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Theorem ixxlb 11686
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
StepHypRef Expression
1 ixx.1 . . . . . . . . 9  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21elixx1 11673 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
323adant3 1034 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
43biimpa 491 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w  e.  RR*  /\  A R w  /\  w S B ) )
54simp1d 1026 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
65ex 440 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
76ssrdv 3450 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
8 infxrcl 11648 . . 3  |-  ( ( A O B ) 
C_  RR*  -> inf ( ( A O B ) , 
RR* ,  <  )  e. 
RR* )
97, 8syl 17 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  -> inf ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
10 simp1 1014 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
11 simprr 771 . . . . . 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* ,  <  ) )
127ad2antrr 737 . . . . . . . 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 11298 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
1413rexrd 9716 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
1514ad2antlr 738 . . . . . . . . 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 769 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  < inf ( ( A O B ) ,  RR* ,  <  )
) )  ->  A  <  w )
1710ad2antrr 737 . . . . . . . . . . 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 ) )
1917, 15, 18syl2anc 671 . . . . . . . . . 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 ) )
2016, 19mpd 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 )
219ad2antrr 737 . . . . . . . . . . 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 1054 . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
24 n0 3753 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
2523, 24sylib 201 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
269adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  -> inf ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
27 simpl2 1018 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
28 infxrlb 11649 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  -> inf ( ( A O B ) ,  RR* ,  <  )  <_  w )
297, 28sylan 478 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  -> inf ( ( A O B ) ,  RR* ,  <  )  <_  w )
304simp3d 1028 . . . . . . . . . . . . . . 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 ) )
325, 27, 31syl2anc 671 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w S B  ->  w  <_  B ) )
3330, 32mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  B )
3426, 5, 27, 29, 33xrletrd 11488 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  -> inf ( ( A O B ) ,  RR* ,  <  )  <_  B )
3525, 34exlimddv 1792 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  -> inf ( ( A O B ) ,  RR* ,  <  )  <_  B )
3635ad2antrr 737 . . . . . . . . . . 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 )
3715, 21, 22, 11, 36xrltletrd 11487 . . . . . . . . . 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 ) )
3915, 22, 38syl2anc 671 . . . . . . . . . 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 ) )
4037, 39mpd 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 )
413ad2antrr 737 . . . . . . . . 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 ) ) )
4215, 20, 40, 41mpbir3and 1197 . . . . . . . 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 ) )
4312, 42, 28syl2anc 671 . . . . . . 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 )
4421, 15xrlenltd 9726 . . . . . . 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* ,  <  ) ) )
4543, 44mpbid 215 . . . . . 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* ,  <  ) )
4611, 45pm2.65da 584 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( A  <  w  /\  w  < inf ( ( A O B ) , 
RR* ,  <  ) ) )
4746nrexdv 2855 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( A  <  w  /\  w  < inf ( ( A O B ) ,  RR* ,  <  ) ) )
48 qbtwnxr 11522 . . . . . 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* ,  <  ) ) )
49483expia 1217 . . . . 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* ,  <  ) ) ) )
5010, 9, 49syl2anc 671 . . . 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* ,  <  ) ) ) )
5147, 50mtod 182 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  A  < inf ( ( A O B ) ,  RR* ,  <  ) )
529, 10, 51xrnltled 9728 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  -> inf ( ( A O B ) ,  RR* ,  <  )  <_  A )
534simp2d 1027 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
5410adantr 471 . . . . . 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 ) )
5654, 5, 55syl2anc 671 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  ( A R w  ->  A  <_  w ) )
5753, 56mpd 15 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  w )
5857ralrimiva 2814 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) A  <_  w
)
59 infxrgelb 11650 . . . 4  |-  ( ( ( A O B )  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_ inf ( ( A O B ) , 
RR* ,  <  )  <->  A. w  e.  ( A O B ) A  <_  w
) )
607, 10, 59syl2anc 671 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A  <_ inf ( ( A O B ) , 
RR* ,  <  )  <->  A. w  e.  ( A O B ) A  <_  w
) )
6158, 60mpbird 240 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_ inf ( ( A O B ) ,  RR* ,  <  ) )
629, 10, 52, 61xrletrid 11481 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  -> inf ( ( A O B ) ,  RR* ,  <  )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753    C_ wss 3416   (/)c0 3743   class class class wbr 4416  (class class class)co 6315    |-> cmpt2 6317  infcinf 7981   RR*cxr 9700    < clt 9701    <_ cle 9702   QQcq 11293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-sup 7982  df-inf 7983  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-n0 10899  df-z 10967  df-uz 11189  df-q 11294
This theorem is referenced by:  ioorf  22574  ioorinv2  22576  ioossioobi  37656
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