Metamath Proof Explorer

Theorem xraddge02

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016)

Ref Expression
Assertion xraddge02 
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ B -> A <_ ( A +e B ) ) )

Proof

Step Hyp Ref Expression
1 xrleid 
 |-  ( A e. RR* -> A <_ A )
2 1 adantr 
 |-  ( ( A e. RR* /\ B e. RR* ) -> A <_ A )
3 simpl 
 |-  ( ( A e. RR* /\ B e. RR* ) -> A e. RR* )
4 0xr 
 |-  0 e. RR*
5 3 4 jctir 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A e. RR* /\ 0 e. RR* ) )
6 xle2add 
 |-  ( ( ( A e. RR* /\ 0 e. RR* ) /\ ( A e. RR* /\ B e. RR* ) ) -> ( ( A <_ A /\ 0 <_ B ) -> ( A +e 0 ) <_ ( A +e B ) ) )
7 5 6 mpancom 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( ( A <_ A /\ 0 <_ B ) -> ( A +e 0 ) <_ ( A +e B ) ) )
8 2 7 mpand 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ B -> ( A +e 0 ) <_ ( A +e B ) ) )
9 xaddid1 
 |-  ( A e. RR* -> ( A +e 0 ) = A )
10 9 breq1d 
 |-  ( A e. RR* -> ( ( A +e 0 ) <_ ( A +e B ) <-> A <_ ( A +e B ) ) )
11 10 adantr 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( ( A +e 0 ) <_ ( A +e B ) <-> A <_ ( A +e B ) ) )
12 8 11 sylibd 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( 0 <_ B -> A <_ ( A +e B ) ) )