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Metamath Proof Explorer

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 ∈ ℂ → - - 𝐴 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		df-neg	⊢ - - 𝐴 = ( 0 − - 𝐴 )
2		0cn	⊢ 0 ∈ ℂ
3		subneg	⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 0 − - 𝐴 ) = ( 0 + 𝐴 ) )
4	2 3	mpan	⊢ ( 𝐴 ∈ ℂ → ( 0 − - 𝐴 ) = ( 0 + 𝐴 ) )
5	1 4	eqtrid	⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = ( 0 + 𝐴 ) )
6		addlid	⊢ ( 𝐴 ∈ ℂ → ( 0 + 𝐴 ) = 𝐴 )
7	5 6	eqtrd	⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 )