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Theorem supminf 10929
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 10928 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
2 ublbneg 10926 . . . . 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 3425 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  C_  RR
4 infmsup 10295 . . . . . 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 1294 . . . . 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 474 . . . 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 1175 . . 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 3103 . . . . . . . 8  |-  ( x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  ->  x  e.  RR )
98adantl 463 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } )  ->  x  e.  RR )
10 ssel2 3339 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
11 negeq 9589 . . . . . . . . . . 11  |-  ( w  =  x  ->  -u w  =  -u x )
1211eleq1d 2499 . . . . . . . . . 10  |-  ( w  =  x  ->  ( -u w  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
1312elrab3 3107 . . . . . . . . 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 9659 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
15 negeq 9589 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
1615eleq1d 2499 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
1716elrab3 3107 . . . . . . . . . 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 9359 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
2019negnegd 9697 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
2120eleq1d 2499 . . . . . . . . 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 463 . . . . . . 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 2432 . . . . . 6  |-  ( A 
C_  RR  ->  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  =  A
)
2524supeq1d 7684 . . . . 5  |-  ( A 
C_  RR  ->  sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
26253ad2ant1 1002 . . . 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 9591 . . 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 2465 . 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 10296 . . . . . 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 1294 . . . . 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 474 . . . 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 1175 . . 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 10277 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
34 recn 9359 . . . 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 9359 . . . 4  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
36 negcon2 9649 . . . 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 474 . . 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 654 . 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3316   (/)c0 3625   class class class wbr 4280   `'ccnv 4826   supcsup 7678   CCcc 9267   RRcr 9268    < clt 9405    <_ cle 9406   -ucneg 9583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585
This theorem is referenced by: (None)
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