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Theorem infrenegsupex 9570
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 8014 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 277 . . . . 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 7015 . . . 4  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  RR )
54recnd 7963 . . 3  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  CC )
65negnegd 8236 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = inf ( A ,  RR ,  <  ) )
7 negeq 8127 . . . . . . . . 9  |-  ( w  =  z  ->  -u w  =  -u z )
87cbvmptv 4096 . . . . . . . 8  |-  ( w  e.  RR  |->  -u w
)  =  ( z  e.  RR  |->  -u z
)
98mptpreima 5117 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
10 eqid 2177 . . . . . . . . . 10  |-  ( w  e.  RR  |->  -u w
)  =  ( w  e.  RR  |->  -u w
)
1110negiso 8888 . . . . . . . . 9  |-  ( ( w  e.  RR  |->  -u w )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
1211simpri 113 . . . . . . . 8  |-  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w )
1312imaeq1i 4962 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  ( ( w  e.  RR  |->  -u w ) " A
)
149, 13eqtr3i 2200 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( w  e.  RR  |->  -u w
) " A )
1514supeq1i 6980 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( w  e.  RR  |->  -u w ) " A ) ,  RR ,  <  )
1611simpli 111 . . . . . . . . 9  |-  ( w  e.  RR  |->  -u w
)  Isom  <  ,  `'  <  ( RR ,  RR )
17 isocnv 5805 . . . . . . . . 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 5795 . . . . . . . . 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 145 . . . . . . 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 7010 . . . . . 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 7011 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 7002 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR  |->  -u w ) " A
) ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) ) )
2715, 26eqtrid 2222 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w
) `  sup ( A ,  RR ,  `'  <  ) ) )
28 df-inf 6977 . . . . . . 7  |- inf ( A ,  RR ,  <  )  =  sup ( A ,  RR ,  `'  <  )
2928eqcomi 2181 . . . . . 6  |-  sup ( A ,  RR ,  `'  <  )  = inf ( A ,  RR ,  <  )
3029fveq2i 5513 . . . . 5  |-  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )
31 eqidd 2178 . . . . . 6  |-  ( ph  ->  ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
32 negeq 8127 . . . . . . 7  |-  ( w  = inf ( A ,  RR ,  <  )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
3332adantl 277 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR ,  <  ) )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
345negcld 8232 . . . . . 6  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  e.  CC )
3531, 33, 4, 34fvmptd 5592 . . . . 5  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3630, 35eqtrid 2222 . . . 4  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3727, 36eqtr2d 2211 . . 3  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  =  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
3837negeqd 8129 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = 
-u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
396, 38eqtr3d 2212 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3129   class class class wbr 4000    |-> cmpt 4061   `'ccnv 4621   "cima 4625   ` cfv 5211    Isom wiso 5212   supcsup 6974  infcinf 6975   CCcc 7787   RRcr 7788    < clt 7969   -ucneg 8106
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-apti 7904  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-sup 6976  df-inf 6977  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-sub 8107  df-neg 8108
This theorem is referenced by:  supminfex  9573  minmax  11209  infssuzcldc  11922
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