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Theorem ruclem13 12536
Description: Lemma for ruc 12537. There is no function that maps  NN onto  RR. (Use nex 1545 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 5470 . . . 4  |-  ( F : NN -onto-> RR  ->  ran 
F  =  RR )
21difeq2d 3307 . . 3  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  ( RR 
\  RR ) )
3 difid 3535 . . 3  |-  ( RR 
\  RR )  =  (/)
42, 3syl6eq 2344 . 2  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  (/) )
5 reex 8844 . . . . . 6  |-  RR  e.  _V
65, 5xpex 4817 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
76, 5mpt2ex 6214 . . . 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 2807 . . 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 5467 . . . . . . . 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 2296 . . . . . . 7  |-  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F
)
13 eqid 2296 . . . . . . 7  |-  seq  0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)  =  seq  0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)
14 eqid 2296 . . . . . . 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 12535 . . . . . 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 3473 . . . . . 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 1626 . . 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 1531    = wceq 1632    e. wcel 1696   [_csb 3094    \ cdif 3162    u. cun 3163   (/)c0 3468   ifcif 3578   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    / cdiv 9439   NNcn 9762   2c2 9811    seq cseq 11062
This theorem is referenced by:  ruc  12537
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  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  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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