Theorem supicc 12320

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

Hypotheses

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

Assertion

Ref	Expression
supicc	|- ( ph -> sup ( A , RR , < ) e. ( B [,] C ) )

Proof of Theorem supicc

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supicc.3	. . . 4 |- ( ph -> A C_ ( B [,] C ) )
2		supicc.1	. . . . 5 |- ( ph -> B e. RR )
3		supicc.2	. . . . 5 |- ( ph -> C e. RR )
4		iccssre 12255	. . . . 5 |- ( ( B e. RR /\ C e. RR ) -> ( B [,] C ) C_ RR )
5	2, 3, 4	syl2anc 693	. . . 4 |- ( ph -> ( B [,] C ) C_ RR )
6	1, 5	sstrd 3613	. . 3 |- ( ph -> A C_ RR )
7		supicc.4	. . 3 |- ( ph -> A =/= (/) )
8	2	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. A ) -> B e. RR )
9	8	rexrd 10089	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. RR* )
10	3	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. A ) -> C e. RR )
11	10	rexrd 10089	. . . . . 6 |- ( ( ph /\ x e. A ) -> C e. RR* )
12	1	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. ( B [,] C ) )
13		iccleub 12229	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> x <_ C )
14	9, 11, 12, 13	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. A ) -> x <_ C )
15	14	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. A x <_ C )
16		breq2 4657	. . . . . 6 |- ( y = C -> ( x <_ y <-> x <_ C ) )
17	16	ralbidv 2986	. . . . 5 |- ( y = C -> ( A. x e. A x <_ y <-> A. x e. A x <_ C ) )
18	17	rspcev 3309	. . . 4 |- ( ( C e. RR /\ A. x e. A x <_ C ) -> E. y e. RR A. x e. A x <_ y )
19	3, 15, 18	syl2anc 693	. . 3 |- ( ph -> E. y e. RR A. x e. A x <_ y )
20		suprcl 10983	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) -> sup ( A , RR , < ) e. RR )
21	6, 7, 19, 20	syl3anc 1326	. 2 |- ( ph -> sup ( A , RR , < ) e. RR )
22	6	sselda 3603	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. RR )
23	1	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. A ) -> A C_ ( B [,] C ) )
24		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. A ) -> x e. A )
25		iccsupr 12266	. . . . . . 7 |- ( ( ( 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 ) )
26	8, 10, 23, 24, 25	syl211anc 1332	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) )
27	26, 20	syl 17	. . . . 5 |- ( ( ph /\ x e. A ) -> sup ( A , RR , < ) e. RR )
28		iccgelb 12230	. . . . . 6 |- ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> B <_ x )
29	9, 11, 12, 28	syl3anc 1326	. . . . 5 |- ( ( ph /\ x e. A ) -> B <_ x )
30		suprub 10984	. . . . . 6 |- ( ( ( A C_ RR /\ A =/= (/) /\ E. y e. RR A. x e. A x <_ y ) /\ x e. A ) -> x <_ sup ( A , RR , < ) )
31	26, 24, 30	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. A ) -> x <_ sup ( A , RR , < ) )
32	8, 22, 27, 29, 31	letrd 10194	. . . 4 |- ( ( ph /\ x e. A ) -> B <_ sup ( A , RR , < ) )
33	32	ralrimiva 2966	. . 3 |- ( ph -> A. x e. A B <_ sup ( A , RR , < ) )
34		r19.3rzv 4064	. . . 4 |- ( A =/= (/) -> ( B <_ sup ( A , RR , < ) <-> A. x e. A B <_ sup ( A , RR , < ) ) )
35	7, 34	syl 17	. . 3 |- ( ph -> ( B <_ sup ( A , RR , < ) <-> A. x e. A B <_ sup ( A , RR , < ) ) )
36	33, 35	mpbird 247	. 2 |- ( ph -> B <_ sup ( A , RR , < ) )
37		suprleub 10989	. . . 4 |- ( ( ( 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 ) )
38	6, 7, 19, 3, 37	syl31anc 1329	. . 3 |- ( ph -> ( sup ( A , RR , < ) <_ C <-> A. x e. A x <_ C ) )
39	15, 38	mpbird 247	. 2 |- ( ph -> sup ( A , RR , < ) <_ C )
40		elicc2 12238	. . 3 |- ( ( 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 ) ) )
41	2, 3, 40	syl2anc 693	. 2 |- ( ph -> ( sup ( A , RR , < ) e. ( B [,] C ) <-> ( sup ( A , RR , < ) e. RR /\ B <_ sup ( A , RR , < ) /\ sup ( A , RR , < ) <_ C ) ) )
42	21, 36, 39, 41	mpbir3and 1245	1 |- ( ph -> sup ( A , RR , < ) e. ( B [,] C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-icc 12182

This theorem is referenced by:  supicclub2  12323  hoidmv1lelem1  40805  hoidmvlelem1  40809