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Theorem rpnnen1lem1 10342
Description: Lemma for rpnnen1 10347. (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
rpnnen1lem1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Distinct variable groups:    k, F, n, x    T, n
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1lem1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nnexALT 9748 . . . 4  |-  NN  e.  _V
21mptex 5746 . . 3  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  e.  _V
3 rpnnen1.2 . . . 4  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
43fvmpt2 5608 . . 3  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
52, 4mpan2 652 . 2  |-  ( x  e.  RR  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
6 rpnnen1.1 . . . . . . 7  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
7 ssrab2 3258 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
86, 7eqsstri 3208 . . . . . 6  |-  T  C_  ZZ
98a1i 10 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
10 nnre 9753 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
11 remulcl 8822 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1211ancoms 439 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1310, 12sylan2 460 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
14 btwnz 10114 . . . . . . . . . . . 12  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
1514simpld 445 . . . . . . . . . . 11  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
1613, 15syl 15 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
17 zre 10028 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  RR )
1817adantl 452 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
19 simpll 730 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
20 nngt0 9775 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
2110, 20jca 518 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
2221ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
23 ltdivmul 9628 . . . . . . . . . . . 12  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2418, 19, 22, 23syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2524rexbidva 2560 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2616, 25mpbird 223 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
27 rabn0 3474 . . . . . . . . 9  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2826, 27sylibr 203 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
296neeq1i 2456 . . . . . . . 8  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
3028, 29sylibr 203 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
316rabeq2i 2785 . . . . . . . . . 10  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
3210ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
3332, 19, 11syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
34 ltle 8910 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3518, 33, 34syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3624, 35sylbid 206 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3736impr 602 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3831, 37sylan2b 461 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3938ralrimiva 2626 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
40 breq2 4027 . . . . . . . . . 10  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
4140ralbidv 2563 . . . . . . . . 9  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
4241rspcev 2884 . . . . . . . 8  |-  ( ( ( k  x.  x
)  e.  RR  /\  A. n  e.  T  n  <_  ( k  x.  x ) )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
4313, 39, 42syl2anc 642 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
44 suprzcl 10091 . . . . . . 7  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
459, 30, 43, 44syl3anc 1182 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
468, 45sseldi 3178 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
47 znq 10320 . . . . 5  |-  ( ( sup ( T ,  RR ,  <  )  e.  ZZ  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k
)  e.  QQ )
4846, 47sylancom 648 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k )  e.  QQ )
49 eqid 2283 . . . 4  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
5048, 49fmptd 5684 . . 3  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
51 qexALT 10331 . . . 4  |-  QQ  e.  _V
5251, 1elmap 6796 . . 3  |-  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) 
<->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
5350, 52sylibr 203 . 2  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) )
545, 53eqeltrd 2357 1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   supcsup 7193   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   ZZcz 10024   QQcq 10316
This theorem is referenced by:  rpnnen1lem3  10344  rpnnen1lem4  10345  rpnnen1lem5  10346  rpnnen1  10347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-q 10317
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