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Theorem pinq 8796
Description: The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pinq  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )

Proof of Theorem pinq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pi 8752 . . . 4  |-  1o  e.  N.
2 opelxpi 4902 . . . 4  |-  ( ( A  e.  N.  /\  1o  e.  N. )  ->  <. A ,  1o >.  e.  ( N.  X.  N. ) )
31, 2mpan2 653 . . 3  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  ( N. 
X.  N. ) )
4 nlt1pi 8775 . . . . . 6  |-  -.  ( 2nd `  y )  <N  1o
51elexi 2957 . . . . . . . 8  |-  1o  e.  _V
6 op2ndg 6352 . . . . . . . 8  |-  ( ( A  e.  N.  /\  1o  e.  _V )  -> 
( 2nd `  <. A ,  1o >. )  =  1o )
75, 6mpan2 653 . . . . . . 7  |-  ( A  e.  N.  ->  ( 2nd `  <. A ,  1o >. )  =  1o )
87breq2d 4216 . . . . . 6  |-  ( A  e.  N.  ->  (
( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )  <->  ( 2nd `  y ) 
<N  1o ) )
94, 8mtbiri 295 . . . . 5  |-  ( A  e.  N.  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  <. A ,  1o >. )
)
109a1d 23 . . . 4  |-  ( A  e.  N.  ->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y )  <N 
( 2nd `  <. A ,  1o >. )
) )
1110ralrimivw 2782 . . 3  |-  ( A  e.  N.  ->  A. y  e.  ( N.  X.  N. ) ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) )
12 breq1 4207 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( x  ~Q  y 
<-> 
<. A ,  1o >.  ~Q  y ) )
13 fveq2 5720 . . . . . . . 8  |-  ( x  =  <. A ,  1o >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  1o >. ) )
1413breq2d 4216 . . . . . . 7  |-  ( x  =  <. A ,  1o >.  ->  ( ( 2nd `  y )  <N  ( 2nd `  x )  <->  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1514notbid 286 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( -.  ( 2nd `  y )  <N 
( 2nd `  x
)  <->  -.  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1612, 15imbi12d 312 . . . . 5  |-  ( x  =  <. A ,  1o >.  ->  ( ( x  ~Q  y  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  x
) )  <->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) ) )
1716ralbidv 2717 . . . 4  |-  ( x  =  <. A ,  1o >.  ->  ( A. y  e.  ( N.  X.  N. ) ( x  ~Q  y  ->  -.  ( 2nd `  y )  <N  ( 2nd `  x ) )  <->  A. y  e.  ( N.  X.  N. ) (
<. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y )  <N 
( 2nd `  <. A ,  1o >. )
) ) )
1817elrab 3084 . . 3  |-  ( <. A ,  1o >.  e.  {
x  e.  ( N. 
X.  N. )  |  A. y  e.  ( N.  X.  N. ) ( x  ~Q  y  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  x
) ) }  <->  ( <. A ,  1o >.  e.  ( N.  X.  N. )  /\  A. y  e.  ( N.  X.  N. )
( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  <. A ,  1o >. )
) ) )
193, 11, 18sylanbrc 646 . 2  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  { x  e.  ( N.  X.  N. )  |  A. y  e.  ( N.  X.  N. ) ( x  ~Q  y  ->  -.  ( 2nd `  y )  <N  ( 2nd `  x ) ) } )
20 df-nq 8781 . 2  |-  Q.  =  { x  e.  ( N.  X.  N. )  | 
A. y  e.  ( N.  X.  N. )
( x  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  x
) ) }
2119, 20syl6eleqr 2526 1  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   <.cop 3809   class class class wbr 4204    X. cxp 4868   ` cfv 5446   2ndc2nd 6340   1oc1o 6709   N.cnpi 8711    <N clti 8714    ~Q ceq 8718   Q.cnq 8719
This theorem is referenced by:  1nq  8797  archnq  8849  prlem934  8902
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-2nd 6342  df-1o 6716  df-ni 8741  df-lti 8744  df-nq 8781
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