Metamath Proof Explorer

Theorem negdi2

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

Ref Expression
Assertion negdi2 
|- ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A - B ) )

Proof

Step Hyp Ref Expression
1 negdi 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A + -u B ) )
2 negcl 
 |-  ( A e. CC -> -u A e. CC )
3 negsub 
 |-  ( ( -u A e. CC /\ B e. CC ) -> ( -u A + -u B ) = ( -u A - B ) )
4 2 3 sylan 
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u A + -u B ) = ( -u A - B ) )
5 1 4 eqtrd 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A + B ) = ( -u A - B ) )