Theorem negcl

Description: Closure law for negative. (Contributed by NM, 6-Aug-2003)

	Ref	Expression
Assertion	negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		df-neg	⊢ - 𝐴 = ( 0 − 𝐴 )
2		0cn	⊢ 0 ∈ ℂ
3		subcl	⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 0 − 𝐴 ) ∈ ℂ )
4	2 3	mpan	⊢ ( 𝐴 ∈ ℂ → ( 0 − 𝐴 ) ∈ ℂ )
5	1 4	eqeltrid	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )