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Theorem supminfex 9627
Description: A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.)
Hypotheses
Ref Expression
supminfex.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
supminfex.ss  |-  ( ph  ->  A  C_  RR )
Assertion
Ref Expression
supminfex  |-  ( ph  ->  sup ( A ,  RR ,  <  )  = 
-uinf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  ) )
Distinct variable groups:    w, A, x, y, z    ph, x, y, z
Allowed substitution hint:    ph( w)

Proof of Theorem supminfex
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supminfex.ex . . . . 5  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  A  y  <  z ) ) )
2 supminfex.ss . . . . 5  |-  ( ph  ->  A  C_  RR )
31, 2supinfneg 9625 . . . 4  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  {
w  e.  RR  |  -u w  e.  A }  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. z  e.  { w  e.  RR  |  -u w  e.  A } z  < 
y ) ) )
4 ssrab2 3255 . . . . 5  |-  { w  e.  RR  |  -u w  e.  A }  C_  RR
54a1i 9 . . . 4  |-  ( ph  ->  { w  e.  RR  |  -u w  e.  A }  C_  RR )
63, 5infrenegsupex 9624 . . 3  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  ) )
7 elrabi 2905 . . . . . . 7  |-  ( x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  ->  x  e.  RR )
87adantl 277 . . . . . 6  |-  ( (
ph  /\  x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } )  ->  x  e.  RR )
92sselda 3170 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
10 negeq 8180 . . . . . . . . . 10  |-  ( z  =  x  ->  -u z  =  -u x )
1110eleq1d 2258 . . . . . . . . 9  |-  ( z  =  x  ->  ( -u z  e.  { w  e.  RR  |  -u w  e.  A }  <->  -u x  e. 
{ w  e.  RR  |  -u w  e.  A } ) )
1211elrab3 2909 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  <->  -u x  e. 
{ w  e.  RR  |  -u w  e.  A } ) )
13 renegcl 8248 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u x  e.  RR )
14 negeq 8180 . . . . . . . . . . 11  |-  ( w  =  -u x  ->  -u w  =  -u -u x )
1514eleq1d 2258 . . . . . . . . . 10  |-  ( w  =  -u x  ->  ( -u w  e.  A  <->  -u -u x  e.  A ) )
1615elrab3 2909 . . . . . . . . 9  |-  ( -u x  e.  RR  ->  (
-u x  e.  {
w  e.  RR  |  -u w  e.  A }  <->  -u -u x  e.  A
) )
1713, 16syl 14 . . . . . . . 8  |-  ( x  e.  RR  ->  ( -u x  e.  { w  e.  RR  |  -u w  e.  A }  <->  -u -u x  e.  A ) )
18 recn 7974 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
1918negnegd 8289 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u -u x  =  x )
2019eleq1d 2258 . . . . . . . 8  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
2112, 17, 203bitrd 214 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  <->  x  e.  A ) )
2221adantl 277 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  <->  x  e.  A
) )
238, 9, 22eqrdav 2188 . . . . 5  |-  ( ph  ->  { z  e.  RR  |  -u z  e.  {
w  e.  RR  |  -u w  e.  A } }  =  A )
2423supeq1d 7016 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
2524negeqd 8182 . . 3  |-  ( ph  -> 
-u sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  = 
-u sup ( A ,  RR ,  <  ) )
266, 25eqtrd 2222 . 2  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  ) )
27 lttri3 8067 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
2827adantl 277 . . . . 5  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
2928, 3infclti 7052 . . . 4  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  RR )
3029recnd 8016 . . 3  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  CC )
3128, 1supclti 7027 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
3231recnd 8016 . . 3  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  CC )
33 negcon2 8240 . . 3  |-  ( (inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  CC  /\ 
sup ( A ,  RR ,  <  )  e.  CC )  ->  (inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  )  <->  sup ( A ,  RR ,  <  )  = 
-uinf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  ) ) )
3430, 32, 33syl2anc 411 . 2  |-  ( ph  ->  (inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  )  <->  sup ( A ,  RR ,  <  )  =  -uinf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )
) )
3526, 34mpbid 147 1  |-  ( ph  ->  sup ( A ,  RR ,  <  )  = 
-uinf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472    C_ wss 3144   class class class wbr 4018   supcsup 7011  infcinf 7012   CCcc 7839   RRcr 7840    < clt 8022   -ucneg 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-apti 7956  ax-pre-ltadd 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161
This theorem is referenced by: (None)
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