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Theorem infmsup 9942
Description: The infimum (expressed as supremum with converse 'less-than') of a set of reals  A is the negative of the supremum of the negatives of its elements. The antecedent ensures that  A is nonempty and has a lower bound. (Contributed by NM, 14-Jun-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
infmsup  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
Distinct variable group:    x, y, z, A

Proof of Theorem infmsup
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ltso 9112 . . . . . . 7  |-  <  Or  RR
2 cnvso 5370 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
31, 2mpbi 200 . . . . . 6  |-  `'  <  Or  RR
43a1i 11 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  `'  <  Or  RR )
5 infm3 9923 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  E. x  e.  RR  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR  ( x  < 
y  ->  E. w  e.  A  w  <  y ) ) )
6 vex 2919 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2919 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7brcnv 5014 . . . . . . . . . 10  |-  ( x `'  <  y  <->  y  <  x )
98notbii 288 . . . . . . . . 9  |-  ( -.  x `'  <  y  <->  -.  y  <  x )
109ralbii 2690 . . . . . . . 8  |-  ( A. y  e.  A  -.  x `'  <  y  <->  A. y  e.  A  -.  y  <  x )
117, 6brcnv 5014 . . . . . . . . . 10  |-  ( y `'  <  x  <->  x  <  y )
12 vex 2919 . . . . . . . . . . . 12  |-  w  e. 
_V
137, 12brcnv 5014 . . . . . . . . . . 11  |-  ( y `'  <  w  <->  w  <  y )
1413rexbii 2691 . . . . . . . . . 10  |-  ( E. w  e.  A  y `'  <  w  <->  E. w  e.  A  w  <  y )
1511, 14imbi12i 317 . . . . . . . . 9  |-  ( ( y `'  <  x  ->  E. w  e.  A  y `'  <  w )  <-> 
( x  <  y  ->  E. w  e.  A  w  <  y ) )
1615ralbii 2690 . . . . . . . 8  |-  ( A. y  e.  RR  (
y `'  <  x  ->  E. w  e.  A  y `'  <  w )  <->  A. y  e.  RR  ( x  <  y  ->  E. w  e.  A  w  <  y ) )
1710, 16anbi12i 679 . . . . . . 7  |-  ( ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR  ( y `'  <  x  ->  E. w  e.  A  y `'  <  w ) )  <->  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR  ( x  < 
y  ->  E. w  e.  A  w  <  y ) ) )
1817rexbii 2691 . . . . . 6  |-  ( E. x  e.  RR  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR  ( y `'  <  x  ->  E. w  e.  A  y `'  <  w ) )  <->  E. x  e.  RR  ( A. y  e.  A  -.  y  <  x  /\  A. y  e.  RR  (
x  <  y  ->  E. w  e.  A  w  <  y ) ) )
195, 18sylibr 204 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  E. x  e.  RR  ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  RR  (
y `'  <  x  ->  E. w  e.  A  y `'  <  w ) ) )
204, 19supcl 7419 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
2120recnd 9070 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  CC )
2221negnegd 9358 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  -u -u sup ( A ,  RR ,  `'  <  )  =  sup ( A ,  RR ,  `'  <  ) )
23 eqid 2404 . . . . . . . 8  |-  ( z  e.  RR  |->  -u z
)  =  ( z  e.  RR  |->  -u z
)
2423mptpreima 5322 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
2523negiso 9940 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z ) )
2625simpri 449 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )
2726imaeq1i 5159 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  ( ( z  e.  RR  |->  -u z ) " A
)
2824, 27eqtr3i 2426 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( z  e.  RR  |->  -u z
) " A )
2928supeq1i 7410 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )
3025simpli 445 . . . . . . . . 9  |-  ( z  e.  RR  |->  -u z
)  Isom  <  ,  `'  <  ( RR ,  RR )
31 isocnv 6009 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  ->  `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR ) )
3230, 31ax-mp 8 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
33 isoeq1 5998 . . . . . . . . 9  |-  ( `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )  ->  ( `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR ) ) )
3426, 33ax-mp 8 . . . . . . . 8  |-  ( `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
)
3532, 34mpbi 200 . . . . . . 7  |-  ( z  e.  RR  |->  -u z
)  Isom  `'  <  ,  <  ( RR ,  RR )
3635a1i 11 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
)
37 simp1 957 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  A  C_  RR )
3836, 37, 19, 4supiso 7433 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )  =  ( ( z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) ) )
3929, 38syl5eq 2448 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  ( ( z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) ) )
40 negeq 9254 . . . . . 6  |-  ( z  =  sup ( A ,  RR ,  `'  <  )  ->  -u z  = 
-u sup ( A ,  RR ,  `'  <  ) )
41 negex 9260 . . . . . 6  |-  -u sup ( A ,  RR ,  `'  <  )  e.  _V
4240, 23, 41fvmpt 5765 . . . . 5  |-  ( sup ( A ,  RR ,  `'  <  )  e.  RR  ->  ( (
z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) )  = 
-u sup ( A ,  RR ,  `'  <  ) )
4320, 42syl 16 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  ( (
z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) )  = 
-u sup ( A ,  RR ,  `'  <  ) )
4439, 43eqtr2d 2437 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  -u sup ( A ,  RR ,  `'  <  )  =  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
4544negeqd 9256 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  -u -u sup ( A ,  RR ,  `'  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
4622, 45eqtr3d 2438 1  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  =  -u sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   (/)c0 3588   class class class wbr 4172    e. cmpt 4226    Or wor 4462   `'ccnv 4836   "cima 4840   ` cfv 5413    Isom wiso 5414   supcsup 7403   RRcr 8945    < clt 9076    <_ cle 9077   -ucneg 9248
This theorem is referenced by:  infmrcl  9943  supminf  10519  mbfinf  19510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250
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