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Theorem supminf 11056
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 11055 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
2 ublbneg 11053 . . . . 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 3548 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  C_  RR
4 infmsup 10422 . . . . . 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 1302 . . . . 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 1183 . . 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 3221 . . . . . . . 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 3462 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
11 negeq 9716 . . . . . . . . . . 11  |-  ( w  =  x  ->  -u w  =  -u x )
1211eleq1d 2523 . . . . . . . . . 10  |-  ( w  =  x  ->  ( -u w  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
1312elrab3 3225 . . . . . . . . 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 9786 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
15 negeq 9716 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
1615eleq1d 2523 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
1716elrab3 3225 . . . . . . . . . 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 9486 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
2019negnegd 9824 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
2120eleq1d 2523 . . . . . . . . 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 2452 . . . . . 6  |-  ( A 
C_  RR  ->  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  =  A
)
2524supeq1d 7810 . . . . 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 9718 . . 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 2495 . 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 10423 . . . . . 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 1302 . . . . 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 1183 . . 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 10404 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
34 recn 9486 . . . 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 9486 . . . 4  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
36 negcon2 9776 . . . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    C_ wss 3439   (/)c0 3748   class class class wbr 4403   `'ccnv 4950   supcsup 7804   CCcc 9394   RRcr 9395    < clt 9532    <_ cle 9533   -ucneg 9710
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712
This theorem is referenced by: (None)
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