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Theorem infxrnegsupex 10871
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 9424 . . . . 5  |-  ( ( f  e.  RR*  /\  g  e.  RR* )  ->  (
f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
21adantl 273 . . . 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 6825 . . 3  |-  ( ph  -> inf ( A ,  RR* ,  <  )  e.  RR* )
5 xnegneg 9457 . . 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 9451 . . . . . . . . 9  |-  ( w  =  z  ->  -e
w  =  -e
z )
87cbvmptv 3964 . . . . . . . 8  |-  ( w  e.  RR*  |->  -e
w )  =  ( z  e.  RR*  |->  -e
z )
98mptpreima 4968 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  {
z  e.  RR*  |  -e z  e.  A }
10 eqid 2100 . . . . . . . . . 10  |-  ( w  e.  RR*  |->  -e
w )  =  ( w  e.  RR*  |->  -e
w )
1110xrnegiso 10870 . . . . . . . . 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 4814 . . . . . . 7  |-  ( `' ( w  e.  RR*  |->  -e w ) " A )  =  ( ( w  e.  RR*  |->  -e w ) " A )
149, 13eqtr3i 2122 . . . . . 6  |-  { z  e.  RR*  |  -e
z  e.  A }  =  ( ( w  e.  RR*  |->  -e
w ) " A
)
1514supeq1i 6790 . . . . 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 5644 . . . . . . . . 9  |-  ( ( w  e.  RR*  |->  -e
w )  Isom  <  ,  `'  <  ( RR* ,  RR* )  ->  `' ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* ) )
1816, 17ax-mp 7 . . . . . . . 8  |-  `' ( w  e.  RR*  |->  -e
w )  Isom  `'  <  ,  <  ( RR* ,  RR* )
19 isoeq1 5634 . . . . . . . . 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 7 . . . . . . . 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 6820 . . . . . 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 6821 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR*  /\  g  e. 
RR* ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 6812 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR*  |->  -e
w ) " A
) ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
2715, 26syl5eq 2144 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )  =  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) ) )
28 df-inf 6787 . . . . . . 7  |- inf ( A ,  RR* ,  <  )  =  sup ( A ,  RR* ,  `'  <  )
2928eqcomi 2104 . . . . . 6  |-  sup ( A ,  RR* ,  `'  <  )  = inf ( A ,  RR* ,  <  )
3029fveq2i 5356 . . . . 5  |-  ( ( w  e.  RR*  |->  -e
w ) `  sup ( A ,  RR* ,  `'  <  ) )  =  ( ( w  e.  RR*  |->  -e w ) ` inf ( A ,  RR* ,  <  ) )
31 eqidd 2101 . . . . . 6  |-  ( ph  ->  ( w  e.  RR*  |->  -e w )  =  ( w  e.  RR*  |->  -e w ) )
32 xnegeq 9451 . . . . . . 7  |-  ( w  = inf ( A ,  RR* ,  <  )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
3332adantl 273 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR* ,  <  ) )  ->  -e w  =  -einf ( A ,  RR* ,  <  ) )
344xnegcld 9479 . . . . . 6  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  e.  RR* )
3531, 33, 4, 34fvmptd 5434 . . . . 5  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
inf ( A ,  RR* ,  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3630, 35syl5eq 2144 . . . 4  |-  ( ph  ->  ( ( w  e. 
RR*  |->  -e w ) `
 sup ( A ,  RR* ,  `'  <  ) )  =  -einf ( A ,  RR* ,  <  ) )
3727, 36eqtr2d 2133 . . 3  |-  ( ph  -> 
-einf ( A ,  RR* ,  <  )  =  sup ( { z  e.  RR*  |  -e
z  e.  A } ,  RR* ,  <  )
)
38 xnegeq 9451 . . 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 2134 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 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {crab 2379    C_ wss 3021   class class class wbr 3875    |-> cmpt 3929   `'ccnv 4476   "cima 4480   ` cfv 5059    Isom wiso 5060   supcsup 6784  infcinf 6785   RR*cxr 7671    < clt 7672    -ecxne 9397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-sub 7806  df-neg 7807  df-xneg 9400
This theorem is referenced by:  xrminmax  10873
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