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Theorem supminf 10934
Description: The supremum of a bounded-above set of reals is the negation of the supremum of that set's image under negation. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
supminf  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) )
Distinct variable group:    x, A, y, z

Proof of Theorem supminf
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 negn0 10933 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
2 ublbneg 10931 . . . . 5  |-  ( E. x  e.  RR  A. y  e.  A  y  <_  x  ->  E. x  e.  RR  A. y  e. 
{ z  e.  RR  |  -u z  e.  A } x  <_  y )
3 ssrab2 3432 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  C_  RR
4 infmsup 10300 . . . . . 6  |-  ( ( { z  e.  RR  |  -u z  e.  A }  C_  RR  /\  {
z  e.  RR  |  -u z  e.  A }  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
{ z  e.  RR  |  -u z  e.  A } x  <_  y )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  ) )
53, 4mp3an1 1301 . . . . 5  |-  ( ( { z  e.  RR  |  -u z  e.  A }  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
{ z  e.  RR  |  -u z  e.  A } x  <_  y )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  ) )
61, 2, 5syl2an 477 . . . 4  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  ) )
763impa 1182 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  ) )
8 elrabi 3109 . . . . . . . 8  |-  ( x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  ->  x  e.  RR )
98adantl 466 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } )  ->  x  e.  RR )
10 ssel2 3346 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
11 negeq 9594 . . . . . . . . . . 11  |-  ( w  =  x  ->  -u w  =  -u x )
1211eleq1d 2504 . . . . . . . . . 10  |-  ( w  =  x  ->  ( -u w  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
1312elrab3 3113 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  -u x  e.  {
z  e.  RR  |  -u z  e.  A }
) )
14 renegcl 9664 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
15 negeq 9594 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
1615eleq1d 2504 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
1716elrab3 3113 . . . . . . . . . 10  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
1814, 17syl 16 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
19 recn 9364 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
2019negnegd 9702 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
2120eleq1d 2504 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
2213, 18, 213bitrd 279 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  x  e.  A
) )
2322adantl 466 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  RR )  ->  (
x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  x  e.  A
) )
249, 10, 23eqrdav 2437 . . . . . 6  |-  ( A 
C_  RR  ->  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  =  A
)
2524supeq1d 7688 . . . . 5  |-  ( A 
C_  RR  ->  sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
26253ad2ant1 1009 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
2726negeqd 9596 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  -u sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  ) )
287, 27eqtrd 2470 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( A ,  RR ,  <  ) )
29 infmrcl 10301 . . . . . 6  |-  ( ( { z  e.  RR  |  -u z  e.  A }  C_  RR  /\  {
z  e.  RR  |  -u z  e.  A }  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
{ z  e.  RR  |  -u z  e.  A } x  <_  y )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR )
303, 29mp3an1 1301 . . . . 5  |-  ( ( { z  e.  RR  |  -u z  e.  A }  =/=  (/)  /\  E. x  e.  RR  A. y  e. 
{ z  e.  RR  |  -u z  e.  A } x  <_  y )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR )
311, 2, 30syl2an 477 . . . 4  |-  ( ( ( A  C_  RR  /\  A  =/=  (/) )  /\  E. x  e.  RR  A. y  e.  A  y  <_  x )  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR )
32313impa 1182 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR )
33 suprcl 10282 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
34 recn 9364 . . . 4  |-  ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  CC )
35 recn 9364 . . . 4  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
36 negcon2 9654 . . . 4  |-  ( ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  CC  /\  sup ( A ,  RR ,  <  )  e.  CC )  ->  ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( A ,  RR ,  <  )  <->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) ) )
3734, 35, 36syl2an 477 . . 3  |-  ( ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR  /\  sup ( A ,  RR ,  <  )  e.  RR )  ->  ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( A ,  RR ,  <  )  <->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) ) )
3832, 33, 37syl2anc 661 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  =  -u sup ( A ,  RR ,  <  )  <->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) ) )
3928, 38mpbid 210 1  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   {crab 2714    C_ wss 3323   (/)c0 3632   class class class wbr 4287   `'ccnv 4834   supcsup 7682   CCcc 9272   RRcr 9273    < clt 9410    <_ cle 9411   -ucneg 9588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590
This theorem is referenced by: (None)
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