Metamath Proof Explorer

Theorem negsubdi2

Description: Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999)

Ref Expression
Assertion negsubdi2 
|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )

Proof

Step Hyp Ref Expression
1 negsubdi 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) )
2 negcl 
 |-  ( A e. CC -> -u A e. CC )
3 addcom 
 |-  ( ( -u A e. CC /\ B e. CC ) -> ( -u A + B ) = ( B + -u A ) )
4 2 3 sylan 
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u A + B ) = ( B + -u A ) )
5 negsub 
 |-  ( ( B e. CC /\ A e. CC ) -> ( B + -u A ) = ( B - A ) )
6 5 ancoms 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B + -u A ) = ( B - A ) )
7 1 4 6 3eqtrd 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )