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Theorem rpnnen1 10605
Description: One half of rpnnen 12826, 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 6106 . 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 10600 . . 3  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 rneq 5095 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
65supeq1d 7451 . . . . 5  |-  ( ( F `  x )  =  ( F `  y )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
72, 3rpnnen1lem5 10604 . . . . . 6  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
8 fveq2 5728 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
98rneqd 5097 . . . . . . . . 9  |-  ( x  =  y  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
109supeq1d 7451 . . . . . . . 8  |-  ( x  =  y  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
11 id 20 . . . . . . . 8  |-  ( x  =  y  ->  x  =  y )
1210, 11eqeq12d 2450 . . . . . . 7  |-  ( x  =  y  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y ) )
1312, 7vtoclga 3017 . . . . . 6  |-  ( y  e.  RR  ->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y )
147, 13eqeqan12d 2451 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  =  sup ( ran  ( F `  y
) ,  RR ,  <  )  <->  x  =  y
) )
156, 14syl5ib 211 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
1615, 8impbid1 195 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
174, 16dom2 7150 . 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 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081    ^m cmap 7018    ~<_ cdom 7107   supcsup 7445   RRcr 8989    < clt 9120    / cdiv 9677   NNcn 10000   ZZcz 10282   QQcq 10574
This theorem is referenced by:  reexALT  10606  rpnnen  12826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-q 10575
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