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Theorem supminfex 8818
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 8816 . . . 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 3088 . . . . 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 8815 . . 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 2754 . . . . . . 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 3008 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
10 negeq 7420 . . . . . . . . . 10  |-  ( z  =  x  ->  -u z  =  -u x )
1110eleq1d 2151 . . . . . . . . 9  |-  ( z  =  x  ->  ( -u z  e.  { w  e.  RR  |  -u w  e.  A }  <->  -u x  e. 
{ w  e.  RR  |  -u w  e.  A } ) )
1211elrab3 2758 . . . . . . . 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 7488 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u x  e.  RR )
14 negeq 7420 . . . . . . . . . . 11  |-  ( w  =  -u x  ->  -u w  =  -u -u x )
1514eleq1d 2151 . . . . . . . . . 10  |-  ( w  =  -u x  ->  ( -u w  e.  A  <->  -u -u x  e.  A ) )
1615elrab3 2758 . . . . . . . . 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 7220 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
1918negnegd 7529 . . . . . . . . 9  |-  ( x  e.  RR  ->  -u -u x  =  x )
2019eleq1d 2151 . . . . . . . 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 2082 . . . . 5  |-  ( ph  ->  { z  e.  RR  |  -u z  e.  {
w  e.  RR  |  -u w  e.  A } }  =  A )
2423supeq1d 6494 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  =  sup ( A ,  RR ,  <  ) )
2524negeqd 7422 . . 3  |-  ( ph  -> 
-u sup ( { z  e.  RR  |  -u z  e.  { w  e.  RR  |  -u w  e.  A } } ,  RR ,  <  )  = 
-u sup ( A ,  RR ,  <  ) )
266, 25eqtrd 2115 . 2  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  =  -u sup ( A ,  RR ,  <  ) )
27 lttri3 7310 . . . . . 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 6530 . . . 4  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  RR )
3029recnd 7261 . . 3  |-  ( ph  -> inf ( { w  e.  RR  |  -u w  e.  A } ,  RR ,  <  )  e.  CC )
3128, 1supclti 6505 . . . 4  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  RR )
3231recnd 7261 . . 3  |-  ( ph  ->  sup ( A ,  RR ,  <  )  e.  CC )
33 negcon2 7480 . . 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 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   {crab 2357    C_ wss 2982   class class class wbr 3805   supcsup 6489  infcinf 6490   CCcc 7093   RRcr 7094    < clt 7267   -ucneg 7399
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-apti 7205  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-ltxr 7272  df-sub 7400  df-neg 7401
This theorem is referenced by: (None)
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