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Theorem rpnnen1 10347
Description: One half of rpnnen 12505, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number  x to the sequence  ( F `  x ) : NN --> QQ such that  ( ( F `  x ) `  k ) is the largest rational number with denominator  k that is strictly less than  x. In this manner, we get a monotonically increasing sequence that converges to  x, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-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
rpnnen1  |-  RR  ~<_  ( QQ 
^m  NN )
Distinct variable groups:    k, F, n, x    T, n
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ovex 5883 . 2  |-  ( QQ 
^m  NN )  e. 
_V
2 rpnnen1.1 . . . 4  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
3 rpnnen1.2 . . . 4  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
42, 3rpnnen1lem1 10342 . . 3  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 rneq 4904 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
65supeq1d 7199 . . . . 5  |-  ( ( F `  x )  =  ( F `  y )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
72, 3rpnnen1lem5 10346 . . . . . 6  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
8 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
98rneqd 4906 . . . . . . . . 9  |-  ( x  =  y  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
109supeq1d 7199 . . . . . . . 8  |-  ( x  =  y  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
11 id 19 . . . . . . . 8  |-  ( x  =  y  ->  x  =  y )
1210, 11eqeq12d 2297 . . . . . . 7  |-  ( x  =  y  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y ) )
1312, 7vtoclga 2849 . . . . . 6  |-  ( y  e.  RR  ->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y )
147, 13eqeqan12d 2298 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  =  sup ( ran  ( F `  y
) ,  RR ,  <  )  <->  x  =  y
) )
156, 14syl5ib 210 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
1615, 8impbid1 194 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
174, 16dom2 6904 . 2  |-  ( ( QQ  ^m  NN )  e.  _V  ->  RR  ~<_  ( QQ  ^m  NN ) )
181, 17ax-mp 8 1  |-  RR  ~<_  ( QQ 
^m  NN )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858    ^m cmap 6772    ~<_ cdom 6861   supcsup 7193   RRcr 8736    < clt 8867    / cdiv 9423   NNcn 9746   ZZcz 10024   QQcq 10316
This theorem is referenced by:  reexALT  10348  rpnnen  12505
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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