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 e. CC
pncan3i.2 
|- B e. CC
Assertion negdii 
|- -u ( A + B ) = ( -u A + -u B )

Proof

Step Hyp Ref Expression
1 negidi.1 
 |-  A e. CC
2 pncan3i.2 
 |-  B e. CC
3 negdi 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
4 1 2 3 mp2an 
 |-  -u ( A + B ) = ( -u A + -u B )