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Theorem rpnnen1lem1 11297
Description: Lemma for rpnnen1 11302. (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 nnex 10622 . . . 4  |-  NN  e.  _V
21mptex 6151 . . 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 5973 . . 3  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
52, 4mpan2 675 . 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 3546 . . . . . . 7  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
86, 7eqsstri 3494 . . . . . 6  |-  T  C_  ZZ
98a1i 11 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
10 nnre 10623 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  k  e.  RR )
11 remulcl 9631 . . . . . . . . . . . . 13  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1211ancoms 454 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
1310, 12sylan2 476 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
14 btwnz 11044 . . . . . . . . . . . 12  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
1514simpld 460 . . . . . . . . . . 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 10948 . . . . . . . . . . . . 13  |-  ( n  e.  ZZ  ->  n  e.  RR )
1817adantl 467 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
19 simpll 758 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
20 nngt0 10645 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
2110, 20jca 534 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
2221ad2antlr 731 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
23 ltdivmul 10487 . . . . . . . . . . . 12  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2418, 19, 22, 23syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2524rexbidva 2933 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2616, 25mpbird 235 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
27 rabn0 3782 . . . . . . . . 9  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2826, 27sylibr 215 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
296neeq1i 2705 . . . . . . . 8  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
3028, 29sylibr 215 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
316rabeq2i 3077 . . . . . . . . . 10  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
3210ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
3332, 19, 11syl2anc 665 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
34 ltle 9729 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3518, 33, 34syl2anc 665 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3624, 35sylbid 218 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3736impr 623 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3831, 37sylan2b 477 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3938ralrimiva 2836 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
40 breq2 4427 . . . . . . . . . 10  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
4140ralbidv 2861 . . . . . . . . 9  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
4241rspcev 3182 . . . . . . . 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 665 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
44 suprzcl 11022 . . . . . . 7  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
459, 30, 43, 44syl3anc 1264 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
468, 45sseldi 3462 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
47 znq 11275 . . . . 5  |-  ( ( sup ( T ,  RR ,  <  )  e.  ZZ  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k
)  e.  QQ )
4846, 47sylancom 671 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  /  k )  e.  QQ )
49 eqid 2422 . . . 4  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
5048, 49fmptd 6061 . . 3  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
51 qex 11283 . . . 4  |-  QQ  e.  _V
5251, 1elmap 7511 . . 3  |-  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) 
<->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) : NN --> QQ )
5350, 52sylibr 215 . 2  |-  ( x  e.  RR  ->  (
k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e.  ( QQ  ^m  NN ) )
545, 53eqeltrd 2507 1  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775   _Vcvv 3080    C_ wss 3436   (/)c0 3761   class class class wbr 4423    |-> cmpt 4482   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7483   supcsup 7963   RRcr 9545   0cc0 9546    x. cmul 9551    < clt 9682    <_ cle 9683    / cdiv 10276   NNcn 10616   ZZcz 10944   QQcq 11271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-q 11272
This theorem is referenced by:  rpnnen1lem3  11299  rpnnen1lem4  11300  rpnnen1lem5  11301  rpnnen1  11302
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