Theorem ordpinq 9765

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

Step	Hyp	Ref	Expression
1		brinxp 5181	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <pQ B <-> A ( <pQ i^i ( Q. X. Q. ) ) B ) )
2		df-ltnq 9740	. . . 4 |- <Q = ( <pQ i^i ( Q. X. Q. ) )
3	2	breqi 4659	. . 3 |- ( A <Q B <-> A ( <pQ i^i ( Q. X. Q. ) ) B )
4	1, 3	syl6bbr 278	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <pQ B <-> A <Q B ) )
5		relxp 5227	. . . . 5 |- Rel ( N. X. N. )
6		elpqn 9747	. . . . 5 |- ( A e. Q. -> A e. ( N. X. N. ) )
7		1st2nd 7214	. . . . 5 |- ( ( Rel ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
8	5, 6, 7	sylancr 695	. . . 4 |- ( A e. Q. -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
9		elpqn 9747	. . . . 5 |- ( B e. Q. -> B e. ( N. X. N. ) )
10		1st2nd 7214	. . . . 5 |- ( ( Rel ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
11	5, 9, 10	sylancr 695	. . . 4 |- ( B e. Q. -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
12	8, 11	breqan12d 4669	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <pQ B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. <pQ <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
13		ordpipq 9764	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. <pQ <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) )
14	12, 13	syl6bb 276	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <pQ B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
15	4, 14	bitr3d 270	1 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   <.cop 4183   class class class wbr 4653   X. cxp 5112   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   <N clti 9669   <pQ cltpq 9672   Q.cnq 9674   <Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-omul 7565  df-ni 9694  df-mi 9696  df-lti 9697  df-ltpq 9732  df-nq 9734  df-ltnq 9740

This theorem is referenced by:  ltsonq  9791  lterpq  9792  ltanq  9793  ltmnq  9794  ltexnq  9797  archnq  9802