Theorem ixxub 11656

Description: Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)

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
ixxub	|- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) = B )

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 ixxub

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 11644	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
3	2	3adant3 1028	. . . . . . 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 487	. . . . . 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	simp3d 1022	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w S B )
6	4	simp1d 1020	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w e. RR* )
7		simp2 1009	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> B e. RR* )
8	7	adantr 467	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> B e. RR* )
9		ixxub.3	. . . . . 6 |- ( ( w e. RR* /\ B e. RR* ) -> ( w S B -> w <_ B ) )
10	6, 8, 9	syl2anc 667	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> ( w S B -> w <_ B ) )
11	5, 10	mpd 15	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w <_ B )
12	11	ralrimiva 2802	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A. w e. ( A O B ) w <_ B )
13	6	ex 436	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( w e. ( A O B ) -> w e. RR* ) )
14	13	ssrdv 3438	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) C_ RR* )
15		supxrleub 11612	. . . 4 |- ( ( ( A O B ) C_ RR* /\ B e. RR* ) -> ( sup ( ( A O B ) , RR* , < ) <_ B <-> A. w e. ( A O B ) w <_ B ) )
16	14, 7, 15	syl2anc 667	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( sup ( ( A O B ) , RR* , < ) <_ B <-> A. w e. ( A O B ) w <_ B ) )
17	12, 16	mpbird 236	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) <_ B )
18		simprl 764	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> sup ( ( A O B ) , RR* , < ) < w )
19	14	ad2antrr 732	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> ( A O B ) C_ RR* )
20		qre 11269	. . . . . . . . . . 11 |- ( w e. QQ -> w e. RR )
21	20	rexrd 9690	. . . . . . . . . 10 |- ( w e. QQ -> w e. RR* )
22	21	ad2antlr 733	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> w e. RR* )
23		simp1 1008	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A e. RR* )
24	23	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> A e. RR* )
25		supxrcl 11600	. . . . . . . . . . . . 13 |- ( ( A O B ) C_ RR* -> sup ( ( A O B ) , RR* , < ) e. RR* )
26	14, 25	syl 17	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) e. RR* )
27	26	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> sup ( ( A O B ) , RR* , < ) e. RR* )
28		simp3 1010	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) =/= (/) )
29		n0 3741	. . . . . . . . . . . . . 14 |- ( ( A O B ) =/= (/) <-> E. w w e. ( A O B ) )
30	28, 29	sylib 200	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> E. w w e. ( A O B ) )
31	23	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A e. RR* )
32	26	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> sup ( ( A O B ) , RR* , < ) e. RR* )
33	4	simp2d 1021	. . . . . . . . . . . . . . 15 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A R w )
34		ixxub.5	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ w e. RR* ) -> ( A R w -> A <_ w ) )
35	31, 6, 34	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> ( A R w -> A <_ w ) )
36	33, 35	mpd 15	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A <_ w )
37		supxrub 11610	. . . . . . . . . . . . . . 15 |- ( ( ( A O B ) C_ RR* /\ w e. ( A O B ) ) -> w <_ sup ( ( A O B ) , RR* , < ) )
38	14, 37	sylan 474	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w <_ sup ( ( A O B ) , RR* , < ) )
39	31, 6, 32, 36, 38	xrletrd 11459	. . . . . . . . . . . . 13 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A <_ sup ( ( A O B ) , RR* , < ) )
40	30, 39	exlimddv 1781	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A <_ sup ( ( A O B ) , RR* , < ) )
41	40	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> A <_ sup ( ( A O B ) , RR* , < ) )
42	24, 27, 22, 41, 18	xrlelttrd 11457	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> A < w )
43		ixxub.4	. . . . . . . . . . 11 |- ( ( A e. RR* /\ w e. RR* ) -> ( A < w -> A R w ) )
44	24, 22, 43	syl2anc 667	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> ( A < w -> A R w ) )
45	42, 44	mpd 15	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> A R w )
46		simprr 766	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> w < B )
47	7	ad2antrr 732	. . . . . . . . . . 11 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> B e. RR* )
48		ixxub.2	. . . . . . . . . . 11 |- ( ( w e. RR* /\ B e. RR* ) -> ( w < B -> w S B ) )
49	22, 47, 48	syl2anc 667	. . . . . . . . . 10 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> ( w < B -> w S B ) )
50	46, 49	mpd 15	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> w S B )
51	3	ad2antrr 732	. . . . . . . . 9 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
52	22, 45, 50, 51	mpbir3and 1191	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> w e. ( A O B ) )
53	19, 52, 37	syl2anc 667	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> w <_ sup ( ( A O B ) , RR* , < ) )
54		xrlenlt 9699	. . . . . . . 8 |- ( ( w e. RR* /\ sup ( ( A O B ) , RR* , < ) e. RR* ) -> ( w <_ sup ( ( A O B ) , RR* , < ) <-> -. sup ( ( A O B ) , RR* , < ) < w ) )
55	22, 27, 54	syl2anc 667	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> ( w <_ sup ( ( A O B ) , RR* , < ) <-> -. sup ( ( A O B ) , RR* , < ) < w ) )
56	53, 55	mpbid 214	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) /\ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) -> -. sup ( ( A O B ) , RR* , < ) < w )
57	18, 56	pm2.65da 580	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) -> -. ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
58	57	nrexdv 2843	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
59		qbtwnxr 11493	. . . . . 6 |- ( ( sup ( ( A O B ) , RR* , < ) e. RR* /\ B e. RR* /\ sup ( ( A O B ) , RR* , < ) < B ) -> E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
60	59	3expia 1210	. . . . 5 |- ( ( sup ( ( A O B ) , RR* , < ) e. RR* /\ B e. RR* ) -> ( sup ( ( A O B ) , RR* , < ) < B -> E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) )
61	26, 7, 60	syl2anc 667	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( sup ( ( A O B ) , RR* , < ) < B -> E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) ) )
62	58, 61	mtod 181	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. sup ( ( A O B ) , RR* , < ) < B )
63		xrlenlt 9699	. . . 4 |- ( ( B e. RR* /\ sup ( ( A O B ) , RR* , < ) e. RR* ) -> ( B <_ sup ( ( A O B ) , RR* , < ) <-> -. sup ( ( A O B ) , RR* , < ) < B ) )
64	7, 26, 63	syl2anc 667	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( B <_ sup ( ( A O B ) , RR* , < ) <-> -. sup ( ( A O B ) , RR* , < ) < B ) )
65	62, 64	mpbird 236	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> B <_ sup ( ( A O B ) , RR* , < ) )
66		xrletri3 11451	. . 3 |- ( ( sup ( ( A O B ) , RR* , < ) e. RR* /\ B e. RR* ) -> ( sup ( ( A O B ) , RR* , < ) = B <-> ( sup ( ( A O B ) , RR* , < ) <_ B /\ B <_ sup ( ( A O B ) , RR* , < ) ) ) )
67	26, 7, 66	syl2anc 667	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( sup ( ( A O B ) , RR* , < ) = B <-> ( sup ( ( A O B ) , RR* , < ) <_ B /\ B <_ sup ( ( A O B ) , RR* , < ) ) ) )
68	17, 65, 67	mpbir2and 933	1 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   C_ wss 3404   (/)c0 3731   class class class wbr 4402  (class class class)co 6290   |-> cmpt2 6292   supcsup 7954   RR*cxr 9674   < clt 9675   <_ cle 9676   QQcq 11264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265

This theorem is referenced by:  ioopnfsup  12091  icopnfsup  12092  bndth  21986  ioorf  22525  ioorinv2  22527  ioorfOLD  22530  ioorinv2OLD  22532  ioossioobi  37618