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Theorem rpnnen1lem1 10984
Description: Lemma for rpnnen1 10989. (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 10329 . . . 4  |-  NN  e.  _V
21mptex 5953 . . 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 5786 . . 3  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
52, 4mpan2 671 . 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 3442 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
86, 7eqsstri 3391 . . . . . 6  |-  T  C_  ZZ
98a1i 11 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
10 nnre 10334 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
11 remulcl 9372 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1211ancoms 453 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1310, 12sylan2 474 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
14 btwnz 10749 . . . . . . . . . . . 12  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
1514simpld 459 . . . . . . . . . . 11  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
1613, 15syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
17 zre 10655 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  RR )
1817adantl 466 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
19 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
20 nngt0 10356 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
2110, 20jca 532 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
2221ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
23 ltdivmul 10209 . . . . . . . . . . . 12  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2418, 19, 22, 23syl3anc 1218 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2524rexbidva 2737 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2616, 25mpbird 232 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
27 rabn0 3662 . . . . . . . . 9  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2826, 27sylibr 212 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
296neeq1i 2623 . . . . . . . 8  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
3028, 29sylibr 212 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
316rabeq2i 2974 . . . . . . . . . 10  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
3210ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
3332, 19, 11syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
34 ltle 9468 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3518, 33, 34syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3624, 35sylbid 215 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3736impr 619 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3831, 37sylan2b 475 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3938ralrimiva 2804 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
40 breq2 4301 . . . . . . . . . 10  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
4140ralbidv 2740 . . . . . . . . 9  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
4241rspcev 3078 . . . . . . . 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 661 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
44 suprzcl 10726 . . . . . . 7  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
459, 30, 43, 44syl3anc 1218 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
468, 45sseldi 3359 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
47 znq 10962 . . . . 5  |-  ( ( sup ( T ,  RR ,  <  )  e.  ZZ  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k
)  e.  QQ )
4846, 47sylancom 667 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k )  e.  QQ )
49 eqid 2443 . . . 4  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
5048, 49fmptd 5872 . . 3  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
51 qexALT 10973 . . . 4  |-  QQ  e.  _V
5251, 1elmap 7246 . . 3  |-  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) 
<->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
5350, 52sylibr 212 . 2  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) )
545, 53eqeltrd 2517 1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977    C_ wss 3333   (/)c0 3642   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^m cmap 7219   supcsup 7695   RRcr 9286   0cc0 9287    x. cmul 9292    < clt 9423    <_ cle 9424    / cdiv 9998   NNcn 10327   ZZcz 10651   QQcq 10958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-q 10959
This theorem is referenced by:  rpnnen1lem3  10986  rpnnen1lem4  10987  rpnnen1lem5  10988  rpnnen1  10989
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