Theorem inficc 39761

Description: The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
inficc.a	|- ( ph -> A e. RR* )
inficc.b	|- ( ph -> B e. RR* )
inficc.s	|- ( ph -> S C_ ( A [,] B ) )
inficc.n0	|- ( ph -> S =/= (/) )

Assertion

Ref	Expression
inficc	|- ( ph -> inf ( S , RR* , < ) e. ( A [,] B ) )

Proof of Theorem inficc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inficc.a	. 2 |- ( ph -> A e. RR* )
2		inficc.b	. 2 |- ( ph -> B e. RR* )
3		inficc.s	. . . 4 |- ( ph -> S C_ ( A [,] B ) )
4		iccssxr 12256	. . . . 5 |- ( A [,] B ) C_ RR*
5	4	a1i 11	. . . 4 |- ( ph -> ( A [,] B ) C_ RR* )
6	3, 5	sstrd 3613	. . 3 |- ( ph -> S C_ RR* )
7		infxrcl 12163	. . 3 |- ( S C_ RR* -> inf ( S , RR* , < ) e. RR* )
8	6, 7	syl 17	. 2 |- ( ph -> inf ( S , RR* , < ) e. RR* )
9	1	adantr 481	. . . . 5 |- ( ( ph /\ x e. S ) -> A e. RR* )
10	2	adantr 481	. . . . 5 |- ( ( ph /\ x e. S ) -> B e. RR* )
11	3	sselda 3603	. . . . 5 |- ( ( ph /\ x e. S ) -> x e. ( A [,] B ) )
12		iccgelb 12230	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
13	9, 10, 11, 12	syl3anc 1326	. . . 4 |- ( ( ph /\ x e. S ) -> A <_ x )
14	13	ralrimiva 2966	. . 3 |- ( ph -> A. x e. S A <_ x )
15		infxrgelb 12165	. . . 4 |- ( ( S C_ RR* /\ A e. RR* ) -> ( A <_ inf ( S , RR* , < ) <-> A. x e. S A <_ x ) )
16	6, 1, 15	syl2anc 693	. . 3 |- ( ph -> ( A <_ inf ( S , RR* , < ) <-> A. x e. S A <_ x ) )
17	14, 16	mpbird 247	. 2 |- ( ph -> A <_ inf ( S , RR* , < ) )
18		inficc.n0	. . . 4 |- ( ph -> S =/= (/) )
19		n0 3931	. . . 4 |- ( S =/= (/) <-> E. x x e. S )
20	18, 19	sylib 208	. . 3 |- ( ph -> E. x x e. S )
21	8	adantr 481	. . . . . 6 |- ( ( ph /\ x e. S ) -> inf ( S , RR* , < ) e. RR* )
22	4, 11	sseldi 3601	. . . . . 6 |- ( ( ph /\ x e. S ) -> x e. RR* )
23	6	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. S ) -> S C_ RR* )
24		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. S ) -> x e. S )
25		infxrlb 12164	. . . . . . 7 |- ( ( S C_ RR* /\ x e. S ) -> inf ( S , RR* , < ) <_ x )
26	23, 24, 25	syl2anc 693	. . . . . 6 |- ( ( ph /\ x e. S ) -> inf ( S , RR* , < ) <_ x )
27		iccleub 12229	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
28	9, 10, 11, 27	syl3anc 1326	. . . . . 6 |- ( ( ph /\ x e. S ) -> x <_ B )
29	21, 22, 10, 26, 28	xrletrd 11993	. . . . 5 |- ( ( ph /\ x e. S ) -> inf ( S , RR* , < ) <_ B )
30	29	ex 450	. . . 4 |- ( ph -> ( x e. S -> inf ( S , RR* , < ) <_ B ) )
31	30	exlimdv 1861	. . 3 |- ( ph -> ( E. x x e. S -> inf ( S , RR* , < ) <_ B ) )
32	20, 31	mpd 15	. 2 |- ( ph -> inf ( S , RR* , < ) <_ B )
33	1, 2, 8, 17, 32	eliccxrd 39753	1 |- ( ph -> inf ( S , RR* , < ) e. ( A [,] B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650  infcinf 8347   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-iun 4522  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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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:  ovnf  40777