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Theorem ruclem13 13847
Description: Lemma for ruc 13848. There is no function that maps  NN onto  RR. (Use nex 1612 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 5784 . . . 4  |-  ( F : NN -onto-> RR  ->  ran 
F  =  RR )
21difeq2d 3604 . . 3  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  ( RR 
\  RR ) )
3 difid 3878 . . 3  |-  ( RR 
\  RR )  =  (/)
42, 3syl6eq 2498 . 2  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  (/) )
5 reex 9581 . . . . . 6  |-  RR  e.  _V
65, 5xpex 6585 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
76, 5mpt2ex 6858 . . . 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 3099 . . 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 5781 . . . . . . . 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 2441 . . . . . . 7  |-  ( {
<. 0 ,  <. 0 ,  1 >. >. }  u.  F )  =  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F
)
13 eqid 2441 . . . . . . 7  |-  seq 0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)  =  seq 0
( d ,  ( { <. 0 ,  <. 0 ,  1 >. >. }  u.  F )
)
14 eqid 2441 . . . . . . 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 13846 . . . . . 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 3772 . . . . . 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 1709 . . 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 1381   E.wex 1597    e. wcel 1802   [_csb 3417    \ cdif 3455    u. cun 3456   (/)c0 3767   ifcif 3922   {csn 4010   <.cop 4016   class class class wbr 4433    X. cxp 4983   ran crn 4986    o. ccom 4989   -->wf 5570   -onto->wfo 5572   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6779   2ndc2nd 6780   supcsup 7898   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    < clt 9626    / cdiv 10207   NNcn 10537   2c2 10586    seqcseq 12081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-seq 12082
This theorem is referenced by:  ruc  13848
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