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Theorem infxrnegsupex 11044
Description: The infimum of a set of extended reals  A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
Hypotheses
Ref Expression
infxrnegsupex.ex  |-  ( ph  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR*  (
x  <  y  ->  E. z  e.  A  z  <  y ) ) )
infxrnegsupex.ss  |-  ( ph  ->  A  C_  RR* )
Assertion
Ref Expression
infxrnegsupex  |-  ( ph  -> inf ( A ,  RR* ,  <  )  =  -e sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )
)
Distinct variable groups:    x, A, y, z    ph, x, y, z

Proof of Theorem infxrnegsupex
Dummy variables  f  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrlttri3 9595 . . . . 5  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 275 . . . 4  |-  ( (
ph  /\  ( f  e.  RR*  /\  g  e. 
RR* ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 infxrnegsupex.ex . . . 4  |-  ( ph  ->  E. x  e.  RR*  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR*  (
x  <  y  ->  E. z  e.  A  z  <  y ) ) )
42, 3infclti 6910 . . 3  |-  ( ph  -> inf ( A ,  RR* ,  <  )  e.  RR* )
5 xnegneg 9628 . . 3  |-  (inf ( A ,  RR* ,  <  )  e.  RR*  ->  -e  -einf ( A ,  RR* ,  <  )  = inf ( A ,  RR* ,  <  ) )
64, 5syl 14 . 2  |-  ( ph  -> 
-e  -einf ( A ,  RR* ,  <  )  = inf ( A ,  RR* ,  <  ) )
7 xnegeq 9622 . . . . . . . . 9  |-  ( w  =  z  ->  -e
w  =  -e
z )
87cbvmptv 4024 . . . . . . . 8  |-  ( w  e.  RR*  |->  -e
w )  =  ( z  e.  RR*  |->  -e
z )
98mptpreima 5032 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  {
z  e.  RR*  |  -e z  e.  A }
10 eqid 2139 . . . . . . . . . 10  |-  ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w )
1110xrnegiso 11043 . . . . . . . . 9  |-  ( ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )  /\  `' ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w ) )
1211simpri 112 . . . . . . . 8  |-  `' ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w )
1312imaeq1i 4878 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  ( ( w  e.  RR*  |->  -e w ) " A )
149, 13eqtr3i 2162 . . . . . 6  |-  { z  e.  RR*  |  -e
z  e.  A }  =  ( ( w  e.  RR*  |->  -e
w ) " A
)
1514supeq1i 6875 . . . . 5  |-  sup ( { z  e.  RR*  | 
-e z  e.  A } ,  RR* ,  <  )  =  sup ( ( ( w  e.  RR*  |->  -e
w ) " A
) ,  RR* ,  <  )
1611simpli 110 . . . . . . . . 9  |-  ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )
17 isocnv 5712 . . . . . . . . 9  |-  ( ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )  ->  `' ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  `' ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* )
19 isoeq1 5702 . . . . . . . . 9  |-  ( `' ( w  e.  RR*  |->  -e w )  =  ( w  e.  RR*  |->  -e w )  -> 
( `' ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* ) 
<->  ( w  e.  RR*  |->  -e w )  Isom  `'  <  ,  <  ( RR* ,  RR* ) ) )
2012, 19ax-mp 5 . . . . . . . 8  |-  ( `' ( w  e.  RR*  |->  -e w )  Isom  `'  <  ,  <  ( RR* ,  RR* )  <->  ( w  e.  RR*  |->  -e w ) 
Isom  `'  <  ,  <  (
RR* ,  RR* ) )
2118, 20mpbi 144 . . . . . . 7  |-  ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* )
2221a1i 9 . . . . . 6  |-  ( ph  ->  ( w  e.  RR*  |->  -e w )  Isom  `'  <  ,  <  ( RR* ,  RR* ) )
23 infxrnegsupex.ss . . . . . 6  |-  ( ph  ->  A  C_  RR* )
243cnvinfex 6905 . . . . . 6  |-  ( ph  ->  E. x  e.  RR*  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR*  ( y `'  <  x  ->  E. z  e.  A  y `'  <  z ) ) )
252cnvti 6906 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR*  /\  g  e. 
RR* ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 6897 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR*  |->  -e
w ) " A
) ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
2715, 26syl5eq 2184 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
28 df-inf 6872 . . . . . . 7  |- inf ( A ,  RR* ,  <  )  =  sup ( A ,  RR* ,  `'  <  )
2928eqcomi 2143 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  = inf ( A ,  RR* ,  <  )
3029fveq2i 5424 . . . . 5  |-  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) )  =  ( ( w  e.  RR*  |->  -e w ) ` inf ( A ,  RR* ,  <  ) )
31 eqidd 2140 . . . . . 6  |-  ( ph  ->  ( w  e.  RR*  |->  -e w )  =  ( w  e.  RR*  |->  -e w ) )
32 xnegeq 9622 . . . . . . 7  |-  ( w  = inf ( A ,  RR* ,  <  )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
3332adantl 275 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR* ,  <  ) )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
344xnegcld 9650 . . . . . 6  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  e.  RR* )
3531, 33, 4, 34fvmptd 5502 . . . . 5  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
inf ( A ,  RR* ,  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3630, 35syl5eq 2184 . . . 4  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
 sup ( A ,  RR* ,  `'  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3727, 36eqtr2d 2173 . . 3  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  =  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )
)
38 xnegeq 9622 . . 3  |-  (  -einf ( A ,  RR* ,  <  )  =  sup ( { z  e.  RR*  | 
-e z  e.  A } ,  RR* ,  <  )  ->  -e  -einf ( A ,  RR* ,  <  )  = 
-e sup ( { z  e.  RR*  | 
-e z  e.  A } ,  RR* ,  <  ) )
3937, 38syl 14 . 2  |-  ( ph  -> 
-e  -einf ( A ,  RR* ,  <  )  =  -e sup ( { z  e. 
RR*  |  -e z  e.  A } ,  RR* ,  <  ) )
406, 39eqtr3d 2174 1  |-  ( ph  -> inf ( A ,  RR* ,  <  )  =  -e sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420    C_ wss 3071   class class class wbr 3929    |-> cmpt 3989   `'ccnv 4538   "cima 4542   ` cfv 5123    Isom wiso 5124   supcsup 6869  infcinf 6870   RR*cxr 7811    < clt 7812    -ecxne 9568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-sub 7947  df-neg 7948  df-xneg 9571
This theorem is referenced by:  xrminmax  11046
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