Theorem ltsonq 9791

Description: 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltsonq	|- <Q Or Q.

Proof of Theorem ltsonq

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

Step	Hyp	Ref	Expression
1		elpqn 9747	. . . . . . 7 |- ( x e. Q. -> x e. ( N. X. N. ) )
2	1	adantr 481	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. ) -> x e. ( N. X. N. ) )
3		xp1st 7198	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
4	2, 3	syl 17	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( 1st ` x ) e. N. )
5		elpqn 9747	. . . . . . 7 |- ( y e. Q. -> y e. ( N. X. N. ) )
6	5	adantl 482	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. ) -> y e. ( N. X. N. ) )
7		xp2nd 7199	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
8	6, 7	syl 17	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( 2nd ` y ) e. N. )
9		mulclpi 9715	. . . . 5 |- ( ( ( 1st ` x ) e. N. /\ ( 2nd ` y ) e. N. ) -> ( ( 1st ` x ) .N ( 2nd ` y ) ) e. N. )
10	4, 8, 9	syl2anc 693	. . . 4 |- ( ( x e. Q. /\ y e. Q. ) -> ( ( 1st ` x ) .N ( 2nd ` y ) ) e. N. )
11		xp1st 7198	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
12	6, 11	syl 17	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( 1st ` y ) e. N. )
13		xp2nd 7199	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
14	2, 13	syl 17	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( 2nd ` x ) e. N. )
15		mulclpi 9715	. . . . 5 |- ( ( ( 1st ` y ) e. N. /\ ( 2nd ` x ) e. N. ) -> ( ( 1st ` y ) .N ( 2nd ` x ) ) e. N. )
16	12, 14, 15	syl2anc 693	. . . 4 |- ( ( x e. Q. /\ y e. Q. ) -> ( ( 1st ` y ) .N ( 2nd ` x ) ) e. N. )
17		ltsopi 9710	. . . . 5 |- <N Or N.
18		sotric 5061	. . . . 5 |- ( ( <N Or N. /\ ( ( ( 1st ` x ) .N ( 2nd ` y ) ) e. N. /\ ( ( 1st ` y ) .N ( 2nd ` x ) ) e. N. ) ) -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) <-> -. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) \/ ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) ) )
19	17, 18	mpan 706	. . . 4 |- ( ( ( ( 1st ` x ) .N ( 2nd ` y ) ) e. N. /\ ( ( 1st ` y ) .N ( 2nd ` x ) ) e. N. ) -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) <-> -. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) \/ ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) ) )
20	10, 16, 19	syl2anc 693	. . 3 |- ( ( x e. Q. /\ y e. Q. ) -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) <-> -. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) \/ ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) ) )
21		ordpinq 9765	. . 3 |- ( ( x e. Q. /\ y e. Q. ) -> ( x <Q y <-> ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
22		fveq2 6191	. . . . . . 7 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
23		fveq2 6191	. . . . . . . 8 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
24	23	eqcomd 2628	. . . . . . 7 |- ( x = y -> ( 2nd ` y ) = ( 2nd ` x ) )
25	22, 24	oveq12d 6668	. . . . . 6 |- ( x = y -> ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) )
26		enqbreq2 9742	. . . . . . . 8 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( x ~Q y <-> ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
27	1, 5, 26	syl2an 494	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x ~Q y <-> ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
28		enqeq 9756	. . . . . . . 8 |- ( ( x e. Q. /\ y e. Q. /\ x ~Q y ) -> x = y )
29	28	3expia 1267	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. ) -> ( x ~Q y -> x = y ) )
30	27, 29	sylbird 250	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. ) -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) -> x = y ) )
31	25, 30	impbid2 216	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x = y <-> ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
32		ordpinq 9765	. . . . . 6 |- ( ( y e. Q. /\ x e. Q. ) -> ( y <Q x <-> ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) )
33	32	ancoms 469	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( y <Q x <-> ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) )
34	31, 33	orbi12d 746	. . . 4 |- ( ( x e. Q. /\ y e. Q. ) -> ( ( x = y \/ y <Q x ) <-> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) \/ ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) ) )
35	34	notbid 308	. . 3 |- ( ( x e. Q. /\ y e. Q. ) -> ( -. ( x = y \/ y <Q x ) <-> -. ( ( ( 1st ` x ) .N ( 2nd ` y ) ) = ( ( 1st ` y ) .N ( 2nd ` x ) ) \/ ( ( 1st ` y ) .N ( 2nd ` x ) ) <N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) ) )
36	20, 21, 35	3bitr4d 300	. 2 |- ( ( x e. Q. /\ y e. Q. ) -> ( x <Q y <-> -. ( x = y \/ y <Q x ) ) )
37	21	3adant3 1081	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) )
38		elpqn 9747	. . . . . . . 8 |- ( z e. Q. -> z e. ( N. X. N. ) )
39	38	3ad2ant3 1084	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> z e. ( N. X. N. ) )
40		xp2nd 7199	. . . . . . 7 |- ( z e. ( N. X. N. ) -> ( 2nd ` z ) e. N. )
41		ltmpi 9726	. . . . . . 7 |- ( ( 2nd ` z ) e. N. -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) <-> ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) ) )
42	39, 40, 41	3syl 18	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( ( 1st ` x ) .N ( 2nd ` y ) ) <N ( ( 1st ` y ) .N ( 2nd ` x ) ) <-> ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) ) )
43	37, 42	bitrd 268	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) ) )
44		ordpinq 9765	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y <Q z <-> ( ( 1st ` y ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) )
45	44	3adant1 1079	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( y <Q z <-> ( ( 1st ` y ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) )
46	1	3ad2ant1 1082	. . . . . . 7 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> x e. ( N. X. N. ) )
47		ltmpi 9726	. . . . . . 7 |- ( ( 2nd ` x ) e. N. -> ( ( ( 1st ` y ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` y ) ) <-> ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) ) )
48	46, 13, 47	3syl 18	. . . . . 6 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( ( 1st ` y ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` y ) ) <-> ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) ) )
49	45, 48	bitrd 268	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( y <Q z <-> ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) ) )
50	43, 49	anbi12d 747	. . . 4 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( x <Q y /\ y <Q z ) <-> ( ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) /\ ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) ) ) )
51		fvex 6201	. . . . . . 7 |- ( 2nd ` x ) e. _V
52		fvex 6201	. . . . . . 7 |- ( 1st ` y ) e. _V
53		fvex 6201	. . . . . . 7 |- ( 2nd ` z ) e. _V
54		mulcompi 9718	. . . . . . 7 |- ( r .N s ) = ( s .N r )
55		mulasspi 9719	. . . . . . 7 |- ( ( r .N s ) .N t ) = ( r .N ( s .N t ) )
56	51, 52, 53, 54, 55	caov13 6864	. . . . . 6 |- ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) = ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) )
57		fvex 6201	. . . . . . 7 |- ( 1st ` z ) e. _V
58		fvex 6201	. . . . . . 7 |- ( 2nd ` y ) e. _V
59	51, 57, 58, 54, 55	caov13 6864	. . . . . 6 |- ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) = ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) )
60	56, 59	breq12i 4662	. . . . 5 |- ( ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) <-> ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
61		fvex 6201	. . . . . . 7 |- ( 1st ` x ) e. _V
62	53, 61, 58, 54, 55	caov13 6864	. . . . . 6 |- ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) = ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) )
63		ltrelpi 9711	. . . . . . 7 |- <N C_ ( N. X. N. )
64	17, 63	sotri 5523	. . . . . 6 |- ( ( ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) /\ ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) -> ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
65	62, 64	syl5eqbrr 4689	. . . . 5 |- ( ( ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) /\ ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) -> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
66	60, 65	sylan2b 492	. . . 4 |- ( ( ( ( 2nd ` z ) .N ( ( 1st ` x ) .N ( 2nd ` y ) ) ) <N ( ( 2nd ` z ) .N ( ( 1st ` y ) .N ( 2nd ` x ) ) ) /\ ( ( 2nd ` x ) .N ( ( 1st ` y ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` x ) .N ( ( 1st ` z ) .N ( 2nd ` y ) ) ) ) -> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
67	50, 66	syl6bi 243	. . 3 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( x <Q y /\ y <Q z ) -> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) )
68		ordpinq 9765	. . . . 5 |- ( ( x e. Q. /\ z e. Q. ) -> ( x <Q z <-> ( ( 1st ` x ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
69	68	3adant2 1080	. . . 4 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q z <-> ( ( 1st ` x ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) )
70	5	3ad2ant2 1083	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> y e. ( N. X. N. ) )
71		ltmpi 9726	. . . . 5 |- ( ( 2nd ` y ) e. N. -> ( ( ( 1st ` x ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` x ) ) <-> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) )
72	70, 7, 71	3syl 18	. . . 4 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( ( 1st ` x ) .N ( 2nd ` z ) ) <N ( ( 1st ` z ) .N ( 2nd ` x ) ) <-> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) )
73	69, 72	bitrd 268	. . 3 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q z <-> ( ( 2nd ` y ) .N ( ( 1st ` x ) .N ( 2nd ` z ) ) ) <N ( ( 2nd ` y ) .N ( ( 1st ` z ) .N ( 2nd ` x ) ) ) ) )
74	67, 73	sylibrd 249	. 2 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( x <Q y /\ y <Q z ) -> x <Q z ) )
75	36, 74	isso2i 5067	1 |- <Q Or Q.

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   Or wor 5034   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   N.cnpi 9666   .N cmi 9668   <N clti 9669   ~Q ceq 9673   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-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-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-ltnq 9740

This theorem is referenced by:  ltbtwnnq  9800  prub  9816  npomex  9818  genpnnp  9827  nqpr  9836  distrlem4pr  9848  prlem934  9855  ltexprlem4  9861  reclem2pr  9870  reclem4pr  9872