Metamath Proof Explorer


Theorem rpgecl

Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

Step Hyp Ref Expression
1 simp2
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> B e. RR )
2 0red
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> 0 e. RR )
3 rpre
 |-  ( A e. RR+ -> A e. RR )
4 3 3ad2ant1
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> A e. RR )
5 rpgt0
 |-  ( A e. RR+ -> 0 < A )
6 5 3ad2ant1
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> 0 < A )
7 simp3
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> A <_ B )
8 2 4 1 6 7 ltletrd
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> 0 < B )
9 elrp
 |-  ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
10 1 8 9 sylanbrc
 |-  ( ( A e. RR+ /\ B e. RR /\ A <_ B ) -> B e. RR+ )