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Theorem infrenegsupex 8991
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 7486 . . . . . 6  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 271 . . . . 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 6639 . . . 4  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  RR )
54recnd 7437 . . 3  |-  ( ph  -> inf ( A ,  RR ,  <  )  e.  CC )
65negnegd 7705 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = inf ( A ,  RR ,  <  ) )
7 negeq 7596 . . . . . . . . 9  |-  ( w  =  z  ->  -u w  =  -u z )
87cbvmptv 3902 . . . . . . . 8  |-  ( w  e.  RR  |->  -u w
)  =  ( z  e.  RR  |->  -u z
)
98mptpreima 4881 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
10 eqid 2085 . . . . . . . . . 10  |-  ( w  e.  RR  |->  -u w
)  =  ( w  e.  RR  |->  -u w
)
1110negiso 8328 . . . . . . . . 9  |-  ( ( w  e.  RR  |->  -u w )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
1211simpri 111 . . . . . . . 8  |-  `' ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w )
1312imaeq1i 4729 . . . . . . 7  |-  ( `' ( w  e.  RR  |->  -u w ) " A
)  =  ( ( w  e.  RR  |->  -u w ) " A
)
149, 13eqtr3i 2107 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( w  e.  RR  |->  -u w
) " A )
1514supeq1i 6604 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( w  e.  RR  |->  -u w ) " A ) ,  RR ,  <  )
1611simpli 109 . . . . . . . . 9  |-  ( w  e.  RR  |->  -u w
)  Isom  <  ,  `'  <  ( RR ,  RR )
17 isocnv 5533 . . . . . . . . 9  |-  ( ( w  e.  RR  |->  -u w )  Isom  <  ,  `'  <  ( RR ,  RR )  ->  `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR ) )
1816, 17ax-mp 7 . . . . . . . 8  |-  `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )
19 isoeq1 5523 . . . . . . . . 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 7 . . . . . . . 8  |-  ( `' ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( w  e.  RR  |->  -u w )  Isom  `'  <  ,  <  ( RR ,  RR )
)
2118, 20mpbi 143 . . . . . . 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 6634 . . . . . 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 6635 . . . . . 6  |-  ( (
ph  /\  ( f  e.  RR  /\  g  e.  RR ) )  -> 
( f  =  g  <-> 
( -.  f `'  <  g  /\  -.  g `'  <  f ) ) )
2622, 23, 24, 25supisoti 6626 . . . . 5  |-  ( ph  ->  sup ( ( ( w  e.  RR  |->  -u w ) " A
) ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) ) )
2715, 26syl5eq 2129 . . . 4  |-  ( ph  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  ( ( w  e.  RR  |->  -u w
) `  sup ( A ,  RR ,  `'  <  ) ) )
28 df-inf 6601 . . . . . . 7  |- inf ( A ,  RR ,  <  )  =  sup ( A ,  RR ,  `'  <  )
2928eqcomi 2089 . . . . . 6  |-  sup ( A ,  RR ,  `'  <  )  = inf ( A ,  RR ,  <  )
3029fveq2i 5259 . . . . 5  |-  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )
31 eqidd 2086 . . . . . 6  |-  ( ph  ->  ( w  e.  RR  |->  -u w )  =  ( w  e.  RR  |->  -u w ) )
32 negeq 7596 . . . . . . 7  |-  ( w  = inf ( A ,  RR ,  <  )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
3332adantl 271 . . . . . 6  |-  ( (
ph  /\  w  = inf ( A ,  RR ,  <  ) )  ->  -u w  =  -uinf ( A ,  RR ,  <  ) )
345negcld 7701 . . . . . 6  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  e.  CC )
3531, 33, 4, 34fvmptd 5333 . . . . 5  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) ` inf ( A ,  RR ,  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3630, 35syl5eq 2129 . . . 4  |-  ( ph  ->  ( ( w  e.  RR  |->  -u w ) `  sup ( A ,  RR ,  `'  <  ) )  =  -uinf ( A ,  RR ,  <  ) )
3727, 36eqtr2d 2118 . . 3  |-  ( ph  -> 
-uinf ( A ,  RR ,  <  )  =  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
3837negeqd 7598 . 2  |-  ( ph  -> 
-u -uinf ( A ,  RR ,  <  )  = 
-u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )
)
396, 38eqtr3d 2119 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 102    <-> wb 103    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356   {crab 2359    C_ wss 2986   class class class wbr 3814    |-> cmpt 3868   `'ccnv 4403   "cima 4407   ` cfv 4972    Isom wiso 4973   supcsup 6598  infcinf 6599   CCcc 7269   RRcr 7270    < clt 7443   -ucneg 7575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-apti 7381  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-isom 4981  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-sup 6600  df-inf 6601  df-pnf 7445  df-mnf 7446  df-ltxr 7448  df-sub 7576  df-neg 7577
This theorem is referenced by:  supminfex  8994  minmax  10500  infssuzcldc  10741
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