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Theorem rpnnen1lem4 10361
Description: Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
Hypotheses
Ref Expression
rpnnen1.1  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
rpnnen1.2  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
Assertion
Ref Expression
rpnnen1lem4  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
Distinct variable groups:    k, F, n, x    T, n
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1lem4
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rpnnen1.1 . . . . 5  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
2 rpnnen1.2 . . . . 5  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
31, 2rpnnen1lem1 10358 . . . 4  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
4 qexALT 10347 . . . . 5  |-  QQ  e.  _V
5 nnexALT 9764 . . . . 5  |-  NN  e.  _V
64, 5elmap 6812 . . . 4  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
73, 6sylib 188 . . 3  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
8 frn 5411 . . . 4  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
9 qssre 10342 . . . 4  |-  QQ  C_  RR
108, 9syl6ss 3204 . . 3  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
117, 10syl 15 . 2  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
12 1nn 9773 . . . . . 6  |-  1  e.  NN
13 ne0i 3474 . . . . . 6  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
1412, 13ax-mp 8 . . . . 5  |-  NN  =/=  (/)
15 fdm 5409 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1615neeq1d 2472 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1714, 16mpbiri 224 . . . 4  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
18 dm0rn0 4911 . . . . 5  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
1918necon3bii 2491 . . . 4  |-  ( dom  ( F `  x
)  =/=  (/)  <->  ran  ( F `
 x )  =/=  (/) )
2017, 19sylib 188 . . 3  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  =/=  (/) )
217, 20syl 15 . 2  |-  ( x  e.  RR  ->  ran  ( F `  x )  =/=  (/) )
221, 2rpnnen1lem3 10360 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
23 breq2 4043 . . . . 5  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2423ralbidv 2576 . . . 4  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2524rspcev 2897 . . 3  |-  ( ( x  e.  RR  /\  A. n  e.  ran  ( F `  x )
n  <_  x )  ->  E. y  e.  RR  A. n  e.  ran  ( F `  x )
n  <_  y )
2622, 25mpdan 649 . 2  |-  ( x  e.  RR  ->  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )
27 suprcl 9730 . 2  |-  ( ( ran  ( F `  x )  C_  RR  /\ 
ran  ( F `  x )  =/=  (/)  /\  E. y  e.  RR  A. n  e.  ran  ( F `  x ) n  <_ 
y )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
2811, 21, 26, 27syl3anc 1182 1  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332
This theorem is referenced by:  rpnnen1lem5  10362
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-inf2 7358  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-map 6790  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-n0 9982  df-z 10041  df-q 10333
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