Metamath Proof Explorer

Theorem npcan

Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step Hyp Ref Expression
1 subcl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2 simpr 
 |-  ( ( A e. CC /\ B e. CC ) -> B e. CC )
3 1 2 addcomd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = ( B + ( A - B ) ) )
4 pncan3 
 |-  ( ( B e. CC /\ A e. CC ) -> ( B + ( A - B ) ) = A )
5 4 ancoms 
 |-  ( ( A e. CC /\ B e. CC ) -> ( B + ( A - B ) ) = A )
6 3 5 eqtrd 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A )