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Theorem rpnnen1lem1 10602
Description: Lemma for rpnnen1 10607. (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 10004 . . . 4  |-  NN  e.  _V
21mptex 5968 . . 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 5814 . . 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 3430 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
86, 7eqsstri 3380 . . . . . 6  |-  T  C_  ZZ
98a1i 11 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
10 nnre 10009 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
11 remulcl 9077 . . . . . . . . . . . . 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 10374 . . . . . . . . . . . 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 16 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
17 zre 10288 . . . . . . . . . . . . 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 10031 . . . . . . . . . . . . . 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 9884 . . . . . . . . . . . 12  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2418, 19, 22, 23syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2524rexbidva 2724 . . . . . . . . . 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 3649 . . . . . . . . 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 2613 . . . . . . . 8  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
3028, 29sylibr 205 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
316rabeq2i 2955 . . . . . . . . . 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 9165 . . . . . . . . . . . . 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 2791 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
40 breq2 4218 . . . . . . . . . 10  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
4140ralbidv 2727 . . . . . . . . 9  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
4241rspcev 3054 . . . . . . . 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 10351 . . . . . . 7  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
459, 30, 43, 44syl3anc 1185 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
468, 45sseldi 3348 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
47 znq 10580 . . . . 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 2438 . . . 4  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
5048, 49fmptd 5895 . . 3  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
51 qexALT 10591 . . . 4  |-  QQ  e.  _V
5251, 1elmap 7044 . . 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 2512 1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   class class class wbr 4214    e. cmpt 4268   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   supcsup 7447   RRcr 8991   0cc0 8992    x. cmul 8997    < clt 9122    <_ cle 9123    / cdiv 9679   NNcn 10002   ZZcz 10284   QQcq 10576
This theorem is referenced by:  rpnnen1lem3  10604  rpnnen1lem4  10605  rpnnen1lem5  10606  rpnnen1  10607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-q 10577
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