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Metamath Proof Explorer

Theorem supicc

Description: Supremum of a bounded set of real numbers. (Contributed by Thierry Arnoux, 17-May-2019)

Ref Expression
Hypotheses supicc.1 
|- ( ph -> B e. RR )
supicc.2 
|- ( ph -> C e. RR )
supicc.3 
|- ( ph -> A C_ ( B [,] C ) )
supicc.4 
|- ( ph -> A =/= (/) )
Assertion supicc 
|- ( ph -> sup ( A , RR , < ) e. ( B [,] C ) )

Proof

Step Hyp Ref Expression
1 supicc.1 
 |-  ( ph -> B e. RR )
2 supicc.2 
 |-  ( ph -> C e. RR )
3 supicc.3 
 |-  ( ph -> A C_ ( B [,] C ) )
4 supicc.4 
 |-  ( ph -> A =/= (/) )
5 iccssre 
 |-  ( ( B e. RR /\ C e. RR ) -> ( B [,] C ) C_ RR )
6 1 2 5 syl2anc 
 |-  ( ph -> ( B [,] C ) C_ RR )
7 3 6 sstrd 
 |-  ( ph -> A C_ RR )
8 1 adantr 
 |-  ( ( ph /\ x e. A ) -> B e. RR )
9 8 rexrd 
 |-  ( ( ph /\ x e. A ) -> B e. RR* )
10 2 adantr 
 |-  ( ( ph /\ x e. A ) -> C e. RR )
11 10 rexrd 
 |-  ( ( ph /\ x e. A ) -> C e. RR* )
12 3 sselda 
 |-  ( ( ph /\ x e. A ) -> x e. ( B [,] C ) )
13 iccleub 
 |-  ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> x <_ C )
14 9 11 12 13 syl3anc 
 |-  ( ( ph /\ x e. A ) -> x <_ C )
15 14 ralrimiva 
 |-  ( ph -> A. x e. A x <_ C )
16 brralrspcev 
 |-  ( ( C e. RR /\ A. x e. A x <_ C ) -> E. y e. RR A. x e. A x <_ y )
17 2 15 16 syl2anc 
 |-  ( ph -> E. y e. RR A. x e. A x <_ y )
18 suprcl 
 |-  ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) -> sup ( A , RR , < ) e. RR )
19 7 4 17 18 syl3anc 
 |-  ( ph -> sup ( A , RR , < ) e. RR )
20 7 sselda 
 |-  ( ( ph /\ x e. A ) -> x e. RR )
21 3 adantr 
 |-  ( ( ph /\ x e. A ) -> A C_ ( B [,] C ) )
22 simpr 
 |-  ( ( ph /\ x e. A ) -> x e. A )
23 iccsupr 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A C_ ( B [,] C ) /\ x e. A ) -> ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) )
24 8 10 21 22 23 syl211anc 
 |-  ( ( ph /\ x e. A ) -> ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) )
25 24 18 syl 
 |-  ( ( ph /\ x e. A ) -> sup ( A , RR , < ) e. RR )
26 iccgelb 
 |-  ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> B <_ x )
27 9 11 12 26 syl3anc 
 |-  ( ( ph /\ x e. A ) -> B <_ x )
28 suprub 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) /\ x e. A ) -> x <_ sup ( A , RR , < ) )
29 24 22 28 syl2anc 
 |-  ( ( ph /\ x e. A ) -> x <_ sup ( A , RR , < ) )
30 8 20 25 27 29 letrd 
 |-  ( ( ph /\ x e. A ) -> B <_ sup ( A , RR , < ) )
31 30 ralrimiva 
 |-  ( ph -> A. x e. A B <_ sup ( A , RR , < ) )
32 r19.3rzv 
 |-  ( A =/= (/) -> ( B <_ sup ( A , RR , < ) <-> A. x e. A B <_ sup ( A , RR , < ) ) )
33 4 32 syl 
 |-  ( ph -> ( B <_ sup ( A , RR , < ) <-> A. x e. A B <_ sup ( A , RR , < ) ) )
34 31 33 mpbird 
 |-  ( ph -> B <_ sup ( A , RR , < ) )
35 suprleub 
 |-  ( ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) /\ C e. RR ) -> ( sup ( A , RR , < ) <_ C <-> A. x e. A x <_ C ) )
36 7 4 17 2 35 syl31anc 
 |-  ( ph -> ( sup ( A , RR , < ) <_ C <-> A. x e. A x <_ C ) )
37 15 36 mpbird 
 |-  ( ph -> sup ( A , RR , < ) <_ C )
38 elicc2 
 |-  ( ( B e. RR /\ C e. RR ) -> ( sup ( A , RR , < ) e. ( B [,] C ) <-> ( sup ( A , RR , < ) e. RR /\ B <_ sup ( A , RR , < ) /\ sup ( A , RR , < ) <_ C ) ) )
39 1 2 38 syl2anc 
 |-  ( ph -> ( sup ( A , RR , < ) e. ( B [,] C ) <-> ( sup ( A , RR , < ) e. RR /\ B <_ sup ( A , RR , < ) /\ sup ( A , RR , < ) <_ C ) ) )
40 19 34 37 39 mpbir3and 
 |-  ( ph -> sup ( A , RR , < ) e. ( B [,] C ) )