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Theorem ordpinq 9322
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 5062 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B ) )
2 df-ltnq 9297 . . . 4  |-  <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
32breqi 4453 . . 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 5110 . . . . 5  |-  Rel  ( N.  X.  N. )
6 elpqn 9304 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
7 1st2nd 6831 . . . . 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 9304 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
10 1st2nd 6831 . . . . 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 4462 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  <pQ  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
13 ordpipq 9321 . . 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 1379    e. wcel 1767    i^i cin 3475   <.cop 4033   class class class wbr 4447    X. cxp 4997   Rel wrel 5004   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   N.cnpi 9223    .N cmi 9225    <N clti 9226    <pQ cltpq 9229   Q.cnq 9231    <Q cltq 9237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-omul 7136  df-ni 9251  df-mi 9253  df-lti 9254  df-ltpq 9289  df-nq 9291  df-ltnq 9297
This theorem is referenced by:  ltsonq  9348  lterpq  9349  ltanq  9350  ltmnq  9351  ltexnq  9354  archnq  9359
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