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Theorem reexALT 10608
Description: The set of real numbers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
reexALT  |-  RR  e.  _V

Proof of Theorem reexALT
Dummy variables  j 
k  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6090 . . . . 5  |-  ( m  =  n  ->  (
m  /  k )  =  ( n  / 
k ) )
21breq1d 4224 . . . 4  |-  ( m  =  n  ->  (
( m  /  k
)  <  x  <->  ( n  /  k )  < 
x ) )
32cbvrabv 2957 . . 3  |-  { m  e.  ZZ  |  ( m  /  k )  < 
x }  =  {
n  e.  ZZ  | 
( n  /  k
)  <  x }
4 oveq2 6091 . . . . . . . . . 10  |-  ( j  =  k  ->  (
m  /  j )  =  ( m  / 
k ) )
54breq1d 4224 . . . . . . . . 9  |-  ( j  =  k  ->  (
( m  /  j
)  <  y  <->  ( m  /  k )  < 
y ) )
65rabbidv 2950 . . . . . . . 8  |-  ( j  =  k  ->  { m  e.  ZZ  |  ( m  /  j )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  y }
)
76supeq1d 7453 . . . . . . 7  |-  ( j  =  k  ->  sup ( { m  e.  ZZ  |  ( m  / 
j )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  ) )
8 id 21 . . . . . . 7  |-  ( j  =  k  ->  j  =  k )
97, 8oveq12d 6101 . . . . . 6  |-  ( j  =  k  ->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
109cbvmptv 4302 . . . . 5  |-  ( j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )
11 breq2 4218 . . . . . . . . 9  |-  ( y  =  x  ->  (
( m  /  k
)  <  y  <->  ( m  /  k )  < 
x ) )
1211rabbidv 2950 . . . . . . . 8  |-  ( y  =  x  ->  { m  e.  ZZ  |  ( m  /  k )  < 
y }  =  {
m  e.  ZZ  | 
( m  /  k
)  <  x }
)
1312supeq1d 7453 . . . . . . 7  |-  ( y  =  x  ->  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  y } ,  RR ,  <  )  =  sup ( { m  e.  ZZ  |  ( m  / 
k )  <  x } ,  RR ,  <  ) )
1413oveq1d 6098 . . . . . 6  |-  ( y  =  x  ->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
)  =  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) )
1514mpteq2dv 4298 . . . . 5  |-  ( y  =  x  ->  (
k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
y } ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
1610, 15syl5eq 2482 . . . 4  |-  ( y  =  x  ->  (
j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) )  =  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
1716cbvmptv 4302 . . 3  |-  ( y  e.  RR  |->  ( j  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  j )  < 
y } ,  RR ,  <  )  /  j
) ) )  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( { m  e.  ZZ  |  ( m  /  k )  < 
x } ,  RR ,  <  )  /  k
) ) )
183, 17rpnnen1 10607 . 2  |-  RR  ~<_  ( QQ 
^m  NN )
19 reldom 7117 . . 3  |-  Rel  ~<_
2019brrelexi 4920 . 2  |-  ( RR  ~<_  ( QQ  ^m  NN )  ->  RR  e.  _V )
2118, 20ax-mp 8 1  |-  RR  e.  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   class class class wbr 4214    e. cmpt 4268  (class class class)co 6083    ^m cmap 7020    ~<_ cdom 7109   supcsup 7447   RRcr 8991    < clt 9122    / cdiv 9679   NNcn 10002   ZZcz 10284   QQcq 10576
This theorem is referenced by:  cnexALT  10610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-n0 10224  df-z 10285  df-q 10577
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