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Theorem infrenegsupex 9415
Description: The infimum of a set of reals  A is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 14-Jan-2022.)
Hypotheses
Ref Expression
infrenegsupex.ex  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. z  e.  A  z  <  y ) ) )
infrenegsupex.ss  |-  ( ph  ->  A  C_  RR )
Assertion
Ref Expression
infrenegsupex  |-  ( ph  -> inf ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
Distinct variable groups:    x, A, y, z    ph, x, y, z

Proof of Theorem infrenegsupex
Dummy variables  f  g  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 7867 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 275 . . . . 5  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
3 infrenegsupex.ex . . . . 5  |-  ( 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 6917 . . . 4  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  RR )
54recnd 7817 . . 3  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  CC )
65negnegd 8087 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = inf ( A ,  RR ,  <  ) )
7 negeq 7978 . . . . . . . . 9  |-  ( w  =  z  ->  -u w  =  -u z )
87cbvmptv 4031 . . . . . . . 8  |-  ( w  e.  RR  |->  -u w
)  =  ( z  e.  RR  |->  -u z
)
98mptpreima 5039 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
10 eqid 2140 . . . . . . . . . 10  |-  ( w  e.  RR  |->  -u w
)  =  ( w  e.  RR  |->  -u w
)
1110negiso 8736 . . . . . . . . 9  |-  ( ( w  e.  RR  |->  -u w )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
1211simpri 112 . . . . . . . 8  |-  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w )
1312imaeq1i 4885 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  ( ( w  e.  RR  |->  -u w ) " A
)
149, 13eqtr3i 2163 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( w  e.  RR  |->  -u w
) " A )
1514supeq1i 6882 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( w  e.  RR  |->  -u w ) " A ) ,  RR ,  <  )
1611simpli 110 . . . . . . . . 9  |-  ( w  e.  RR  |->  -u w
)  Isom  <  ,  `'  <  ( RR ,  RR )
17 isocnv 5719 . . . . . . . . 9  |-  ( ( w  e.  RR  |->  -u w )  Isom  <  ,  `'  <  ( RR ,  RR )  ->  `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR ) )
1816, 17ax-mp 5 . . . . . . . 8  |-  `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )
19 isoeq1 5709 . . . . . . . . 9  |-  ( `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w )  ->  ( `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR ) ) )
2012, 19ax-mp 5 . . . . . . . 8  |-  ( `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )
)
2118, 20mpbi 144 . . . . . . 7  |-  ( w  e.  RR  |->  -u w
)  Isom  `'  <  ,  <  ( RR ,  RR )
2221a1i 9 . . . . . 6  |-  ( ph  ->  ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR ) )
23 infrenegsupex.ss . . . . . 6  |-  ( ph  ->  A  C_  RR )
243cnvinfex 6912 . . . . . 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 6913 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 6904 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR  |->  -u w ) " A
) ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) ) )
2715, 26syl5eq 2185 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w
) `  sup ( A ,  RR ,  `'  <  ) ) )
28 df-inf 6879 . . . . . . 7  |- inf ( A ,  RR ,  <  )  =  sup ( A ,  RR ,  `'  <  )
2928eqcomi 2144 . . . . . 6  |-  sup ( A ,  RR ,  `'  <  )  = inf ( A ,  RR ,  <  )
3029fveq2i 5431 . . . . 5  |-  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )
31 eqidd 2141 . . . . . 6  |-  ( ph  ->  ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
32 negeq 7978 . . . . . . 7  |-  ( w  = inf ( A ,  RR ,  <  )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
3332adantl 275 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR ,  <  ) )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
345negcld 8083 . . . . . 6  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  e.  CC )
3531, 33, 4, 34fvmptd 5509 . . . . 5  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3630, 35syl5eq 2185 . . . 4  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3727, 36eqtr2d 2174 . . 3  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  =  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
3837negeqd 7980 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = 
-u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
396, 38eqtr3d 2175 1  |-  ( ph  -> inf ( A ,  RR ,  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421    C_ wss 3075   class class class wbr 3936    |-> cmpt 3996   `'ccnv 4545   "cima 4549   ` cfv 5130    Isom wiso 5131   supcsup 6876  infcinf 6877   CCcc 7641   RRcr 7642    < clt 7823   -ucneg 7957
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-apti 7758  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-sup 6878  df-inf 6879  df-pnf 7825  df-mnf 7826  df-ltxr 7828  df-sub 7958  df-neg 7959
This theorem is referenced by:  supminfex  9418  minmax  11032  infssuzcldc  11678
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