Theorem neg2sub

Description: Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007)

	Ref	Expression
Assertion	neg2sub	|- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( B - A ) )

Proof

Step	Hyp	Ref	Expression
1		negcl	|- ( A e. CC -> -u A e. CC )
2		subneg	|- ( ( -u A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( -u A + B ) )
3	1 2	sylan	|- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( -u A + B ) )
4		negsubdi	|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( -u A + B ) )
5		negsubdi2	|- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
6	3 4 5	3eqtr2d	|- ( ( A e. CC /\ B e. CC ) -> ( -u A - -u B ) = ( B - A ) )