Theorem lterpq 9421

Description: Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
lterpq	|- ( A <pQ B <-> ( /Q ` A ) <Q ( /Q ` B ) )

Proof of Theorem lterpq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltpq 9361	. . . 4 |- <pQ = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) }
2		opabssxp 4928	. . . 4 |- { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) } C_ ( ( N. X. N. ) X. ( N. X. N. ) )
3	1, 2	eqsstri 3474	. . 3 |- <pQ C_ ( ( N. X. N. ) X. ( N. X. N. ) )
4	3	brel 4902	. 2 |- ( A <pQ B -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
5		ltrelnq 9377	. . . 4 |- <Q C_ ( Q. X. Q. )
6	5	brel 4902	. . 3 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) )
7		elpqn 9376	. . . 4 |- ( ( /Q ` A ) e. Q. -> ( /Q ` A ) e. ( N. X. N. ) )
8		elpqn 9376	. . . 4 |- ( ( /Q ` B ) e. Q. -> ( /Q ` B ) e. ( N. X. N. ) )
9		nqerf 9381	. . . . . . 7 |- /Q : ( N. X. N. ) --> Q.
10	9	fdmi 5757	. . . . . 6 |- dom /Q = ( N. X. N. )
11		0nelxp 4881	. . . . . 6 |- -. (/) e. ( N. X. N. )
12	10, 11	ndmfvrcl 5913	. . . . 5 |- ( ( /Q ` A ) e. ( N. X. N. ) -> A e. ( N. X. N. ) )
13	10, 11	ndmfvrcl 5913	. . . . 5 |- ( ( /Q ` B ) e. ( N. X. N. ) -> B e. ( N. X. N. ) )
14	12, 13	anim12i 574	. . . 4 |- ( ( ( /Q ` A ) e. ( N. X. N. ) /\ ( /Q ` B ) e. ( N. X. N. ) ) -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
15	7, 8, 14	syl2an 484	. . 3 |- ( ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
16	6, 15	syl 17	. 2 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
17		xp1st 6850	. . . . 5 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
18		xp2nd 6851	. . . . 5 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
19		mulclpi 9344	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
20	17, 18, 19	syl2an 484	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
21		ltmpi 9355	. . . 4 |- ( ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. -> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) <N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) <-> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) <N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) ) )
22	20, 21	syl 17	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) <N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) <-> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) <N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) ) )
23		nqercl 9382	. . . 4 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
24		nqercl 9382	. . . 4 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
25		ordpinq 9394	. . . 4 |- ( ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) -> ( ( /Q ` A ) <Q ( /Q ` B ) <-> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) <N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) )
26	23, 24, 25	syl2an 484	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( /Q ` A ) <Q ( /Q ` B ) <-> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) <N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) )
27		1st2nd2 6857	. . . . . 6 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
28		1st2nd2 6857	. . . . . 6 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
29	27, 28	breqan12d 4432	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A <pQ B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. <pQ <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
30		ordpipq 9393	. . . . 5 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. <pQ <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) )
31	29, 30	syl6bb 269	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A <pQ B <-> ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) )
32		xp1st 6850	. . . . . . 7 |- ( ( /Q ` A ) e. ( N. X. N. ) -> ( 1st ` ( /Q ` A ) ) e. N. )
33	23, 7, 32	3syl 18	. . . . . 6 |- ( A e. ( N. X. N. ) -> ( 1st ` ( /Q ` A ) ) e. N. )
34		xp2nd 6851	. . . . . . 7 |- ( ( /Q ` B ) e. ( N. X. N. ) -> ( 2nd ` ( /Q ` B ) ) e. N. )
35	24, 8, 34	3syl 18	. . . . . 6 |- ( B e. ( N. X. N. ) -> ( 2nd ` ( /Q ` B ) ) e. N. )
36		mulclpi 9344	. . . . . 6 |- ( ( ( 1st ` ( /Q ` A ) ) e. N. /\ ( 2nd ` ( /Q ` B ) ) e. N. ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) e. N. )
37	33, 35, 36	syl2an 484	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) e. N. )
38		ltmpi 9355	. . . . 5 |- ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) e. N. -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) )
39	37, 38	syl 17	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) <N ( ( 1st ` B ) .N ( 2nd ` A ) ) <-> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) ) )
40		mulcompi 9347	. . . . . 6 |- ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) )
41	40	a1i 11	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) )
42		nqerrel 9383	. . . . . . . . 9 |- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )
43	23, 7	syl 17	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. ( N. X. N. ) )
44		enqbreq2 9371	. . . . . . . . . 10 |- ( ( A e. ( N. X. N. ) /\ ( /Q ` A ) e. ( N. X. N. ) ) -> ( A ~Q ( /Q ` A ) <-> ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) = ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) ) )
45	43, 44	mpdan 679	. . . . . . . . 9 |- ( A e. ( N. X. N. ) -> ( A ~Q ( /Q ` A ) <-> ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) = ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) ) )
46	42, 45	mpbid 215	. . . . . . . 8 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) = ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) )
47	46	eqcomd 2468	. . . . . . 7 |- ( A e. ( N. X. N. ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) = ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) )
48		nqerrel 9383	. . . . . . . 8 |- ( B e. ( N. X. N. ) -> B ~Q ( /Q ` B ) )
49	24, 8	syl 17	. . . . . . . . 9 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. ( N. X. N. ) )
50		enqbreq2 9371	. . . . . . . . 9 |- ( ( B e. ( N. X. N. ) /\ ( /Q ` B ) e. ( N. X. N. ) ) -> ( B ~Q ( /Q ` B ) <-> ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) = ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) ) )
51	49, 50	mpdan 679	. . . . . . . 8 |- ( B e. ( N. X. N. ) -> ( B ~Q ( /Q ` B ) <-> ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) = ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) ) )
52	48, 51	mpbid 215	. . . . . . 7 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) = ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) )
53	47, 52	oveqan12d 6334	. . . . . 6 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) ) )
54		mulcompi 9347	. . . . . . 7 |- ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) )
55		fvex 5898	. . . . . . . 8 |- ( 1st ` B ) e. _V
56		fvex 5898	. . . . . . . 8 |- ( 2nd ` A ) e. _V
57		fvex 5898	. . . . . . . 8 |- ( 1st ` ( /Q ` A ) ) e. _V
58		mulcompi 9347	. . . . . . . 8 |- ( x .N y ) = ( y .N x )
59		mulasspi 9348	. . . . . . . 8 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
60		fvex 5898	. . . . . . . 8 |- ( 2nd ` ( /Q ` B ) ) e. _V
61	55, 56, 57, 58, 59, 60	caov411 6528	. . . . . . 7 |- ( ( ( 1st ` B ) .N ( 2nd ` A ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) = ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) )
62	54, 61	eqtri 2484	. . . . . 6 |- ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) .N ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) )
63		mulcompi 9347	. . . . . . 7 |- ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) = ( ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) )
64		fvex 5898	. . . . . . . 8 |- ( 1st ` ( /Q ` B ) ) e. _V
65		fvex 5898	. . . . . . . 8 |- ( 2nd ` ( /Q ` A ) ) e. _V
66		fvex 5898	. . . . . . . 8 |- ( 1st ` A ) e. _V
67		fvex 5898	. . . . . . . 8 |- ( 2nd ` B ) e. _V
68	64, 65, 66, 58, 59, 67	caov411 6528	. . . . . . 7 |- ( ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) )
69	63, 68	eqtri 2484	. . . . . 6 |- ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) )
70	53, 62, 69	3eqtr4g 2521	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) = ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) )
71	41, 70	breq12d 4429	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` A ) .N ( 2nd ` B ) ) ) <N ( ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) .N ( ( 1st ` B ) .N ( 2nd ` A ) ) ) <-> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) <N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) ) )
72	31, 39, 71	3bitrd 287	. . 3 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A <pQ B <-> ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) ) <N ( ( ( 1st ` A ) .N ( 2nd ` B ) ) .N ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` ( /Q ` A ) ) ) ) ) )
73	22, 26, 72	3bitr4rd 294	. 2 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( A <pQ B <-> ( /Q ` A ) <Q ( /Q ` B ) ) )
74	4, 16, 73	pm5.21nii 359	1 |- ( A <pQ B <-> ( /Q ` A ) <Q ( /Q ` B ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   <.cop 3986   class class class wbr 4416   {copab 4474   X. cxp 4851   ` cfv 5601  (class class class)co 6315   1stc1st 6818   2ndc2nd 6819   N.cnpi 9295   .N cmi 9297   <N clti 9298   <pQ cltpq 9301   ~Q ceq 9302   Q.cnq 9303   /Qcerq 9305   <Q cltq 9309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  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-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-omul 7213  df-er 7389  df-ni 9323  df-mi 9325  df-lti 9326  df-ltpq 9361  df-enq 9362  df-nq 9363  df-erq 9364  df-1nq 9367  df-ltnq 9369

This theorem is referenced by:  ltanq  9422  ltmnq  9423  1lt2nq  9424