This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem infxrge0lb

Description: A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020) (Revised by AV, 4-Oct-2020)

Ref Expression
Hypotheses infxrge0lb.a 
|- ( ph -> A C_ ( 0 [,] +oo ) )
infxrge0lb.b 
|- ( ph -> B e. A )
Assertion infxrge0lb 
|- ( ph -> inf ( A , ( 0 [,] +oo ) , < ) <_ B )

Proof

Step Hyp Ref Expression
1 infxrge0lb.a 
 |-  ( ph -> A C_ ( 0 [,] +oo ) )
2 infxrge0lb.b 
 |-  ( ph -> B e. A )
3 iccssxr 
 |-  ( 0 [,] +oo ) C_ RR*
4 xrltso 
 |-  < Or RR*
5 soss 
 |-  ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
6 3 4 5 mp2 
 |-  < Or ( 0 [,] +oo )
7 6 a1i 
 |-  ( ph -> < Or ( 0 [,] +oo ) )
8 xrge0infss 
 |-  ( A C_ ( 0 [,] +oo ) -> E. x e. ( 0 [,] +oo ) ( A. y e. A -. y < x /\ A. y e. ( 0 [,] +oo ) ( x < y -> E. z e. A z < y ) ) )
9 1 8 syl 
 |-  ( ph -> E. x e. ( 0 [,] +oo ) ( A. y e. A -. y < x /\ A. y e. ( 0 [,] +oo ) ( x < y -> E. z e. A z < y ) ) )
10 7 9 infcl 
 |-  ( ph -> inf ( A , ( 0 [,] +oo ) , < ) e. ( 0 [,] +oo ) )
11 3 10 sselid 
 |-  ( ph -> inf ( A , ( 0 [,] +oo ) , < ) e. RR* )
12 1 2 sseldd 
 |-  ( ph -> B e. ( 0 [,] +oo ) )
13 3 12 sselid 
 |-  ( ph -> B e. RR* )
14 7 9 inflb 
 |-  ( ph -> ( B e. A -> -. B < inf ( A , ( 0 [,] +oo ) , < ) ) )
15 2 14 mpd 
 |-  ( ph -> -. B < inf ( A , ( 0 [,] +oo ) , < ) )
16 11 13 15 xrnltled 
 |-  ( ph -> inf ( A , ( 0 [,] +oo ) , < ) <_ B )