Theorem ltaddpos2

Description: Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005)

	Ref	Expression
Assertion	ltaddpos2	|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltaddpos	|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) )
2		recn	|- ( A e. RR -> A e. CC )
3		recn	|- ( B e. RR -> B e. CC )
4		addcom	|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
5	2 3 4	syl2an	|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
6	5	breq2d	|- ( ( A e. RR /\ B e. RR ) -> ( B < ( A + B ) <-> B < ( B + A ) ) )
7	1 6	bitr4d	|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) )