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Theorem infmsup 9748
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 8919 . . . . . . 7  |-  <  Or  RR
2 cnvso 5230 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
31, 2mpbi 199 . . . . . 6  |-  `'  <  Or  RR
43a1i 10 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  `'  <  Or  RR )
5 infm3 9729 . . . . . 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 2804 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7brcnv 4880 . . . . . . . . . 10  |-  ( x `'  <  y  <->  y  <  x )
98notbii 287 . . . . . . . . 9  |-  ( -.  x `'  <  y  <->  -.  y  <  x )
109ralbii 2580 . . . . . . . 8  |-  ( A. y  e.  A  -.  x `'  <  y  <->  A. y  e.  A  -.  y  <  x )
117, 6brcnv 4880 . . . . . . . . . 10  |-  ( y `'  <  x  <->  x  <  y )
12 vex 2804 . . . . . . . . . . . 12  |-  w  e. 
_V
137, 12brcnv 4880 . . . . . . . . . . 11  |-  ( y `'  <  w  <->  w  <  y )
1413rexbii 2581 . . . . . . . . . 10  |-  ( E. w  e.  A  y `'  <  w  <->  E. w  e.  A  w  <  y )
1511, 14imbi12i 316 . . . . . . . . 9  |-  ( ( y `'  <  x  ->  E. w  e.  A  y `'  <  w )  <-> 
( x  <  y  ->  E. w  e.  A  w  <  y ) )
1615ralbii 2580 . . . . . . . 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 678 . . . . . . 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 2581 . . . . . 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 203 . . . . 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 7225 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
2120recnd 8877 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  CC )
2221negnegd 9164 . 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 2296 . . . . . . . 8  |-  ( z  e.  RR  |->  -u z
)  =  ( z  e.  RR  |->  -u z
)
2423mptpreima 5182 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
2523negiso 9746 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z ) )
2625simpri 448 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )
2726imaeq1i 5025 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  ( ( z  e.  RR  |->  -u z ) " A
)
2824, 27eqtr3i 2318 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( z  e.  RR  |->  -u z
) " A )
2928supeq1i 7216 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )
3025simpli 444 . . . . . . . . 9  |-  ( z  e.  RR  |->  -u z
)  Isom  <  ,  `'  <  ( RR ,  RR )
31 isocnv 5843 . . . . . . . . 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 5832 . . . . . . . . 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 199 . . . . . . 7  |-  ( z  e.  RR  |->  -u z
)  Isom  `'  <  ,  <  ( RR ,  RR )
3635a1i 10 . . . . . 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 955 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  A  C_  RR )
3836, 37, 19, 4supiso 7239 . . . . 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 2340 . . . 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 9060 . . . . . 6  |-  ( z  =  sup ( A ,  RR ,  `'  <  )  ->  -u z  = 
-u sup ( A ,  RR ,  `'  <  ) )
41 negex 9066 . . . . . 6  |-  -u sup ( A ,  RR ,  `'  <  )  e.  _V
4240, 23, 41fvmpt 5618 . . . . 5  |-  ( sup ( A ,  RR ,  `'  <  )  e.  RR  ->  ( (
z  e.  RR  |->  -u z ) `  sup ( A ,  RR ,  `'  <  ) )  = 
-u sup ( A ,  RR ,  `'  <  ) )
4320, 42syl 15 . . . 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 2329 . . 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 9062 . 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 2330 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093    Or wor 4329   `'ccnv 4704   "cima 4708   ` cfv 5271    Isom wiso 5272   supcsup 7209   RRcr 8752    < clt 8883    <_ cle 8884   -ucneg 9054
This theorem is referenced by:  infmrcl  9749  supminf  10321  mbfinf  19036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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