Metamath Proof Explorer

Theorem negdi

Description: Distribution of negative over addition. (Contributed by NM, 10-May-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion negdi 
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )

Proof

Step Hyp Ref Expression
1 subneg 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A - -u B ) = ( A + B ) )
2 1 negeqd 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A - -u B ) = -u ( A + B ) )
3 negcl 
 |-  ( B e. CC -> -u B e. CC )
4 negsubdi 
 |-  ( ( A e. CC /\ -u B e. CC ) -> -u ( A - -u B ) = ( -u A + -u B ) )
5 3 4 sylan2 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A - -u B ) = ( -u A + -u B ) )
6 2 5 eqtr3d 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )