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Theorem ruclem13 13520
Description: Lemma for ruc 13521. 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 5620 . . . 4  |-  ( F : NN -onto-> RR  ->  ran 
F  =  RR )
21difeq2d 3471 . . 3  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  ( RR 
\  RR ) )
3 difid 3744 . . 3  |-  ( RR 
\  RR )  =  (/)
42, 3syl6eq 2489 . 2  |-  ( F : NN -onto-> RR  ->  ( RR  \  ran  F
)  =  (/) )
5 reex 9369 . . . . . 6  |-  RR  e.  _V
65, 5xpex 6507 . . . . 5  |-  ( RR 
X.  RR )  e. 
_V
76, 5mpt2ex 6649 . . . 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 2976 . . 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 5617 . . . . . . . 8  |-  ( F : NN -onto-> RR  ->  F : NN --> RR )
109adantr 462 . . . . . . 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 458 . . . . . . 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 13519 . . . . . 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 3639 . . . . . 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 1364   E.wex 1591    e. wcel 1761   [_csb 3285    \ cdif 3322    u. cun 3323   (/)c0 3634   ifcif 3788   {csn 3874   <.cop 3880   class class class wbr 4289    X. cxp 4834   ran crn 4837    o. ccom 4840   -->wf 5411   -onto->wfo 5413   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   supcsup 7686   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    / cdiv 9989   NNcn 10318   2c2 10367    seqcseq 11802
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803
This theorem is referenced by:  ruc  13521
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