MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supminf Unicode version

Theorem supminf 10519
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 10518 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
2 ublbneg 10516 . . . . 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 3388 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  C_  RR
4 infmsup 9942 . . . . . 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 1266 . . . . 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 464 . . . 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 1148 . . 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 3050 . . . . . . . 8  |-  ( x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  ->  x  e.  RR )
98adantl 453 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } )  ->  x  e.  RR )
10 ssel2 3303 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
11 negeq 9254 . . . . . . . . . . 11  |-  ( w  =  x  ->  -u w  =  -u x )
1211eleq1d 2470 . . . . . . . . . 10  |-  ( w  =  x  ->  ( -u w  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
1312elrab3 3053 . . . . . . . . 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 9320 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
15 negeq 9254 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
1615eleq1d 2470 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
1716elrab3 3053 . . . . . . . . . 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 9036 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
2019negnegd 9358 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
2120eleq1d 2470 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
2213, 18, 213bitrd 271 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  x  e.  A
) )
2322adantl 453 . . . . . . 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 2403 . . . . . 6  |-  ( A 
C_  RR  ->  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  =  A
)
2524supeq1d 7409 . . . . 5  |-  ( A 
C_  RR  ->  sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
26253ad2ant1 978 . . . 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 9256 . . 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 2436 . 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 9943 . . . . . 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 1266 . . . . 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 464 . . . 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 1148 . . 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 9924 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
34 recn 9036 . . . 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 9036 . . . 4  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
36 negcon2 9310 . . . 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 464 . . 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 643 . 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 202 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   class class class wbr 4172   `'ccnv 4836   supcsup 7403   CCcc 8944   RRcr 8945    < clt 9076    <_ cle 9077   -ucneg 9248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250
  Copyright terms: Public domain W3C validator