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Theorem infmsup 10304
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 gtso 9452 . . . . . 6  |-  `'  <  Or  RR
21a1i 11 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  `'  <  Or  RR )
3 infm3 10285 . . . . . 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 ) ) )
4 vex 2973 . . . . . . . . . . 11  |-  x  e. 
_V
5 vex 2973 . . . . . . . . . . 11  |-  y  e. 
_V
64, 5brcnv 5018 . . . . . . . . . 10  |-  ( x `'  <  y  <->  y  <  x )
76notbii 296 . . . . . . . . 9  |-  ( -.  x `'  <  y  <->  -.  y  <  x )
87ralbii 2737 . . . . . . . 8  |-  ( A. y  e.  A  -.  x `'  <  y  <->  A. y  e.  A  -.  y  <  x )
95, 4brcnv 5018 . . . . . . . . . 10  |-  ( y `'  <  x  <->  x  <  y )
10 vex 2973 . . . . . . . . . . . 12  |-  w  e. 
_V
115, 10brcnv 5018 . . . . . . . . . . 11  |-  ( y `'  <  w  <->  w  <  y )
1211rexbii 2738 . . . . . . . . . 10  |-  ( E. w  e.  A  y `'  <  w  <->  E. w  e.  A  w  <  y )
139, 12imbi12i 326 . . . . . . . . 9  |-  ( ( y `'  <  x  ->  E. w  e.  A  y `'  <  w )  <-> 
( x  <  y  ->  E. w  e.  A  w  <  y ) )
1413ralbii 2737 . . . . . . . 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 ) )
158, 14anbi12i 692 . . . . . . 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 ) ) )
1615rexbii 2738 . . . . . 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 ) ) )
173, 16sylibr 212 . . . . 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 ) ) )
182, 17supcl 7704 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
1918recnd 9408 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  CC )
2019negnegd 9706 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  -u -u sup ( A ,  RR ,  `'  <  )  =  sup ( A ,  RR ,  `'  <  ) )
21 eqid 2441 . . . . . . . 8  |-  ( z  e.  RR  |->  -u z
)  =  ( z  e.  RR  |->  -u z
)
2221mptpreima 5328 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
2321negiso 10302 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z ) )
2423simpri 459 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )
2524imaeq1i 5163 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  ( ( z  e.  RR  |->  -u z ) " A
)
2622, 25eqtr3i 2463 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( z  e.  RR  |->  -u z
) " A )
2726supeq1i 7693 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )
2823simpli 455 . . . . . . . . 9  |-  ( z  e.  RR  |->  -u z
)  Isom  <  ,  `'  <  ( RR ,  RR )
29 isocnv 6018 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  ->  `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR ) )
3028, 29ax-mp 5 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
31 isoeq1 6007 . . . . . . . . 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 ) ) )
3224, 31ax-mp 5 . . . . . . . 8  |-  ( `' ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )  <->  ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
)
3330, 32mpbi 208 . . . . . . 7  |-  ( z  e.  RR  |->  -u z
)  Isom  `'  <  ,  <  ( RR ,  RR )
3433a1i 11 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  ( z  e.  RR  |->  -u z )  Isom  `'  <  ,  <  ( RR ,  RR )
)
35 simp1 983 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  A  C_  RR )
3634, 35, 17, 2supiso 7718 . . . . 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 ,  `'  <  ) ) )
3727, 36syl5eq 2485 . . . 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 ,  `'  <  ) ) )
38 negeq 9598 . . . . . 6  |-  ( z  =  sup ( A ,  RR ,  `'  <  )  ->  -u z  = 
-u sup ( A ,  RR ,  `'  <  ) )
39 negex 9604 . . . . . 6  |-  -u sup ( A ,  RR ,  `'  <  )  e.  _V
4038, 21, 39fvmpt 5771 . . . . 5  |-  ( sup ( A ,  RR ,  `'  <  )  e.  RR  ->  ( (
z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) )  = 
-u sup ( A ,  RR ,  `'  <  ) )
4118, 40syl 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 ,  `'  <  ) )
4237, 41eqtr2d 2474 . . 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 ,  <  ) )
4342negeqd 9600 . 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 ,  <  ) )
4420, 43eqtr3d 2475 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717    C_ wss 3325   (/)c0 3634   class class class wbr 4289    e. cmpt 4347    Or wor 4636   `'ccnv 4835   "cima 4839   ` cfv 5415    Isom wiso 5416   supcsup 7686   RRcr 9277    < clt 9414    <_ cle 9415   -ucneg 9592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594
This theorem is referenced by:  infmrcl  10305  supminf  10938  mbfinf  21102
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