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Theorem pinq 8551
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 8507 . . . 4  |-  1o  e.  N.
2 opelxpi 4721 . . . 4  |-  ( ( A  e.  N.  /\  1o  e.  N. )  ->  <. A ,  1o >.  e.  ( N.  X.  N. ) )
31, 2mpan2 652 . . 3  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  ( N. 
X.  N. ) )
4 nlt1pi 8530 . . . . . 6  |-  -.  ( 2nd `  y )  <N  1o
51elexi 2797 . . . . . . . 8  |-  1o  e.  _V
6 op2ndg 6133 . . . . . . . 8  |-  ( ( A  e.  N.  /\  1o  e.  _V )  -> 
( 2nd `  <. A ,  1o >. )  =  1o )
75, 6mpan2 652 . . . . . . 7  |-  ( A  e.  N.  ->  ( 2nd `  <. A ,  1o >. )  =  1o )
87breq2d 4035 . . . . . 6  |-  ( A  e.  N.  ->  (
( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )  <->  ( 2nd `  y ) 
<N  1o ) )
94, 8mtbiri 294 . . . . 5  |-  ( A  e.  N.  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  <. A ,  1o >. )
)
109a1d 22 . . . 4  |-  ( A  e.  N.  ->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y )  <N 
( 2nd `  <. A ,  1o >. )
) )
1110ralrimivw 2627 . . 3  |-  ( A  e.  N.  ->  A. y  e.  ( N.  X.  N. ) ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) )
12 breq1 4026 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( x  ~Q  y 
<-> 
<. A ,  1o >.  ~Q  y ) )
13 fveq2 5525 . . . . . . . 8  |-  ( x  =  <. A ,  1o >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  1o >. ) )
1413breq2d 4035 . . . . . . 7  |-  ( x  =  <. A ,  1o >.  ->  ( ( 2nd `  y )  <N  ( 2nd `  x )  <->  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1514notbid 285 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( -.  ( 2nd `  y )  <N 
( 2nd `  x
)  <->  -.  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1612, 15imbi12d 311 . . . . 5  |-  ( x  =  <. A ,  1o >.  ->  ( ( x  ~Q  y  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  x
) )  <->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) ) )
1716ralbidv 2563 . . . 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 2923 . . 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 645 . 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 8536 . 2  |-  Q.  =  { x  e.  ( N.  X.  N. )  | 
A. y  e.  ( N.  X.  N. )
( x  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  x
) ) }
2119, 20syl6eleqr 2374 1  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687   ` cfv 5255   2ndc2nd 6121   1oc1o 6472   N.cnpi 8466    <N clti 8469    ~Q ceq 8473   Q.cnq 8474
This theorem is referenced by:  1nq  8552  archnq  8604  prlem934  8657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-2nd 6123  df-1o 6479  df-ni 8496  df-lti 8499  df-nq 8536
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