Theorem negdii

Description: Distribution of negative over addition. (Contributed by NM, 28-Jul-1999) (Proof shortened by OpenAI, 25-Mar-2011)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
Assertion	negdii	⊢ - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		negdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 ) )
4	1 2 3	mp2an	⊢ - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 )