Metamath Proof Explorer


Theorem negdi2

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

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

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 ) = ( - 𝐴𝐵 ) )
4 2 3 sylan ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 + - 𝐵 ) = ( - 𝐴𝐵 ) )
5 1 4 eqtrd ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴𝐵 ) )