Metamath Proof Explorer


Theorem resubidaddid1

Description: Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023)

Ref Expression
Assertion resubidaddid1
|- ( ( A e. RR /\ B e. RR ) -> ( ( A -R A ) + B ) = B )

Proof

Step Hyp Ref Expression
1 readdsub
 |-  ( ( A e. RR /\ B e. RR /\ A e. RR ) -> ( ( A + B ) -R A ) = ( ( A -R A ) + B ) )
2 1 3anidm13
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) -R A ) = ( ( A -R A ) + B ) )
3 repncan2
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) -R A ) = B )
4 2 3 eqtr3d
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A -R A ) + B ) = B )