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Theorem ordpinq 9367
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 4917 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B ) )
2 df-ltnq 9342 . . . 4  |-  <Q  =  (  <pQ  i^i  ( Q.  X.  Q. ) )
32breqi 4432 . . 3  |-  ( A 
<Q  B  <->  A (  <pQ  i^i  ( Q.  X.  Q. ) ) B )
41, 3syl6bbr 266 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  A  <Q  B ) )
5 relxp 4962 . . . . 5  |-  Rel  ( N.  X.  N. )
6 elpqn 9349 . . . . 5  |-  ( A  e.  Q.  ->  A  e.  ( N.  X.  N. ) )
7 1st2nd 6853 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  A  e.  ( N.  X.  N. ) )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
85, 6, 7sylancr 667 . . . 4  |-  ( A  e.  Q.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
9 elpqn 9349 . . . . 5  |-  ( B  e.  Q.  ->  B  e.  ( N.  X.  N. ) )
10 1st2nd 6853 . . . . 5  |-  ( ( Rel  ( N.  X.  N. )  /\  B  e.  ( N.  X.  N. ) )  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
115, 9, 10sylancr 667 . . . 4  |-  ( B  e.  Q.  ->  B  =  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
128, 11breqan12d 4441 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  <pQ  <. ( 1st `  B
) ,  ( 2nd `  B ) >. )
)
13 ordpipq 9366 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  <pQ  <. ( 1st `  B ) ,  ( 2nd `  B
) >. 
<->  ( ( 1st `  A
)  .N  ( 2nd `  B ) )  <N 
( ( 1st `  B
)  .N  ( 2nd `  A ) ) )
1412, 13syl6bb 264 . 2  |-  ( ( A  e.  Q.  /\  B  e.  Q. )  ->  ( A  <pQ  B  <->  ( ( 1st `  A )  .N  ( 2nd `  B
) )  <N  (
( 1st `  B
)  .N  ( 2nd `  A ) ) ) )
154, 14bitr3d 258 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 187    /\ wa 370    = wceq 1437    e. wcel 1870    i^i cin 3441   <.cop 4008   class class class wbr 4426    X. cxp 4852   Rel wrel 4859   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   N.cnpi 9268    .N cmi 9270    <N clti 9271    <pQ cltpq 9274   Q.cnq 9276    <Q cltq 9282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-omul 7195  df-ni 9296  df-mi 9298  df-lti 9299  df-ltpq 9334  df-nq 9336  df-ltnq 9342
This theorem is referenced by:  ltsonq  9393  lterpq  9394  ltanq  9395  ltmnq  9396  ltexnq  9399  archnq  9404
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