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

Theorem ordpinq 9112
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 4901 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B ) )
2 df-ltnq 9087 . . . 4  |-  <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
32breqi 4298 . . 3  |-  ( A 
<Q  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B )
41, 3syl6bbr 263 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A  <Q  B ) )
5 relxp 4947 . . . . 5  |-  Rel  ( N.  X.  N. )
6 elpqn 9094 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
7 1st2nd 6620 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
85, 6, 7sylancr 663 . . . 4  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
9 elpqn 9094 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
10 1st2nd 6620 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
115, 9, 10sylancr 663 . . . 4  |-  ( B  e.  Q.  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
128, 11breqan12d 4307 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  <pQ  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
13 ordpipq 9111 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
1412, 13syl6bb 261 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
154, 14bitr3d 255 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3327   <.cop 3883   class class class wbr 4292    X. cxp 4838   Rel wrel 4845   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   N.cnpi 9011    .N cmi 9013    <N clti 9014    <pQ cltpq 9017   Q.cnq 9019    <Q cltq 9025
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-omul 6925  df-ni 9041  df-mi 9043  df-lti 9044  df-ltpq 9079  df-nq 9081  df-ltnq 9087
This theorem is referenced by:  ltsonq  9138  lterpq  9139  ltanq  9140  ltmnq  9141  ltexnq  9144  archnq  9149
  Copyright terms: Public domain W3C validator