Theorem resubcl

Description: Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997)

	Ref	Expression
Assertion	resubcl	|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		recn	|- ( B e. RR -> B e. CC )
3		negsub	|- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
4	1 2 3	syl2an	|- ( ( A e. RR /\ B e. RR ) -> ( A + -u B ) = ( A - B ) )
5		renegcl	|- ( B e. RR -> -u B e. RR )
6		readdcl	|- ( ( A e. RR /\ -u B e. RR ) -> ( A + -u B ) e. RR )
7	5 6	sylan2	|- ( ( A e. RR /\ B e. RR ) -> ( A + -u B ) e. RR )
8	4 7	eqeltrrd	|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )