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Theorem rpnnen1lem4 10535
Description: Lemma for rpnnen1 10537. (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 10532 . . . 4  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
4 qexALT 10521 . . . . 5  |-  QQ  e.  _V
5 nnexALT 9934 . . . . 5  |-  NN  e.  _V
64, 5elmap 6978 . . . 4  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
73, 6sylib 189 . . 3  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
8 frn 5537 . . . 4  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
9 qssre 10516 . . . 4  |-  QQ  C_  RR
108, 9syl6ss 3303 . . 3  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
117, 10syl 16 . 2  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
12 1nn 9943 . . . . . 6  |-  1  e.  NN
13 ne0i 3577 . . . . . 6  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
1412, 13ax-mp 8 . . . . 5  |-  NN  =/=  (/)
15 fdm 5535 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1615neeq1d 2563 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1714, 16mpbiri 225 . . . 4  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
18 dm0rn0 5026 . . . . 5  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
1918necon3bii 2582 . . . 4  |-  ( dom  ( F `  x
)  =/=  (/)  <->  ran  ( F `
 x )  =/=  (/) )
2017, 19sylib 189 . . 3  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  =/=  (/) )
217, 20syl 16 . 2  |-  ( x  e.  RR  ->  ran  ( F `  x )  =/=  (/) )
221, 2rpnnen1lem3 10534 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
23 breq2 4157 . . . . 5  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2423ralbidv 2669 . . . 4  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2524rspcev 2995 . . 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 650 . 2  |-  ( x  e.  RR  ->  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )
27 suprcl 9900 . 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 1184 1  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653    C_ wss 3263   (/)c0 3571   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020    ^m cmap 6954   supcsup 7380   RRcr 8922   1c1 8924    < clt 9053    <_ cle 9054    / cdiv 9609   NNcn 9932   ZZcz 10214   QQcq 10506
This theorem is referenced by:  rpnnen1lem5  10536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-n0 10154  df-z 10215  df-q 10507
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