mpan	|- ( A e. CC -> ( 0 - -u A ) = ( 0 + A ) )
5	1 4	eqtrid	|- ( A e. CC -> -u -u A = ( 0 + A ) )
6		addid2	|- ( A e. CC -> ( 0 + A ) = A )
7	5 6	eqtrd	|- ( A e. CC -> -u -u A = A )