Theorem negdi2

Description: Distribution of negative over addition. (Contributed by NM, 1-Jan-2006)

	Ref	Expression
Assertion	negdi2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		negdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 ) )
2		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
3		negsub	⊢ ( ( - 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 + - 𝐵 ) = ( - 𝐴 − 𝐵 ) )
4	2 3	sylan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 + - 𝐵 ) = ( - 𝐴 − 𝐵 ) )
5	1 4	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴 − 𝐵 ) )