Theorem ixxub 11521

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 11509	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
3	2	3adant3 1017	. . . . . . 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 482	. . . . . 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 1011	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w S B )
6	4	simp1d 1009	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w e. RR* )
7		simp2 998	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> B e. RR* )
8	7	adantr 463	. . . . . 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 659	. . . . 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 2817	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A. w e. ( A O B ) w <_ B )
13	6	ex 432	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( w e. ( A O B ) -> w e. RR* ) )
14	13	ssrdv 3447	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) C_ RR* )
15		supxrleub 11489	. . . 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 659	. . 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 756	. . . . . 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 724	. . . . . . . 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 11150	. . . . . . . . . . 11 |- ( w e. QQ -> w e. RR )
21	20	rexrd 9593	. . . . . . . . . 10 |- ( w e. QQ -> w e. RR* )
22	21	ad2antlr 725	. . . . . . . . 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 997	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A e. RR* )
24	23	ad2antrr 724	. . . . . . . . . . 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 11477	. . . . . . . . . . . . 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 724	. . . . . . . . . . 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 999	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) =/= (/) )
29		n0 3747	. . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> A e. RR* )
32	26	adantr 463	. . . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . . 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 659	. . . . . . . . . . . . . . 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 11487	. . . . . . . . . . . . . . 15 |- ( ( ( A O B ) C_ RR* /\ w e. ( A O B ) ) -> w <_ sup ( ( A O B ) , RR* , < ) )
38	14, 37	sylan 469	. . . . . . . . . . . . . 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 11336	. . . . . . . . . . . . 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 1747	. . . . . . . . . . . 12 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> A <_ sup ( ( A O B ) , RR* , < ) )
41	40	ad2antrr 724	. . . . . . . . . . 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 11334	. . . . . . . . . 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 659	. . . . . . . . . 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 758	. . . . . . . . . 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 724	. . . . . . . . . . 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 659	. . . . . . . . . 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 724	. . . . . . . . 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 1180	. . . . . . . 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 659	. . . . . . 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 9602	. . . . . . . 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 659	. . . . . . 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 574	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. QQ ) -> -. ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
58	57	nrexdv 2859	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
59		qbtwnxr 11370	. . . . . 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 1199	. . . . 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 659	. . . 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 9602	. . . 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 659	. . 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 11329	. . 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 659	. 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 923	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 367   /\ w3a 974   = wceq 1405   E.wex 1633   e. wcel 1842   =/= wne 2598   A.wral 2753   E.wrex 2754   {crab 2757   C_ wss 3413   (/)c0 3737   class class class wbr 4394  (class class class)co 6234   |-> cmpt2 6236   supcsup 7854   RR*cxr 9577   < clt 9578   <_ cle 9579   QQcq 11145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-n0 10757  df-z 10826  df-uz 11046  df-q 11146

This theorem is referenced by:  ioopnfsup  11942  icopnfsup  11943  bndth  21642  ioorf  22166  ioorinv2  22168  ioossioobi  36907