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Theorem ruclem13 13827
Description: Lemma for ruc 13828. There is no function that maps  NN onto  RR. (Use nex 1605 if you want this in the form  -.  E. f
f : NN -onto-> RR.) (Contributed by NM, 14-Oct-2004.) (Proof shortened by Fan Zheng, 6-Jun-2016.)
Assertion
Ref Expression
ruclem13  |-  -.  F : NN -onto-> RR

Proof of Theorem ruclem13
Dummy variables  d  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 forn 5791 . . . 4  |-  ( F : NN -onto-> RR  ->  ran 
F  =  RR )
21difeq2d 3617 . . 3  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  ( RR 
\  RR ) )
3 difid 3890 . . 3  |-  ( RR 
\  RR )  =  (/)
42, 3syl6eq 2519 . 2  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  (/) )
5 reex 9574 . . . . . 6  |-  RR  e.  _V
65, 5xpex 6706 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
76, 5mpt2ex 6852 . . . 4  |-  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )  e. 
_V
87isseti 3114 . . 3  |-  E. d 
d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )
9 fof 5788 . . . . . . . 8  |-  ( F : NN -onto-> RR  ->  F : NN --> RR )
109adantr 465 . . . . . . 7  |-  ( ( F : NN -onto-> RR  /\  d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )  ->  F : NN --> RR )
11 simpr 461 . . . . . . 7  |-  ( ( F : NN -onto-> RR  /\  d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )  ->  d  =  ( x  e.  ( RR 
X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
12 eqid 2462 . . . . . . 7  |-  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F
)
13 eqid 2462 . . . . . . 7  |-  seq 0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)  =  seq 0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)
14 eqid 2462 . . . . . . 7  |-  sup ( ran  ( 1st  o.  seq 0 ( d ,  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
) ) ) ,  RR ,  <  )  =  sup ( ran  ( 1st  o.  seq 0 ( d ,  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )
) ) ,  RR ,  <  )
1510, 11, 12, 13, 14ruclem12 13826 . . . . . 6  |-  ( ( F : NN -onto-> RR  /\  d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )  ->  sup ( ran  ( 1st  o.  seq 0 ( d ,  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )
) ) ,  RR ,  <  )  e.  ( RR  \  ran  F
) )
16 n0i 3785 . . . . . 6  |-  ( sup ( ran  ( 1st 
o.  seq 0 ( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F
) ) ) ,  RR ,  <  )  e.  ( RR  \  ran  F )  ->  -.  ( RR  \  ran  F )  =  (/) )
1715, 16syl 16 . . . . 5  |-  ( ( F : NN -onto-> RR  /\  d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )  ->  -.  ( RR  \  ran  F )  =  (/) )
1817ex 434 . . . 4  |-  ( F : NN -onto-> RR  ->  ( d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )  ->  -.  ( RR  \  ran  F )  =  (/) ) )
1918exlimdv 1695 . . 3  |-  ( F : NN -onto-> RR  ->  ( E. d  d  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x ) )  /  2 )  /  m ]_ if ( m  <  y , 
<. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) )  ->  -.  ( RR  \  ran  F )  =  (/) ) )
208, 19mpi 17 . 2  |-  ( F : NN -onto-> RR  ->  -.  ( RR  \  ran  F )  =  (/) )
214, 20pm2.65i 173 1  |-  -.  F : NN -onto-> RR
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   [_csb 3430    \ cdif 3468    u. cun 3469   (/)c0 3780   ifcif 3934   {csn 4022   <.cop 4028   class class class wbr 4442    X. cxp 4992   ran crn 4995    o. ccom 4998   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    < clt 9619    / cdiv 10197   NNcn 10527   2c2 10576    seqcseq 12065
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-seq 12066
This theorem is referenced by:  ruc  13828
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