Theorem lterpq 9226

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 9166	. . . 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 4995	. . . 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 3470	. . 3 |- <pQ C_ ( ( N. X. N. ) X. ( N. X. N. ) )
4	3	brel 4971	. 2 |- ( A <pQ B -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
5		ltrelnq 9182	. . . 4 |- <Q C_ ( Q. X. Q. )
6	5	brel 4971	. . 3 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) )
7		elpqn 9181	. . . 4 |- ( ( /Q ` A ) e. Q. -> ( /Q ` A ) e. ( N. X. N. ) )
8		elpqn 9181	. . . 4 |- ( ( /Q ` B ) e. Q. -> ( /Q ` B ) e. ( N. X. N. ) )
9		nqerf 9186	. . . . . . 7 |- /Q : ( N. X. N. ) --> Q.
10	9	fdmi 5648	. . . . . 6 |- dom /Q = ( N. X. N. )
11		0nelxp 4951	. . . . . 6 |- -. (/) e. ( N. X. N. )
12	10, 11	ndmfvrcl 5800	. . . . 5 |- ( ( /Q ` A ) e. ( N. X. N. ) -> A e. ( N. X. N. ) )
13	10, 11	ndmfvrcl 5800	. . . . 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 6692	. . . . 5 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
18		xp2nd 6693	. . . . 5 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
19		mulclpi 9149	. . . . 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 9160	. . . 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 9187	. . . 4 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
24		nqercl 9187	. . . 4 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
25		ordpinq 9199	. . . 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 6699	. . . . . 6 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
28		1st2nd2 6699	. . . . . 6 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
29	27, 28	breqan12d 4391	. . . . 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 9198	. . . . 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 6692	. . . . . . 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 6693	. . . . . . 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 9149	. . . . . 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 9160	. . . . 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 9152	. . . . . 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 9188	. . . . . . . . 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 9176	. . . . . . . . . 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 2457	. . . . . . 7 |- ( A e. ( N. X. N. ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) = ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) )
48		nqerrel 9188	. . . . . . . 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 9176	. . . . . . . . 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 6195	. . . . . 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 9152	. . . . . . 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 5785	. . . . . . . 8 |- ( 1st ` B ) e. _V
56		fvex 5785	. . . . . . . 8 |- ( 2nd ` A ) e. _V
57		fvex 5785	. . . . . . . 8 |- ( 1st ` ( /Q ` A ) ) e. _V
58		mulcompi 9152	. . . . . . . 8 |- ( x .N y ) = ( y .N x )
59		mulasspi 9153	. . . . . . . 8 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
60		fvex 5785	. . . . . . . 8 |- ( 2nd ` ( /Q ` B ) ) e. _V
61	55, 56, 57, 58, 59, 60	caov411 6381	. . . . . . 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 2478	. . . . . 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 9152	. . . . . . 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 5785	. . . . . . . 8 |- ( 1st ` ( /Q ` B ) ) e. _V
65		fvex 5785	. . . . . . . 8 |- ( 2nd ` ( /Q ` A ) ) e. _V
66		fvex 5785	. . . . . . . 8 |- ( 1st ` A ) e. _V
67		fvex 5785	. . . . . . . 8 |- ( 2nd ` B ) e. _V
68	64, 65, 66, 58, 59, 67	caov411 6381	. . . . . . 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 2478	. . . . . 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 2515	. . . . 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 4389	. . . 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 1370   e. wcel 1757   <.cop 3967   class class class wbr 4376   {copab 4433   X. cxp 4922   ` cfv 5502  (class class class)co 6176   1stc1st 6661   2ndc2nd 6662   N.cnpi 9098   .N cmi 9100   <N clti 9101   <pQ cltpq 9104   ~Q ceq 9105   Q.cnq 9106   /Qcerq 9108   <Q cltq 9112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-omul 7011  df-er 7187  df-ni 9128  df-mi 9130  df-lti 9131  df-ltpq 9166  df-enq 9167  df-nq 9168  df-erq 9169  df-1nq 9172  df-ltnq 9174

This theorem is referenced by:  ltanq  9227  ltmnq  9228  1lt2nq  9229