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Theorem infmsup 10313
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 9461 . . . . . 6  |-  `'  <  Or  RR
21a1i 11 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  `'  <  Or  RR )
3 infm3 10294 . . . . . 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 2980 . . . . . . . . . . 11  |-  x  e. 
_V
5 vex 2980 . . . . . . . . . . 11  |-  y  e. 
_V
64, 5brcnv 5027 . . . . . . . . . 10  |-  ( x `'  <  y  <->  y  <  x )
76notbii 296 . . . . . . . . 9  |-  ( -.  x `'  <  y  <->  -.  y  <  x )
87ralbii 2744 . . . . . . . 8  |-  ( A. y  e.  A  -.  x `'  <  y  <->  A. y  e.  A  -.  y  <  x )
95, 4brcnv 5027 . . . . . . . . . 10  |-  ( y `'  <  x  <->  x  <  y )
10 vex 2980 . . . . . . . . . . . 12  |-  w  e. 
_V
115, 10brcnv 5027 . . . . . . . . . . 11  |-  ( y `'  <  w  <->  w  <  y )
1211rexbii 2745 . . . . . . . . . 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 2744 . . . . . . . 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 697 . . . . . . 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 2745 . . . . . 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 7713 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
1918recnd 9417 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  CC )
2019negnegd 9715 . 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 2443 . . . . . . . 8  |-  ( z  e.  RR  |->  -u z
)  =  ( z  e.  RR  |->  -u z
)
2221mptpreima 5336 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
2321negiso 10311 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z ) )
2423simpri 462 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )
2524imaeq1i 5171 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  ( ( z  e.  RR  |->  -u z ) " A
)
2622, 25eqtr3i 2465 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( z  e.  RR  |->  -u z
) " A )
2726supeq1i 7702 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )
2823simpli 458 . . . . . . . . 9  |-  ( z  e.  RR  |->  -u z
)  Isom  <  ,  `'  <  ( RR ,  RR )
29 isocnv 6026 . . . . . . . . 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 6015 . . . . . . . . 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 988 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  A  C_  RR )
3634, 35, 17, 2supiso 7727 . . . . 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 2487 . . . 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 9607 . . . . . 6  |-  ( z  =  sup ( A ,  RR ,  `'  <  )  ->  -u z  = 
-u sup ( A ,  RR ,  `'  <  ) )
39 negex 9613 . . . . . 6  |-  -u sup ( A ,  RR ,  `'  <  )  e.  _V
4038, 21, 39fvmpt 5779 . . . . 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 2476 . . 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 9609 . 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 2477 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724    C_ wss 3333   (/)c0 3642   class class class wbr 4297    e. cmpt 4355    Or wor 4645   `'ccnv 4844   "cima 4848   ` cfv 5423    Isom wiso 5424   supcsup 7695   RRcr 9286    < clt 9423    <_ cle 9424   -ucneg 9601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603
This theorem is referenced by:  infmrcl  10314  supminf  10947  mbfinf  21148
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