Metamath Proof Explorer
Description: Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
negidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
pncand.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
negdid |
⊢ ( 𝜑 → - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
negidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
pncand.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
negdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → - ( 𝐴 + 𝐵 ) = ( - 𝐴 + - 𝐵 ) ) |