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Theorem ordpinq 8755
Description: Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ordpinq  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )

Proof of Theorem ordpinq
StepHypRef Expression
1 brinxp 4882 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B ) )
2 df-ltnq 8730 . . . 4  |-  <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
32breqi 4161 . . 3  |-  ( A 
<Q  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B )
41, 3syl6bbr 255 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A  <Q  B ) )
5 relxp 4925 . . . . 5  |-  Rel  ( N.  X.  N. )
6 elpqn 8737 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
7 1st2nd 6334 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
85, 6, 7sylancr 645 . . . 4  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
9 elpqn 8737 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
10 1st2nd 6334 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
115, 9, 10sylancr 645 . . . 4  |-  ( B  e.  Q.  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
128, 11breqan12d 4170 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  <pQ  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
13 ordpipq 8754 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
1412, 13syl6bb 253 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
154, 14bitr3d 247 1  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <Q  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    i^i cin 3264   <.cop 3762   class class class wbr 4155    X. cxp 4818   Rel wrel 4825   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   N.cnpi 8654    .N cmi 8656    <N clti 8657    <pQ cltpq 8660   Q.cnq 8662    <Q cltq 8668
This theorem is referenced by:  ltsonq  8781  lterpq  8782  ltanq  8783  ltmnq  8784  ltexnq  8787  archnq  8792
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-omul 6667  df-ni 8684  df-mi 8686  df-lti 8687  df-ltpq 8722  df-nq 8724  df-ltnq 8730
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