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Theorem supminf 10321
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 10320 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  { z  e.  RR  |  -u z  e.  A }  =/=  (/) )
2 ublbneg 10318 . . . . 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 3271 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  C_  RR
4 infmsup 9748 . . . . . 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 1264 . . . . 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 463 . . . 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 1146 . . 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 negeq 9060 . . . . . . . . . . 11  |-  ( w  =  x  ->  -u w  =  -u x )
98eleq1d 2362 . . . . . . . . . 10  |-  ( w  =  x  ->  ( -u w  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u x  e.  { z  e.  RR  |  -u z  e.  A }
) )
109elrab 2936 . . . . . . . . 9  |-  ( x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  ( x  e.  RR  /\  -u x  e.  { z  e.  RR  |  -u z  e.  A } ) )
1110simplbi 446 . . . . . . . 8  |-  ( x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  ->  x  e.  RR )
1211adantl 452 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } )  ->  x  e.  RR )
13 ssel2 3188 . . . . . . 7  |-  ( ( A  C_  RR  /\  x  e.  A )  ->  x  e.  RR )
149elrab3 2937 . . . . . . . . 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 }
) )
15 renegcl 9126 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u x  e.  RR )
16 negeq 9060 . . . . . . . . . . . 12  |-  ( z  =  -u x  ->  -u z  =  -u -u x )
1716eleq1d 2362 . . . . . . . . . . 11  |-  ( z  =  -u x  ->  ( -u z  e.  A  <->  -u -u x  e.  A ) )
1817elrab3 2937 . . . . . . . . . 10  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
1915, 18syl 15 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u x  e.  { z  e.  RR  |  -u z  e.  A }  <->  -u -u x  e.  A
) )
20 recn 8843 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  x  e.  CC )
2120negnegd 9164 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -u -u x  =  x )
2221eleq1d 2362 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
2314, 19, 223bitrd 270 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  e.  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  <->  x  e.  A
) )
2423adantl 452 . . . . . . 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
) )
2512, 13, 24eqrdav 2295 . . . . . 6  |-  ( A 
C_  RR  ->  { w  e.  RR  |  -u w  e.  { z  e.  RR  |  -u z  e.  A } }  =  A
)
2625supeq1d 7215 . . . . 5  |-  ( A 
C_  RR  ->  sup ( { w  e.  RR  |  -u w  e.  {
z  e.  RR  |  -u z  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
27263ad2ant1 976 . . . 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 ,  <  ) )
2827negeqd 9062 . . 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 ,  <  ) )
297, 28eqtrd 2328 . 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 ,  <  ) )
30 infmrcl 9749 . . . . . 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 )
313, 30mp3an1 1264 . . . . 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 )
321, 2, 31syl2an 463 . . . 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 )
33323impa 1146 . . 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 )
34 suprcl 9730 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
35 recn 8843 . . . 4  |-  ( sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  RR  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  `'  <  )  e.  CC )
36 recn 8843 . . . 4  |-  ( sup ( A ,  RR ,  <  )  e.  RR  ->  sup ( A ,  RR ,  <  )  e.  CC )
37 negcon2 9116 . . . 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 ,  `'  <  ) ) )
3835, 36, 37syl2an 463 . . 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 ,  `'  <  ) ) )
3933, 34, 38syl2anc 642 . 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 ,  `'  <  ) ) )
4029, 39mpbid 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039   `'ccnv 4704   supcsup 7209   CCcc 8751   RRcr 8752    < clt 8883    <_ cle 8884   -ucneg 9054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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