Theorem ixxub 11546

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 11534	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
3	2	3adant3 1016	. . . . . . 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 484	. . . . . 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 1010	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w S B )
6	4	simp1d 1008	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w e. RR* )
7		simp2 997	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> B e. RR* )
8	7	adantr 465	. . . . . 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 661	. . . . 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 2878	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A. w e. ( A O B ) w <_ B )
13	6	ex 434	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( w e. ( A O B ) -> w e. RR* ) )
14	13	ssrdv 3510	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) C_ RR* )
15		supxrleub 11514	. . . 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 661	. . 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 232	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) <_ B )
18		simprl 755	. . . . . 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 725	. . . . . . . 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 11183	. . . . . . . . . . 11 |- ( w e. QQ -> w e. RR )
21	20	rexrd 9639	. . . . . . . . . 10 |- ( w e. QQ -> w e. RR* )
22	21	ad2antlr 726	. . . . . . . . 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 996	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A e. RR* )
24	23	ad2antrr 725	. . . . . . . . . . 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 11502	. . . . . . . . . . . . 13 |- ( ( A O B ) C_ RR* -> sup ( ( A O B ) , RR* , < ) e. RR* )
26	14, 25	syl 16	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) e. RR* )
27	26	ad2antrr 725	. . . . . . . . . . 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 998	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) =/= (/) )
29		n0 3794	. . . . . . . . . . . . . 14 |- ( ( A O B ) =/= (/) <-> E. w w e. ( A O B ) )
30	28, 29	sylib 196	. . . . . . . . . . . . 13 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> E. w w e. ( A O B ) )
31	23	adantr 465	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A e. RR* )
32	26	adantr 465	. . . . . . . . . . . . . 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 1009	. . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . 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 11512	. . . . . . . . . . . . . . 15 |- ( ( ( A O B ) C_ RR* /\ w e. ( A O B ) ) -> w <_ sup ( ( A O B ) , RR* , < ) )
38	14, 37	sylan 471	. . . . . . . . . . . . . 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 11361	. . . . . . . . . . . . 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 1702	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A <_ sup ( ( A O B ) , RR* , < ) )
41	40	ad2antrr 725	. . . . . . . . . . 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 11359	. . . . . . . . . 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 661	. . . . . . . . . 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 756	. . . . . . . . . 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 725	. . . . . . . . . . 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 661	. . . . . . . . . 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 725	. . . . . . . . 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 1179	. . . . . . . 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 661	. . . . . . 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 9648	. . . . . . . 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 661	. . . . . . 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 210	. . . . . 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 576	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) -> -. ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
58	57	nrexdv 2920	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
59		qbtwnxr 11395	. . . . . 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 1198	. . . . 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 661	. . . 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 177	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. sup ( ( A O B ) , RR* , < ) < B )
63		xrlenlt 9648	. . . 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 661	. . 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 232	. 2 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> B <_ sup ( ( A O B ) , RR* , < ) )
66		xrletri3 11354	. . 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 661	. 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 920	1 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> sup ( ( A O B ) , RR* , < ) = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6282   |-> cmpt2 6284   supcsup 7896   RR*cxr 9623   < clt 9624   <_ cle 9625   QQcq 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179

This theorem is referenced by:  ioopnfsup  11954  icopnfsup  11955  bndth  21190  ioorf  21714  ioorinv2  21716  ioossioobi  31121