Theorem lterpq 9792

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 9732	. . . 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 5193	. . . 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 3635	. . 3 |- <pQ C_ ( ( N. X. N. ) X. ( N. X. N. ) )
4	3	brel 5168	. 2 |- ( A <pQ B -> ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) )
5		ltrelnq 9748	. . . 4 |- <Q C_ ( Q. X. Q. )
6	5	brel 5168	. . 3 |- ( ( /Q ` A ) <Q ( /Q ` B ) -> ( ( /Q ` A ) e. Q. /\ ( /Q ` B ) e. Q. ) )
7		elpqn 9747	. . . 4 |- ( ( /Q ` A ) e. Q. -> ( /Q ` A ) e. ( N. X. N. ) )
8		elpqn 9747	. . . 4 |- ( ( /Q ` B ) e. Q. -> ( /Q ` B ) e. ( N. X. N. ) )
9		nqerf 9752	. . . . . . 7 |- /Q : ( N. X. N. ) --> Q.
10	9	fdmi 6052	. . . . . 6 |- dom /Q = ( N. X. N. )
11		0nelxp 5143	. . . . . 6 |- -. (/) e. ( N. X. N. )
12	10, 11	ndmfvrcl 6219	. . . . 5 |- ( ( /Q ` A ) e. ( N. X. N. ) -> A e. ( N. X. N. ) )
13	10, 11	ndmfvrcl 6219	. . . . 5 |- ( ( /Q ` B ) e. ( N. X. N. ) -> B e. ( N. X. N. ) )
14	12, 13	anim12i 590	. . . 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 494	. . 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 7198	. . . . 5 |- ( A e. ( N. X. N. ) -> ( 1st ` A ) e. N. )
18		xp2nd 7199	. . . . 5 |- ( B e. ( N. X. N. ) -> ( 2nd ` B ) e. N. )
19		mulclpi 9715	. . . . 5 |- ( ( ( 1st ` A ) e. N. /\ ( 2nd ` B ) e. N. ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
20	17, 18, 19	syl2an 494	. . . 4 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` A ) .N ( 2nd ` B ) ) e. N. )
21		ltmpi 9726	. . . 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 9753	. . . 4 |- ( A e. ( N. X. N. ) -> ( /Q ` A ) e. Q. )
24		nqercl 9753	. . . 4 |- ( B e. ( N. X. N. ) -> ( /Q ` B ) e. Q. )
25		ordpinq 9765	. . . 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 494	. . 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 7205	. . . . . 6 |- ( A e. ( N. X. N. ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
28		1st2nd2 7205	. . . . . 6 |- ( B e. ( N. X. N. ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
29	27, 28	breqan12d 4669	. . . . 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 9764	. . . . 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 276	. . . 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 7198	. . . . . . 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 7199	. . . . . . 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 9715	. . . . . 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 494	. . . . 5 |- ( ( A e. ( N. X. N. ) /\ B e. ( N. X. N. ) ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` ( /Q ` B ) ) ) e. N. )
38		ltmpi 9726	. . . . 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 9718	. . . . . 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 9754	. . . . . . . . 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 9742	. . . . . . . . . 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 702	. . . . . . . . 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 222	. . . . . . . 8 |- ( A e. ( N. X. N. ) -> ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) = ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) )
47	46	eqcomd 2628	. . . . . . 7 |- ( A e. ( N. X. N. ) -> ( ( 1st ` ( /Q ` A ) ) .N ( 2nd ` A ) ) = ( ( 1st ` A ) .N ( 2nd ` ( /Q ` A ) ) ) )
48		nqerrel 9754	. . . . . . . 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 9742	. . . . . . . . 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 702	. . . . . . . 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 222	. . . . . . 7 |- ( B e. ( N. X. N. ) -> ( ( 1st ` B ) .N ( 2nd ` ( /Q ` B ) ) ) = ( ( 1st ` ( /Q ` B ) ) .N ( 2nd ` B ) ) )
53	47, 52	oveqan12d 6669	. . . . . 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 9718	. . . . . . 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 6201	. . . . . . . 8 |- ( 1st ` B ) e. _V
56		fvex 6201	. . . . . . . 8 |- ( 2nd ` A ) e. _V
57		fvex 6201	. . . . . . . 8 |- ( 1st ` ( /Q ` A ) ) e. _V
58		mulcompi 9718	. . . . . . . 8 |- ( x .N y ) = ( y .N x )
59		mulasspi 9719	. . . . . . . 8 |- ( ( x .N y ) .N z ) = ( x .N ( y .N z ) )
60		fvex 6201	. . . . . . . 8 |- ( 2nd ` ( /Q ` B ) ) e. _V
61	55, 56, 57, 58, 59, 60	caov411 6866	. . . . . . 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 2644	. . . . . 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 9718	. . . . . . 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 6201	. . . . . . . 8 |- ( 1st ` ( /Q ` B ) ) e. _V
65		fvex 6201	. . . . . . . 8 |- ( 2nd ` ( /Q ` A ) ) e. _V
66		fvex 6201	. . . . . . . 8 |- ( 1st ` A ) e. _V
67		fvex 6201	. . . . . . . 8 |- ( 2nd ` B ) e. _V
68	64, 65, 66, 58, 59, 67	caov411 6866	. . . . . . 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 2644	. . . . . 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 2681	. . . . 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 4666	. . . 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 294	. . 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 301	. 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 368	1 |- ( A <pQ B <-> ( /Q ` A ) <Q ( /Q ` B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   <N clti 9669   <pQ cltpq 9672   ~Q ceq 9673   Q.cnq 9674   /Qcerq 9676   <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-reu 2919  df-rmo 2920  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-pw 4160  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-pred 5680  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltanq  9793  ltmnq  9794  1lt2nq  9795