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Theorem infxrnegsupex 11284
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 9810 . . . . 5  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 277 . . . 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 7035 . . 3  |-  ( ph  -> inf ( A ,  RR* ,  <  )  e.  RR* )
5 xnegneg 9846 . . 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 9840 . . . . . . . . 9  |-  ( w  =  z  ->  -e
w  =  -e
z )
87cbvmptv 4111 . . . . . . . 8  |-  ( w  e.  RR*  |->  -e
w )  =  ( z  e.  RR*  |->  -e
z )
98mptpreima 5134 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  {
z  e.  RR*  |  -e z  e.  A }
10 eqid 2187 . . . . . . . . . 10  |-  ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w )
1110xrnegiso 11283 . . . . . . . . 9  |-  ( ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )  /\  `' ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w ) )
1211simpri 113 . . . . . . . 8  |-  `' ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w )
1312imaeq1i 4979 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  ( ( w  e.  RR*  |->  -e w ) " A )
149, 13eqtr3i 2210 . . . . . 6  |-  { z  e.  RR*  |  -e
z  e.  A }  =  ( ( w  e.  RR*  |->  -e
w ) " A
)
1514supeq1i 7000 . . . . 5  |-  sup ( { z  e.  RR*  | 
-e z  e.  A } ,  RR* ,  <  )  =  sup ( ( ( w  e.  RR*  |->  -e
w ) " A
) ,  RR* ,  <  )
1611simpli 111 . . . . . . . . 9  |-  ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )
17 isocnv 5825 . . . . . . . . 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 5815 . . . . . . . . 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 145 . . . . . . 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 7030 . . . . . 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 7031 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR*  /\  g  e. 
RR* ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 7022 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR*  |->  -e
w ) " A
) ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
2715, 26eqtrid 2232 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
28 df-inf 6997 . . . . . . 7  |- inf ( A ,  RR* ,  <  )  =  sup ( A ,  RR* ,  `'  <  )
2928eqcomi 2191 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  = inf ( A ,  RR* ,  <  )
3029fveq2i 5530 . . . . 5  |-  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) )  =  ( ( w  e.  RR*  |->  -e w ) ` inf ( A ,  RR* ,  <  ) )
31 eqidd 2188 . . . . . 6  |-  ( ph  ->  ( w  e.  RR*  |->  -e w )  =  ( w  e.  RR*  |->  -e w ) )
32 xnegeq 9840 . . . . . . 7  |-  ( w  = inf ( A ,  RR* ,  <  )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
3332adantl 277 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR* ,  <  ) )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
344xnegcld 9868 . . . . . 6  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  e.  RR* )
3531, 33, 4, 34fvmptd 5610 . . . . 5  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
inf ( A ,  RR* ,  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3630, 35eqtrid 2232 . . . 4  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
 sup ( A ,  RR* ,  `'  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3727, 36eqtr2d 2221 . . 3  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  =  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )
)
38 xnegeq 9840 . . 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 2222 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466   {crab 2469    C_ wss 3141   class class class wbr 4015    |-> cmpt 4076   `'ccnv 4637   "cima 4641   ` cfv 5228    Isom wiso 5229   supcsup 6994  infcinf 6995   RR*cxr 8004    < clt 8005    -ecxne 9782
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-apti 7939  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-sup 6996  df-inf 6997  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-sub 8143  df-neg 8144  df-xneg 9785
This theorem is referenced by:  xrminmax  11286
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