Theorem lterpq 9344

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 9284	. . . 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 5072	. . . 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 3534	. . 3 |- <pQ C_ ( ( N. X. N. ) X. ( N. X. N. ) )
4	3	brel 5047	. 2 |- ( A <pQ B -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
5		ltrelnq 9300	. . . 4 |- <Q C_ ( Q. X. Q. )
6	5	brel 5047	. . 3 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) )
7		elpqn 9299	. . . 4 |- ( ( /Q ` A ) e. Q. -> ( /Q ` A ) e. ( N. X. N. ) )
8		elpqn 9299	. . . 4 |- ( ( /Q ` B ) e. Q. -> ( /Q ` B ) e. ( N. X. N. ) )
9		nqerf 9304	. . . . . . 7 |- /Q : ( N. X. N. ) --> Q.
10	9	fdmi 5734	. . . . . 6 |- dom /Q = ( N. X. N. )
11		0nelxp 5026	. . . . . 6 |- -. (/) e. ( N. X. N. )
12	10, 11	ndmfvrcl 5889	. . . . 5 |- ( ( /Q ` A ) e. ( N. X. N. ) -> A e. ( N. X. N. ) )
13	10, 11	ndmfvrcl 5889	. . . . 5 |- ( ( /Q ` B ) e. ( N. X. N. ) -> B e. ( N. X. N. ) )
14	12, 13	anim12i 566	. . . 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 477	. . 3 |- ( ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
16	6, 15	syl 16	. 2 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
17		xp1st 6811	. . . . 5 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
18		xp2nd 6812	. . . . 5 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
19		mulclpi 9267	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
20	17, 18, 19	syl2an 477	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
21		ltmpi 9278	. . . 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 16	. . 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 9305	. . . 4 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
24		nqercl 9305	. . . 4 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
25		ordpinq 9317	. . . 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 477	. . 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 6818	. . . . . 6 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
28		1st2nd2 6818	. . . . . 6 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
29	27, 28	breqan12d 4462	. . . . 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 9316	. . . . 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 261	. . . 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 6811	. . . . . . 7 |- ( ( /Q ` A ) e. ( N. X. N. ) -> ( 1st ` ( /Q ` A ) ) e. N. )
33	23, 7, 32	3syl 20	. . . . . 6 |- ( A e. ( N. X. N. ) -> ( 1st ` ( /Q ` A ) ) e. N. )
34		xp2nd 6812	. . . . . . 7 |- ( ( /Q ` B ) e. ( N. X. N. ) -> ( 2nd ` ( /Q ` B ) ) e. N. )
35	24, 8, 34	3syl 20	. . . . . 6 |- ( B e. ( N. X. N. ) -> ( 2nd ` ( /Q ` B ) ) e. N. )
36		mulclpi 9267	. . . . . 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 477	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) e. N. )
38		ltmpi 9278	. . . . 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 16	. . . 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 9270	. . . . . 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 9306	. . . . . . . . 9 |- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )
43	23, 7	syl 16	. . . . . . . . . 10 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. ( N. X. N. ) )
44		enqbreq2 9294	. . . . . . . . . 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 668	. . . . . . . . 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 210	. . . . . . . 8 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) = ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) )
47	46	eqcomd 2475	. . . . . . 7 |- ( A e. ( N. X. N. ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) = ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) )
48		nqerrel 9306	. . . . . . . 8 |- ( B e. ( N. X. N. ) -> B ~Q ( /Q ` B ) )
49	24, 8	syl 16	. . . . . . . . 9 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. ( N. X. N. ) )
50		enqbreq2 9294	. . . . . . . . 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 668	. . . . . . . 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 210	. . . . . . 7 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) = ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) )
53	47, 52	oveqan12d 6301	. . . . . 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 9270	. . . . . . 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 5874	. . . . . . . 8 |- ( 1st ` B ) e. _V
56		fvex 5874	. . . . . . . 8 |- ( 2nd ` A ) e. _V
57		fvex 5874	. . . . . . . 8 |- ( 1st ` ( /Q ` A ) ) e. _V
58		mulcompi 9270	. . . . . . . 8 |- ( x .N y ) = ( y .N x )
59		mulasspi 9271	. . . . . . . 8 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
60		fvex 5874	. . . . . . . 8 |- ( 2nd ` ( /Q ` B ) ) e. _V
61	55, 56, 57, 58, 59, 60	caov411 6489	. . . . . . 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 2496	. . . . . 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 9270	. . . . . . 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 5874	. . . . . . . 8 |- ( 1st ` ( /Q ` B ) ) e. _V
65		fvex 5874	. . . . . . . 8 |- ( 2nd ` ( /Q ` A ) ) e. _V
66		fvex 5874	. . . . . . . 8 |- ( 1st ` A ) e. _V
67		fvex 5874	. . . . . . . 8 |- ( 2nd ` B ) e. _V
68	64, 65, 66, 58, 59, 67	caov411 6489	. . . . . . 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 2496	. . . . . 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 2533	. . . . 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 4460	. . . 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 279	. . 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 286	. 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 353	1 |- ( A <pQ B <-> ( /Q ` A ) <Q ( /Q ` B ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   <.cop 4033   class class class wbr 4447   {copab 4504   X. cxp 4997   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   N.cnpi 9218   .N cmi 9220   <N clti 9221   <pQ cltpq 9224   ~Q ceq 9225   Q.cnq 9226   /Qcerq 9228   <Q cltq 9232

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 6574

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-reu 2821  df-rmo 2822  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-pw 4012  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ni 9246  df-mi 9248  df-lti 9249  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-1nq 9290  df-ltnq 9292

This theorem is referenced by:  ltanq  9345  ltmnq  9346  1lt2nq  9347