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Theorem rpnnen1 10498
Description: One half of rpnnen 12713, 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 6006 . 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 10493 . . 3  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 rneq 5007 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
65supeq1d 7346 . . . . 5  |-  ( ( F `  x )  =  ( F `  y )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
72, 3rpnnen1lem5 10497 . . . . . 6  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
8 fveq2 5632 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
98rneqd 5009 . . . . . . . . 9  |-  ( x  =  y  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
109supeq1d 7346 . . . . . . . 8  |-  ( x  =  y  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
11 id 19 . . . . . . . 8  |-  ( x  =  y  ->  x  =  y )
1210, 11eqeq12d 2380 . . . . . . 7  |-  ( x  =  y  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y ) )
1312, 7vtoclga 2934 . . . . . 6  |-  ( y  e.  RR  ->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y )
147, 13eqeqan12d 2381 . . . . 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 7047 . 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 1647    e. wcel 1715   {crab 2632   _Vcvv 2873   class class class wbr 4125    e. cmpt 4179   ran crn 4793   ` cfv 5358  (class class class)co 5981    ^m cmap 6915    ~<_ cdom 7004   supcsup 7340   RRcr 8883    < clt 9014    / cdiv 9570   NNcn 9893   ZZcz 10175   QQcq 10467
This theorem is referenced by:  reexALT  10499  rpnnen  12713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-n0 10115  df-z 10176  df-q 10468
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