Theorem xnpcan

Description: Extended real version of npcan . (Contributed by Mario Carneiro, 20-Aug-2015)

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

Proof

Step	Hyp	Ref	Expression
1		rexr	|- ( B e. RR -> B e. RR* )
2		xnegneg	|- ( B e. RR* -> -e -e B = B )
3	1 2	syl	|- ( B e. RR -> -e -e B = B )
4	3	adantl	|- ( ( A e. RR* /\ B e. RR ) -> -e -e B = B )
5	4	oveq2d	|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e -e -e B ) = ( ( A +e -e B ) +e B ) )
6		rexneg	|- ( B e. RR -> -e B = -u B )
7		renegcl	|- ( B e. RR -> -u B e. RR )
8	6 7	eqeltrd	|- ( B e. RR -> -e B e. RR )
9		xpncan	|- ( ( A e. RR* /\ -e B e. RR ) -> ( ( A +e -e B ) +e -e -e B ) = A )
10	8 9	sylan2	|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e -e -e B ) = A )
11	5 10	eqtr3d	|- ( ( A e. RR* /\ B e. RR ) -> ( ( A +e -e B ) +e B ) = A )