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Theorem pinq 8567
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 8523 . . . 4  |-  1o  e.  N.
2 opelxpi 4737 . . . 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 8546 . . . . . 6  |-  -.  ( 2nd `  y )  <N  1o
51elexi 2810 . . . . . . . 8  |-  1o  e.  _V
6 op2ndg 6149 . . . . . . . 8  |-  ( ( A  e.  N.  /\  1o  e.  _V )  -> 
( 2nd `  <. A ,  1o >. )  =  1o )
75, 6mpan2 652 . . . . . . 7  |-  ( A  e.  N.  ->  ( 2nd `  <. A ,  1o >. )  =  1o )
87breq2d 4051 . . . . . 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 2640 . . 3  |-  ( A  e.  N.  ->  A. y  e.  ( N.  X.  N. ) ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) )
12 breq1 4042 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( x  ~Q  y 
<-> 
<. A ,  1o >.  ~Q  y ) )
13 fveq2 5541 . . . . . . . 8  |-  ( x  =  <. A ,  1o >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  1o >. ) )
1413breq2d 4051 . . . . . . 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 2576 . . . 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 2936 . . 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 8552 . 2  |-  Q.  =  { x  e.  ( N.  X.  N. )  | 
A. y  e.  ( N.  X.  N. )
( x  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  x
) ) }
2119, 20syl6eleqr 2387 1  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703   ` cfv 5271   2ndc2nd 6137   1oc1o 6488   N.cnpi 8482    <N clti 8485    ~Q ceq 8489   Q.cnq 8490
This theorem is referenced by:  1nq  8568  archnq  8620  prlem934  8673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-2nd 6139  df-1o 6495  df-ni 8512  df-lti 8515  df-nq 8552
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