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Theorem rpnnen1 11154
Description: One half of rpnnen 13985, 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 6246 . 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 11149 . . 3  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 rneq 5158 . . . . . 6  |-  ( ( F `  x )  =  ( F `  y )  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
65supeq1d 7842 . . . . 5  |-  ( ( F `  x )  =  ( F `  y )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
72, 3rpnnen1lem5 11153 . . . . . 6  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
8 fveq2 5791 . . . . . . . . . 10  |-  ( x  =  y  ->  ( F `  x )  =  ( F `  y ) )
98rneqd 5160 . . . . . . . . 9  |-  ( x  =  y  ->  ran  ( F `  x )  =  ran  ( F `
 y ) )
109supeq1d 7842 . . . . . . . 8  |-  ( x  =  y  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  sup ( ran  ( F `  y ) ,  RR ,  <  ) )
11 id 22 . . . . . . . 8  |-  ( x  =  y  ->  x  =  y )
1210, 11eqeq12d 2418 . . . . . . 7  |-  ( x  =  y  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y ) )
1312, 7vtoclga 3115 . . . . . 6  |-  ( y  e.  RR  ->  sup ( ran  ( F `  y ) ,  RR ,  <  )  =  y )
147, 13eqeqan12d 2419 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  =  sup ( ran  ( F `  y
) ,  RR ,  <  )  <->  x  =  y
) )
156, 14syl5ib 219 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
1615, 8impbid1 203 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
x  =  y ) )
174, 16dom2 7499 . 2  |-  ( ( QQ  ^m  NN )  e.  _V  ->  RR  ~<_  ( QQ  ^m  NN ) )
181, 17ax-mp 5 1  |-  RR  ~<_  ( QQ 
^m  NN )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1836   {crab 2750   _Vcvv 3051   class class class wbr 4384    |-> cmpt 4442   ran crn 4931   ` cfv 5513  (class class class)co 6218    ^m cmap 7360    ~<_ cdom 7455   supcsup 7837   RRcr 9424    < clt 9561    / cdiv 10145   NNcn 10474   ZZcz 10803   QQcq 11123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-inf2 7994  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-map 7362  df-en 7458  df-dom 7459  df-sdom 7460  df-sup 7838  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-n0 10735  df-z 10804  df-q 11124
This theorem is referenced by:  reexALT  11155  rpnnen  13985
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