MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pinq Unicode version

Theorem pinq 8730
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 8686 . . . 4  |-  1o  e.  N.
2 opelxpi 4843 . . . 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 8709 . . . . . 6  |-  -.  ( 2nd `  y )  <N  1o
51elexi 2901 . . . . . . . 8  |-  1o  e.  _V
6 op2ndg 6292 . . . . . . . 8  |-  ( ( A  e.  N.  /\  1o  e.  _V )  -> 
( 2nd `  <. A ,  1o >. )  =  1o )
75, 6mpan2 653 . . . . . . 7  |-  ( A  e.  N.  ->  ( 2nd `  <. A ,  1o >. )  =  1o )
87breq2d 4158 . . . . . 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 2726 . . 3  |-  ( A  e.  N.  ->  A. y  e.  ( N.  X.  N. ) ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) )
12 breq1 4149 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( x  ~Q  y 
<-> 
<. A ,  1o >.  ~Q  y ) )
13 fveq2 5661 . . . . . . . 8  |-  ( x  =  <. A ,  1o >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  1o >. ) )
1413breq2d 4158 . . . . . . 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 2662 . . . 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 3028 . . 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 8715 . 2  |-  Q.  =  { x  e.  ( N.  X.  N. )  | 
A. y  e.  ( N.  X.  N. )
( x  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  x
) ) }
2119, 20syl6eleqr 2471 1  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   _Vcvv 2892   <.cop 3753   class class class wbr 4146    X. cxp 4809   ` cfv 5387   2ndc2nd 6280   1oc1o 6646   N.cnpi 8645    <N clti 8648    ~Q ceq 8652   Q.cnq 8653
This theorem is referenced by:  1nq  8731  archnq  8783  prlem934  8836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fv 5395  df-2nd 6282  df-1o 6653  df-ni 8675  df-lti 8678  df-nq 8715
  Copyright terms: Public domain W3C validator