Metamath Proof Explorer

Theorem ltaddpos

Description: Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004)

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

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 ltadd2 
 |-  ( ( 0 e. RR /\ A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B + 0 ) < ( B + A ) ) )
3 1 2 mp3an1 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> ( B + 0 ) < ( B + A ) ) )
4 recn 
 |-  ( B e. RR -> B e. CC )
5 4 addid1d 
 |-  ( B e. RR -> ( B + 0 ) = B )
6 5 adantl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B + 0 ) = B )
7 6 breq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( B + 0 ) < ( B + A ) <-> B < ( B + A ) ) )
8 3 7 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) )