Theorem nppcan

Description: Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005)

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

Proof

Step	Hyp	Ref	Expression
1		subcl	|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2	1	3adant3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC )
3		simp3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
4		simp2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
5	2 3 4	add32d	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + C ) + B ) = ( ( ( A - B ) + B ) + C ) )
6		npcan	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A )
7	6	oveq1d	|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - B ) + B ) + C ) = ( A + C ) )
8	7	3adant3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + B ) + C ) = ( A + C ) )
9	5 8	eqtrd	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + C ) + B ) = ( A + C ) )