Metamath Proof Explorer

Theorem xadd0ge

Description: A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses xadd0ge.a 
|- ( ph -> A e. RR* )
xadd0ge.b 
|- ( ph -> B e. ( 0 [,] +oo ) )
Assertion xadd0ge 
|- ( ph -> A <_ ( A +e B ) )

Proof

Step Hyp Ref Expression
1 xadd0ge.a 
 |-  ( ph -> A e. RR* )
2 xadd0ge.b 
 |-  ( ph -> B e. ( 0 [,] +oo ) )
3 xaddid1 
 |-  ( A e. RR* -> ( A +e 0 ) = A )
4 1 3 syl 
 |-  ( ph -> ( A +e 0 ) = A )
5 4 eqcomd 
 |-  ( ph -> A = ( A +e 0 ) )
6 0xr 
 |-  0 e. RR*
7 6 a1i 
 |-  ( ph -> 0 e. RR* )
8 1 7 jca 
 |-  ( ph -> ( A e. RR* /\ 0 e. RR* ) )
9 iccssxr 
 |-  ( 0 [,] +oo ) C_ RR*
10 9 2 sseldi 
 |-  ( ph -> B e. RR* )
11 1 10 jca 
 |-  ( ph -> ( A e. RR* /\ B e. RR* ) )
12 8 11 jca 
 |-  ( ph -> ( ( A e. RR* /\ 0 e. RR* ) /\ ( A e. RR* /\ B e. RR* ) ) )
13 1 xrleidd 
 |-  ( ph -> A <_ A )
14 pnfxr 
 |-  +oo e. RR*
15 14 a1i 
 |-  ( ph -> +oo e. RR* )
16 iccgelb 
 |-  ( ( 0 e. RR* /\ +oo e. RR* /\ B e. ( 0 [,] +oo ) ) -> 0 <_ B )
17 7 15 2 16 syl3anc 
 |-  ( ph -> 0 <_ B )
18 13 17 jca 
 |-  ( ph -> ( A <_ A /\ 0 <_ B ) )
19 xle2add 
 |-  ( ( ( A e. RR* /\ 0 e. RR* ) /\ ( A e. RR* /\ B e. RR* ) ) -> ( ( A <_ A /\ 0 <_ B ) -> ( A +e 0 ) <_ ( A +e B ) ) )
20 12 18 19 sylc 
 |-  ( ph -> ( A +e 0 ) <_ ( A +e B ) )
21 5 20 eqbrtrd 
 |-  ( ph -> A <_ ( A +e B ) )