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Theorem rpnnen1lem1 10337
Description: Lemma for rpnnen1 10342. (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
Dummy variable  y is distinct from all other variables.
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1lem1
StepHypRef Expression
1 nnexALT 9743 . . . 4  |-  NN  e.  _V
21mptex 5707 . . 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 5569 . . 3  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
52, 4mpan2 654 . 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 3259 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
86, 7eqsstri 3209 . . . . . 6  |-  T  C_  ZZ
98a1i 12 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
10 nnre 9748 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
11 remulcl 8817 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1211ancoms 441 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1310, 12sylan2 462 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
14 btwnz 10109 . . . . . . . . . . . 12  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
1514simpld 447 . . . . . . . . . . 11  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
1613, 15syl 17 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
17 zre 10023 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  RR )
1817adantl 454 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
19 simpll 732 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
20 nngt0 9770 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
2110, 20jca 520 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
2221ad2antlr 709 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
23 ltdivmul 9623 . . . . . . . . . . . 12  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2418, 19, 22, 23syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2524rexbidva 2561 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2616, 25mpbird 225 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
27 rabn0 3475 . . . . . . . . 9  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2826, 27sylibr 205 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
296neeq1i 2457 . . . . . . . 8  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
3028, 29sylibr 205 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
316rabeq2i 2786 . . . . . . . . . 10  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
3210ad2antlr 709 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
3332, 19, 11syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
34 ltle 8905 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3518, 33, 34syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3624, 35sylbid 208 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3736impr 604 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3831, 37sylan2b 463 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3938ralrimiva 2627 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
40 breq2 4028 . . . . . . . . . 10  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
4140ralbidv 2564 . . . . . . . . 9  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
4241rspcev 2885 . . . . . . . 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 644 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
44 suprzcl 10086 . . . . . . 7  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
459, 30, 43, 44syl3anc 1184 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
468, 45sseldi 3179 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
47 znq 10315 . . . . 5  |-  ( ( sup ( T ,  RR ,  <  )  e.  ZZ  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k
)  e.  QQ )
4846, 47sylancom 650 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k )  e.  QQ )
49 eqid 2284 . . . 4  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
5048, 49fmptd 5645 . . 3  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
51 qexALT 10326 . . . 4  |-  QQ  e.  _V
5251, 1elmap 6791 . . 3  |-  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) 
<->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
5350, 52sylibr 205 . 2  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) )
545, 53eqeltrd 2358 1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   {crab 2548   _Vcvv 2789    C_ wss 3153   (/)c0 3456   class class class wbr 4024    e. cmpt 4078   -->wf 5217   ` cfv 5221  (class class class)co 5819    ^m cmap 6767   supcsup 7188   RRcr 8731   0cc0 8732    x. cmul 8737    < clt 8862    <_ cle 8863    / cdiv 9418   NNcn 9741   ZZcz 10019   QQcq 10311
This theorem is referenced by:  rpnnen1lem3  10339  rpnnen1lem4  10340  rpnnen1lem5  10341  rpnnen1  10342
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-n0 9961  df-z 10020  df-q 10312
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