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Theorem ruclem13 12520
Description: Lemma for ruc 12521. There is no function that maps  NN onto  RR. (Use nex 1542 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 5454 . . . 4  |-  ( F : NN -onto-> RR  ->  ran 
F  =  RR )
21difeq2d 3294 . . 3  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  ( RR 
\  RR ) )
3 difid 3522 . . 3  |-  ( RR 
\  RR )  =  (/)
42, 3syl6eq 2331 . 2  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  (/) )
5 reex 8828 . . . . . 6  |-  RR  e.  _V
65, 5xpex 4801 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
76, 5mpt2ex 6198 . . . 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 2794 . . 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 5451 . . . . . . . 8  |-  ( F : NN -onto-> RR  ->  F : NN --> RR )
109adantr 451 . . . . . . 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 447 . . . . . . 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 2283 . . . . . . 7  |-  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F
)
13 eqid 2283 . . . . . . 7  |-  seq  0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)  =  seq  0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)
14 eqid 2283 . . . . . . 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 12519 . . . . . 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 3460 . . . . . 6  |-  ( sup ( ran  ( 1st 
o.  seq  0 ( d ,  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )
) ) ,  RR ,  <  )  e.  ( RR  \  ran  F
)  ->  -.  ( RR  \  ran  F )  =  (/) )
1715, 16syl 15 . . . . 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 423 . . . 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 1664 . . 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 16 . 2  |-  ( F : NN -onto-> RR  ->  -.  ( RR  \  ran  F )  =  (/) )
214, 20pm2.65i 165 1  |-  -.  F : NN -onto-> RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   [_csb 3081    \ cdif 3149    u. cun 3150   (/)c0 3455   ifcif 3565   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   ran crn 4690    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    / cdiv 9423   NNcn 9746   2c2 9795    seq cseq 11046
This theorem is referenced by:  ruc  12521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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