Theorem ltaddrp

Description: Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007)

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

Proof

Step	Hyp	Ref	Expression
1		elrp	|- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
2		ltaddpos	|- ( ( B e. RR /\ A e. RR ) -> ( 0 < B <-> A < ( A + B ) ) )
3	2	biimpd	|- ( ( B e. RR /\ A e. RR ) -> ( 0 < B -> A < ( A + B ) ) )
4	3	expcom	|- ( A e. RR -> ( B e. RR -> ( 0 < B -> A < ( A + B ) ) ) )
5	4	imp32	|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> A < ( A + B ) )
6	1 5	sylan2b	|- ( ( A e. RR /\ B e. RR+ ) -> A < ( A + B ) )