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Theorem supminfex 9054
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 9052 . . . 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 3104 . . . . 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 9051 . . 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 2766 . . . . . . 7  |-  ( x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  ->  x  e.  RR )
87adantl 271 . . . . . 6  |-  ( (
ph  /\  x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } )  ->  x  e.  RR )
92sselda 3023 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
10 negeq 7654 . . . . . . . . . 10  |-  ( z  =  x  ->  -u z  =  -u x )
1110eleq1d 2156 . . . . . . . . 9  |-  ( z  =  x  ->  ( -u z  e.  { w  e.  RR  |  -u w  e.  A }  <->  -u x  e. 
{ w  e.  RR  |  -u w  e.  A } ) )
1211elrab3 2770 . . . . . . . 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 7722 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u x  e.  RR )
14 negeq 7654 . . . . . . . . . . 11  |-  ( w  =  -u x  ->  -u w  =  -u -u x )
1514eleq1d 2156 . . . . . . . . . 10  |-  ( w  =  -u x  ->  ( -u w  e.  A  <->  -u -u x  e.  A ) )
1615elrab3 2770 . . . . . . . . 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 7454 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
1918negnegd 7763 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u -u x  =  x )
2019eleq1d 2156 . . . . . . . 8  |-  ( x  e.  RR  ->  ( -u -u x  e.  A  <->  x  e.  A ) )
2112, 17, 203bitrd 212 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  e.  { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } }  <->  x  e.  A ) )
2221adantl 271 . . . . . 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 2087 . . . . 5  |-  ( ph  ->  { z  e.  RR  |  -u z  e.  {
w  e.  RR  |  -u w  e.  A } }  =  A )
2423supeq1d 6661 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
2524negeqd 7656 . . 3  |-  ( ph  -> 
-u sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  = 
-u sup ( A ,  RR ,  <  ) )
266, 25eqtrd 2120 . 2  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  ) )
27 lttri3 7544 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
2827adantl 271 . . . . 5  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
2928, 3infclti 6697 . . . 4  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  RR )
3029recnd 7495 . . 3  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  CC )
3128, 1supclti 6672 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
3231recnd 7495 . . 3  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  CC )
33 negcon2 7714 . . 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 403 . 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 145 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2997   class class class wbr 3837   supcsup 6656  infcinf 6657   CCcc 7327   RRcr 7328    < clt 7501   -ucneg 7633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-apti 7439  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-sup 6658  df-inf 6659  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635
This theorem is referenced by: (None)
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