Theorem ixxlb 12197

Description: Extract the lower bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by AV, 12-Sep-2020.)

Hypotheses

Ref	Expression
ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
ixxub.2	|- ( ( w e. RR* /\ B e. RR* ) -> ( w < B -> w S B ) )
ixxub.3	|- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w <_ B ) )
ixxub.4	|- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A R w ) )
ixxub.5	|- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A <_ w ) )

Assertion

Ref	Expression
ixxlb	|- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> inf ( ( A O B ) , RR* , < ) = A )

Distinct variable groups:   x, w, y, z, A   w, O   w, B, x, y, z   x, R, y, z   x, S, y, z

Allowed substitution hints:   R( w)   S( w)   O( x, y, z)

Proof of Theorem ixxlb

Step	Hyp	Ref	Expression
1		ixx.1	. . . . . . . . 9 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
2	1	elixx1 12184	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
3	2	3adant3 1081	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
4	3	biimpa 501	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> ( w e. RR* /\ A R w /\ w S B ) )
5	4	simp1d 1073	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w e. RR* )
6	5	ex 450	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( w e. ( A O B ) -> w e. RR* ) )
7	6	ssrdv 3609	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) C_ RR* )
8		infxrcl 12163	. . 3 |- ( ( A O B ) C_ RR* -> inf ( ( A O B ) , RR* , < ) e. RR* )
9	7, 8	syl 17	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> inf ( ( A O B ) , RR* , < ) e. RR* )
10		simp1 1061	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A e. RR* )
11		simprr 796	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> w < inf ( ( A O B ) , RR* , < ) )
12	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> ( A O B ) C_ RR* )
13		qre 11793	. . . . . . . . . . 11 |- ( w e. QQ -> w e. RR )
14	13	rexrd 10089	. . . . . . . . . 10 |- ( w e. QQ -> w e. RR* )
15	14	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> w e. RR* )
16		simprl 794	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> A < w )
17	10	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> A e. RR* )
18		ixxub.4	. . . . . . . . . . 11 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A R w ) )
19	17, 15, 18	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> ( A < w -> A R w ) )
20	16, 19	mpd 15	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> A R w )
21	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> inf ( ( A O B ) , RR* , < ) e. RR* )
22		simpll2 1101	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> B e. RR* )
23		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) =/= (/) )
24		n0 3931	. . . . . . . . . . . . . 14 |- ( ( A O B ) =/= (/) <-> E. w w e. ( A O B ) )
25	23, 24	sylib 208	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> E. w w e. ( A O B ) )
26	9	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> inf ( ( A O B ) , RR* , < ) e. RR* )
27		simpl2 1065	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> B e. RR* )
28		infxrlb 12164	. . . . . . . . . . . . . . 15 |- ( ( ( A O B ) C_ RR* /\ w e. ( A O B ) ) -> inf ( ( A O B ) , RR* , < ) <_ w )
29	7, 28	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> inf ( ( A O B ) , RR* , < ) <_ w )
30	4	simp3d 1075	. . . . . . . . . . . . . . 15 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w S B )
31		ixxub.3	. . . . . . . . . . . . . . . 16 |- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w <_ B ) )
32	5, 27, 31	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> ( w S B -> w <_ B ) )
33	30, 32	mpd 15	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w <_ B )
34	26, 5, 27, 29, 33	xrletrd 11993	. . . . . . . . . . . . 13 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> inf ( ( A O B ) , RR* , < ) <_ B )
35	25, 34	exlimddv 1863	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> inf ( ( A O B ) , RR* , < ) <_ B )
36	35	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> inf ( ( A O B ) , RR* , < ) <_ B )
37	15, 21, 22, 11, 36	xrltletrd 11992	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> w < B )
38		ixxub.2	. . . . . . . . . . 11 |- ( ( w e. RR* /\ B e. RR* ) -> ( w < B -> w S B ) )
39	15, 22, 38	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> ( w < B -> w S B ) )
40	37, 39	mpd 15	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> w S B )
41	3	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
42	15, 20, 40, 41	mpbir3and 1245	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> w e. ( A O B ) )
43	12, 42, 28	syl2anc 693	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> inf ( ( A O B ) , RR* , < ) <_ w )
44	21, 15	xrlenltd 10104	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> (inf ( ( A O B ) , RR* , < ) <_ w <-> -. w < inf ( ( A O B ) , RR* , < ) ) )
45	43, 44	mpbid 222	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) -> -. w < inf ( ( A O B ) , RR* , < ) )
46	11, 45	pm2.65da 600	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) -> -. ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) )
47	46	nrexdv 3001	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. E. w e. QQ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) )
48		qbtwnxr 12031	. . . . . 6 |- ( ( A e. RR* /\ inf ( ( A O B ) , RR* , < ) e. RR* /\ A < inf ( ( A O B ) , RR* , < ) ) -> E. w e. QQ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) )
49	48	3expia 1267	. . . . 5 |- ( ( A e. RR* /\ inf ( ( A O B ) , RR* , < ) e. RR* ) -> ( A < inf ( ( A O B ) , RR* , < ) -> E. w e. QQ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) )
50	10, 9, 49	syl2anc 693	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A < inf ( ( A O B ) , RR* , < ) -> E. w e. QQ ( A < w /\ w < inf ( ( A O B ) , RR* , < ) ) ) )
51	47, 50	mtod 189	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. A < inf ( ( A O B ) , RR* , < ) )
52	9, 10, 51	xrnltled 10106	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> inf ( ( A O B ) , RR* , < ) <_ A )
53	4	simp2d 1074	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A R w )
54	10	adantr 481	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A e. RR* )
55		ixxub.5	. . . . . 6 |- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A <_ w ) )
56	54, 5, 55	syl2anc 693	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> ( A R w -> A <_ w ) )
57	53, 56	mpd 15	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A <_ w )
58	57	ralrimiva 2966	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A. w e. ( A O B ) A <_ w )
59		infxrgelb 12165	. . . 4 |- ( ( ( A O B ) C_ RR* /\ A e. RR* ) -> ( A <_ inf ( ( A O B ) , RR* , < ) <-> A. w e. ( A O B ) A <_ w ) )
60	7, 10, 59	syl2anc 693	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A <_ inf ( ( A O B ) , RR* , < ) <-> A. w e. ( A O B ) A <_ w ) )
61	58, 60	mpbird 247	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A <_ inf ( ( A O B ) , RR* , < ) )
62	9, 10, 52, 61	xrletrid 11986	1 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> inf ( ( A O B ) , RR* , < ) = A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652  infcinf 8347   RR*cxr 10073   < clt 10074   <_ cle 10075   QQcq 11788

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789

This theorem is referenced by:  ioorf  23341  ioorinv2  23343  ioossioobi  39743