Theorem ixxub 11408

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 11396	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( w e. ( A O B ) <-> ( w e. RR* /\ A R w /\ w S B ) ) )
3	2	3adant3 1008	. . . . . . 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 1002	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w S B )
6	4	simp1d 1000	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) /\ w e. ( A O B ) ) -> w e. RR* )
7		simp2 989	. . . . . . 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 2881	. . 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 3446	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) C_ RR* )
15		supxrleub 11376	. . . 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 11045	. . . . . . . . . . 11 |- ( w e. QQ -> w e. RR )
21	20	rexrd 9520	. . . . . . . . . 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 988	. . . . . . . . . . . 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 11364	. . . . . . . . . . . . 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 990	. . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> ( A O B ) =/= (/) )
29		n0 3730	. . . . . . . . . . . . . 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 1001	. . . . . . . . . . . . . . 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 11374	. . . . . . . . . . . . . . 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 11223	. . . . . . . . . . . . 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 1693	. . . . . . . . . . . 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 11221	. . . . . . . . . 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 1171	. . . . . . . 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 9529	. . . . . . . 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 2901	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( A O B ) =/= (/) ) -> -. E. w e. QQ ( sup ( ( A O B ) , RR* , < ) < w /\ w < B ) )
59		qbtwnxr 11257	. . . . . 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 1190	. . . . 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 9529	. . . 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 11216	. . 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 913	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 965   = wceq 1370   E.wex 1587   e. wcel 1757   =/= wne 2641   A.wral 2792   E.wrex 2793   {crab 2796   C_ wss 3412   (/)c0 3721   class class class wbr 4376  (class class class)co 6176   |-> cmpt2 6178   supcsup 7777   RR*cxr 9504   < clt 9505   <_ cle 9506   QQcq 11040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-q 11041

This theorem is referenced by:  ioopnfsup  11790  icopnfsup  11791  bndth  20632  ioorf  21155  ioorinv2  21157