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Theorem infmsup 9988
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 9158 . . . . . . 7  |-  <  Or  RR
2 cnvso 5413 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
31, 2mpbi 201 . . . . . 6  |-  `'  <  Or  RR
43a1i 11 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  `'  <  Or  RR )
5 infm3 9969 . . . . . 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 2961 . . . . . . . . . . 11  |-  x  e. 
_V
7 vex 2961 . . . . . . . . . . 11  |-  y  e. 
_V
86, 7brcnv 5057 . . . . . . . . . 10  |-  ( x `'  <  y  <->  y  <  x )
98notbii 289 . . . . . . . . 9  |-  ( -.  x `'  <  y  <->  -.  y  <  x )
109ralbii 2731 . . . . . . . 8  |-  ( A. y  e.  A  -.  x `'  <  y  <->  A. y  e.  A  -.  y  <  x )
117, 6brcnv 5057 . . . . . . . . . 10  |-  ( y `'  <  x  <->  x  <  y )
12 vex 2961 . . . . . . . . . . . 12  |-  w  e. 
_V
137, 12brcnv 5057 . . . . . . . . . . 11  |-  ( y `'  <  w  <->  w  <  y )
1413rexbii 2732 . . . . . . . . . 10  |-  ( E. w  e.  A  y `'  <  w  <->  E. w  e.  A  w  <  y )
1511, 14imbi12i 318 . . . . . . . . 9  |-  ( ( y `'  <  x  ->  E. w  e.  A  y `'  <  w )  <-> 
( x  <  y  ->  E. w  e.  A  w  <  y ) )
1615ralbii 2731 . . . . . . . 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 680 . . . . . . 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 2732 . . . . . 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 205 . . . . 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 7465 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  RR )
2120recnd 9116 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  sup ( A ,  RR ,  `'  <  )  e.  CC )
2221negnegd 9404 . 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 2438 . . . . . . . 8  |-  ( z  e.  RR  |->  -u z
)  =  ( z  e.  RR  |->  -u z
)
2423mptpreima 5365 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  { z  e.  RR  |  -u z  e.  A }
2523negiso 9986 . . . . . . . . 9  |-  ( ( z  e.  RR  |->  -u z )  Isom  <  ,  `'  <  ( RR ,  RR )  /\  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z ) )
2625simpri 450 . . . . . . . 8  |-  `' ( z  e.  RR  |->  -u z )  =  ( z  e.  RR  |->  -u z )
2726imaeq1i 5202 . . . . . . 7  |-  ( `' ( z  e.  RR  |->  -u z ) " A
)  =  ( ( z  e.  RR  |->  -u z ) " A
)
2824, 27eqtr3i 2460 . . . . . 6  |-  { z  e.  RR  |  -u z  e.  A }  =  ( ( z  e.  RR  |->  -u z
) " A )
2928supeq1i 7454 . . . . 5  |-  sup ( { z  e.  RR  |  -u z  e.  A } ,  RR ,  <  )  =  sup (
( ( z  e.  RR  |->  -u z ) " A ) ,  RR ,  <  )
3025simpli 446 . . . . . . . . 9  |-  ( z  e.  RR  |->  -u z
)  Isom  <  ,  `'  <  ( RR ,  RR )
31 isocnv 6052 . . . . . . . . 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 6041 . . . . . . . . 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 201 . . . . . . 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 958 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  x  <_  y
)  ->  A  C_  RR )
3836, 37, 19, 4supiso 7479 . . . . 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 2482 . . . 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 9300 . . . . . 6  |-  ( z  =  sup ( A ,  RR ,  `'  <  )  ->  -u z  = 
-u sup ( A ,  RR ,  `'  <  ) )
41 negex 9306 . . . . . 6  |-  -u sup ( A ,  RR ,  `'  <  )  e.  _V
4240, 23, 41fvmpt 5808 . . . . 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 2471 . . 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 9302 . 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 2472 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   class class class wbr 4214    e. cmpt 4268    Or wor 4504   `'ccnv 4879   "cima 4883   ` cfv 5456    Isom wiso 5457   supcsup 7447   RRcr 8991    < clt 9122    <_ cle 9123   -ucneg 9294
This theorem is referenced by:  infmrcl  9989  supminf  10565  mbfinf  19559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296
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