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Metamath Proof Explorer

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 )