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

Theorem pinq 9111
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 9067 . . . 4  |-  1o  e.  N.
2 opelxpi 4886 . . . 4  |-  ( ( A  e.  N.  /\  1o  e.  N. )  ->  <. A ,  1o >.  e.  ( N.  X.  N. ) )
31, 2mpan2 671 . . 3  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  ( N. 
X.  N. ) )
4 nlt1pi 9090 . . . . . 6  |-  -.  ( 2nd `  y )  <N  1o
51elexi 2997 . . . . . . . 8  |-  1o  e.  _V
6 op2ndg 6605 . . . . . . . 8  |-  ( ( A  e.  N.  /\  1o  e.  _V )  -> 
( 2nd `  <. A ,  1o >. )  =  1o )
75, 6mpan2 671 . . . . . . 7  |-  ( A  e.  N.  ->  ( 2nd `  <. A ,  1o >. )  =  1o )
87breq2d 4319 . . . . . 6  |-  ( A  e.  N.  ->  (
( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )  <->  ( 2nd `  y ) 
<N  1o ) )
94, 8mtbiri 303 . . . . 5  |-  ( A  e.  N.  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  <. A ,  1o >. )
)
109a1d 25 . . . 4  |-  ( A  e.  N.  ->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y )  <N 
( 2nd `  <. A ,  1o >. )
) )
1110ralrimivw 2815 . . 3  |-  ( A  e.  N.  ->  A. y  e.  ( N.  X.  N. ) ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) )
12 breq1 4310 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( x  ~Q  y 
<-> 
<. A ,  1o >.  ~Q  y ) )
13 fveq2 5706 . . . . . . . 8  |-  ( x  =  <. A ,  1o >.  ->  ( 2nd `  x
)  =  ( 2nd `  <. A ,  1o >. ) )
1413breq2d 4319 . . . . . . 7  |-  ( x  =  <. A ,  1o >.  ->  ( ( 2nd `  y )  <N  ( 2nd `  x )  <->  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1514notbid 294 . . . . . 6  |-  ( x  =  <. A ,  1o >.  ->  ( -.  ( 2nd `  y )  <N 
( 2nd `  x
)  <->  -.  ( 2nd `  y )  <N  ( 2nd `  <. A ,  1o >. ) ) )
1612, 15imbi12d 320 . . . . 5  |-  ( x  =  <. A ,  1o >.  ->  ( ( x  ~Q  y  ->  -.  ( 2nd `  y ) 
<N  ( 2nd `  x
) )  <->  ( <. A ,  1o >.  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  <. A ,  1o >. )
) ) )
1716ralbidv 2750 . . . 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 3132 . . 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 664 . 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 9096 . 2  |-  Q.  =  { x  e.  ( N.  X.  N. )  | 
A. y  e.  ( N.  X.  N. )
( x  ~Q  y  ->  -.  ( 2nd `  y
)  <N  ( 2nd `  x
) ) }
2119, 20syl6eleqr 2534 1  |-  ( A  e.  N.  ->  <. A ,  1o >.  e.  Q. )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2730   {crab 2734   _Vcvv 2987   <.cop 3898   class class class wbr 4307    X. cxp 4853   ` cfv 5433   2ndc2nd 6591   1oc1o 6928   N.cnpi 9026    <N clti 9029    ~Q ceq 9033   Q.cnq 9034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-iota 5396  df-fun 5435  df-fv 5441  df-om 6492  df-2nd 6593  df-1o 6935  df-ni 9056  df-lti 9059  df-nq 9096
This theorem is referenced by:  1nq  9112  archnq  9164  prlem934  9217
  Copyright terms: Public domain W3C validator