Theorem ltsosr 6692

Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

Assertion

Ref	Expression
ltsosr	|- <R Or R.

Proof of Theorem ltsosr

Dummy variables a b c d e f r s t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltposr 6691	. 2 |- <R Po R.
2		df-nr 6655	. . . 4 |- R. = (( P. X. P.) /. ~R )
3		breq1 3758	. . . . 5 |- ([ <. a , b >.] ~R = x -> ([ <. a , b >.] ~R <R [ <. c , d >.] ~R <-> x <R [ <. c , d >.] ~R ))
4		breq1 3758	. . . . . 6 |- ([ <. a , b >.] ~R = x -> ([ <. a , b >.] ~R <R [ <. e , f >.] ~R <-> x <R [ <. e , f >.] ~R ))
5	4	orbi1d 704	. . . . 5 |- ([ <. a , b >.] ~R = x -> (([ <. a , b >.] ~R <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R ) <-> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R )))
6	3, 5	imbi12d 223	. . . 4 |- ([ <. a , b >.] ~R = x -> (([ <. a , b >.] ~R <R [ <. c , d >.] ~R -> ([ <. a , b >.] ~R <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R )) <-> (x <R [ <. c , d >.] ~R -> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R ))))
7		breq2 3759	. . . . 5 |- ([ <. c , d >.] ~R = y -> (x <R [ <. c , d >.] ~R <-> x <R y))
8		breq2 3759	. . . . . 6 |- ([ <. c , d >.] ~R = y -> ([ <. e , f >.] ~R <R [ <. c , d >.] ~R <-> [ <. e , f >.] ~R <R y))
9	8	orbi2d 703	. . . . 5 |- ([ <. c , d >.] ~R = y -> ((x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R ) <-> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R y)))
10	7, 9	imbi12d 223	. . . 4 |- ([ <. c , d >.] ~R = y -> ((x <R [ <. c , d >.] ~R -> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R )) <-> (x <R y -> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R y))))
11		breq2 3759	. . . . . 6 |- ([ <. e , f >.] ~R = z -> (x <R [ <. e , f >.] ~R <-> x <R z))
12		breq1 3758	. . . . . 6 |- ([ <. e , f >.] ~R = z -> ([ <. e , f >.] ~R <R y <-> z <R y))
13	11, 12	orbi12d 706	. . . . 5 |- ([ <. e , f >.] ~R = z -> ((x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R y) <-> (x <R z \/ z <R y)))
14	13	imbi2d 219	. . . 4 |- ([ <. e , f >.] ~R = z -> ((x <R y -> (x <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R y)) <-> (x <R y -> (x <R z \/ z <R y))))
15		simp1l 927	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> a e. P.)
16		simp3r 932	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> f e. P.)
17		addclpr 6520	. . . . . . . . 9 |- (( a e. P. /\ f e. P.) -> ( a +P. f) e. P.)
18	15, 16, 17	syl2anc 391	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( a +P. f) e. P.)
19		simp2r 930	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> d e. P.)
20		addclpr 6520	. . . . . . . 8 |- ((( a +P. f) e. P. /\ d e. P.) -> (( a +P. f) +P. d) e. P.)
21	18, 19, 20	syl2anc 391	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. f) +P. d) e. P.)
22		simp2l 929	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> c e. P.)
23		addclpr 6520	. . . . . . . . 9 |- ((f e. P. /\ c e. P.) -> (f +P. c) e. P.)
24	16, 22, 23	syl2anc 391	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (f +P. c) e. P.)
25		simp1r 928	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> b e. P.)
26		addclpr 6520	. . . . . . . 8 |- (((f +P. c) e. P. /\ b e. P.) -> ((f +P. c) +P. b) e. P.)
27	24, 25, 26	syl2anc 391	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((f +P. c) +P. b) e. P.)
28		simp3l 931	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> e e. P.)
29		addclpr 6520	. . . . . . . . 9 |- (( b e. P. /\ e e. P.) -> ( b +P. e) e. P.)
30	25, 28, 29	syl2anc 391	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( b +P. e) e. P.)
31		addclpr 6520	. . . . . . . 8 |- ((( b +P. e) e. P. /\ d e. P.) -> (( b +P. e) +P. d) e. P.)
32	30, 19, 31	syl2anc 391	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( b +P. e) +P. d) e. P.)
33		ltsopr 6570	. . . . . . . 8 |- <P Or P.
34		sowlin 4048	. . . . . . . 8 |- (( <P Or P. /\ ((( a +P. f) +P. d) e. P. /\ ((f +P. c) +P. b) e. P. /\ (( b +P. e) +P. d) e. P.)) -> ((( a +P. f) +P. d) <P ((f +P. c) +P. b) -> ((( a +P. f) +P. d) <P (( b +P. e) +P. d) \/ (( b +P. e) +P. d) <P ((f +P. c) +P. b))))
35	33, 34	mpan 400	. . . . . . 7 |- (((( a +P. f) +P. d) e. P. /\ ((f +P. c) +P. b) e. P. /\ (( b +P. e) +P. d) e. P.) -> ((( a +P. f) +P. d) <P ((f +P. c) +P. b) -> ((( a +P. f) +P. d) <P (( b +P. e) +P. d) \/ (( b +P. e) +P. d) <P ((f +P. c) +P. b))))
36	21, 27, 32, 35	syl3anc 1134	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((( a +P. f) +P. d) <P ((f +P. c) +P. b) -> ((( a +P. f) +P. d) <P (( b +P. e) +P. d) \/ (( b +P. e) +P. d) <P ((f +P. c) +P. b))))
37		addclpr 6520	. . . . . . . . 9 |- (( a e. P. /\ d e. P.) -> ( a +P. d) e. P.)
38	15, 19, 37	syl2anc 391	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( a +P. d) e. P.)
39		addclpr 6520	. . . . . . . . 9 |- (( b e. P. /\ c e. P.) -> ( b +P. c) e. P.)
40	25, 22, 39	syl2anc 391	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( b +P. c) e. P.)
41		ltaprg 6592	. . . . . . . 8 |- ((( a +P. d) e. P. /\ ( b +P. c) e. P. /\ f e. P.) -> (( a +P. d) <P ( b +P. c) <-> (f +P. ( a +P. d)) <P (f +P. ( b +P. c))))
42	38, 40, 16, 41	syl3anc 1134	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. d) <P ( b +P. c) <-> (f +P. ( a +P. d)) <P (f +P. ( b +P. c))))
43		addcomprg 6554	. . . . . . . . . . 11 |- (( r e. P. /\ s e. P.) -> ( r +P. s) = ( s +P. r))
44	43	adantl 262	. . . . . . . . . 10 |- (((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) /\ ( r e. P. /\ s e. P.)) -> ( r +P. s) = ( s +P. r))
45		addassprg 6555	. . . . . . . . . . 11 |- (( r e. P. /\ s e. P. /\ t e. P.) -> (( r +P. s) +P. t) = ( r +P. ( s +P. t)))
46	45	adantl 262	. . . . . . . . . 10 |- (((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) /\ ( r e. P. /\ s e. P. /\ t e. P.)) -> (( r +P. s) +P. t) = ( r +P. ( s +P. t)))
47	16, 15, 19, 44, 46	caov12d 5624	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (f +P. ( a +P. d)) = ( a +P. (f +P. d)))
48	46, 15, 16, 19	caovassd 5602	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. f) +P. d) = ( a +P. (f +P. d)))
49	47, 48	eqtr4d 2072	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (f +P. ( a +P. d)) = (( a +P. f) +P. d))
50	46, 16, 25, 22	caovassd 5602	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((f +P. b) +P. c) = (f +P. ( b +P. c)))
51	16, 25, 22, 44, 46	caov32d 5623	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((f +P. b) +P. c) = ((f +P. c) +P. b))
52	50, 51	eqtr3d 2071	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (f +P. ( b +P. c)) = ((f +P. c) +P. b))
53	49, 52	breq12d 3768	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((f +P. ( a +P. d)) <P (f +P. ( b +P. c)) <-> (( a +P. f) +P. d) <P ((f +P. c) +P. b)))
54	42, 53	bitrd 177	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. d) <P ( b +P. c) <-> (( a +P. f) +P. d) <P ((f +P. c) +P. b)))
55		ltaprg 6592	. . . . . . . . 9 |- (( r e. P. /\ s e. P. /\ t e. P.) -> ( r <P s <-> ( t +P. r) <P ( t +P. s)))
56	55	adantl 262	. . . . . . . 8 |- (((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) /\ ( r e. P. /\ s e. P. /\ t e. P.)) -> ( r <P s <-> ( t +P. r) <P ( t +P. s)))
57	56, 18, 30, 19, 44	caovord2d 5612	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. f) <P ( b +P. e) <-> (( a +P. f) +P. d) <P (( b +P. e) +P. d)))
58		addclpr 6520	. . . . . . . . . 10 |- (( e e. P. /\ d e. P.) -> ( e +P. d) e. P.)
59	28, 19, 58	syl2anc 391	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( e +P. d) e. P.)
60	56, 59, 24, 25, 44	caovord2d 5612	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( e +P. d) <P (f +P. c) <-> (( e +P. d) +P. b) <P ((f +P. c) +P. b)))
61	46, 25, 28, 19	caovassd 5602	. . . . . . . . . 10 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( b +P. e) +P. d) = ( b +P. ( e +P. d)))
62	44, 25, 59	caovcomd 5599	. . . . . . . . . 10 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ( b +P. ( e +P. d)) = (( e +P. d) +P. b))
63	61, 62	eqtrd 2069	. . . . . . . . 9 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( b +P. e) +P. d) = (( e +P. d) +P. b))
64	63	breq1d 3765	. . . . . . . 8 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((( b +P. e) +P. d) <P ((f +P. c) +P. b) <-> (( e +P. d) +P. b) <P ((f +P. c) +P. b)))
65	60, 64	bitr4d 180	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( e +P. d) <P (f +P. c) <-> (( b +P. e) +P. d) <P ((f +P. c) +P. b)))
66	57, 65	orbi12d 706	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ((( a +P. f) <P ( b +P. e) \/ ( e +P. d) <P (f +P. c)) <-> ((( a +P. f) +P. d) <P (( b +P. e) +P. d) \/ (( b +P. e) +P. d) <P ((f +P. c) +P. b))))
67	36, 54, 66	3imtr4d 192	. . . . 5 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (( a +P. d) <P ( b +P. c) -> (( a +P. f) <P ( b +P. e) \/ ( e +P. d) <P (f +P. c))))
68		ltsrprg 6675	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.)) -> ([ <. a , b >.] ~R <R [ <. c , d >.] ~R <-> ( a +P. d) <P ( b +P. c)))
69	68	3adant3 923	. . . . 5 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. a , b >.] ~R <R [ <. c , d >.] ~R <-> ( a +P. d) <P ( b +P. c)))
70		ltsrprg 6675	. . . . . . 7 |- ((( a e. P. /\ b e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. a , b >.] ~R <R [ <. e , f >.] ~R <-> ( a +P. f) <P ( b +P. e)))
71	70	3adant2 922	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. a , b >.] ~R <R [ <. e , f >.] ~R <-> ( a +P. f) <P ( b +P. e)))
72		ltsrprg 6675	. . . . . . . 8 |- ((( e e. P. /\ f e. P.) /\ ( c e. P. /\ d e. P.)) -> ([ <. e , f >.] ~R <R [ <. c , d >.] ~R <-> ( e +P. d) <P (f +P. c)))
73	72	ancoms 255	. . . . . . 7 |- ((( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. e , f >.] ~R <R [ <. c , d >.] ~R <-> ( e +P. d) <P (f +P. c)))
74	73	3adant1 921	. . . . . 6 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. e , f >.] ~R <R [ <. c , d >.] ~R <-> ( e +P. d) <P (f +P. c)))
75	71, 74	orbi12d 706	. . . . 5 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> (([ <. a , b >.] ~R <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R ) <-> (( a +P. f) <P ( b +P. e) \/ ( e +P. d) <P (f +P. c))))
76	67, 69, 75	3imtr4d 192	. . . 4 |- ((( a e. P. /\ b e. P.) /\ ( c e. P. /\ d e. P.) /\ ( e e. P. /\ f e. P.)) -> ([ <. a , b >.] ~R <R [ <. c , d >.] ~R -> ([ <. a , b >.] ~R <R [ <. e , f >.] ~R \/ [ <. e , f >.] ~R <R [ <. c , d >.] ~R )))
77	2, 6, 10, 14, 76	3ecoptocl 6131	. . 3 |- ((x e. R. /\ y e. R. /\ z e. R.) -> (x <R y -> (x <R z \/ z <R y)))
78	77	rgen3 2400	. 2 |- A.x e. R. A.y e. R. A.z e. R. (x <R y -> (x <R z \/ z <R y))
79		df-iso 4025	. 2 |- ( <R Or R. <-> ( <R Po R. /\ A.x e. R. A.y e. R. A.z e. R. (x <R y -> (x <R z \/ z <R y))))
80	1, 78, 79	mpbir2an 848	1 |- <R Or R.

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628   /\ w3a 884   = wceq 1242   e. wcel 1390  A.wral 2300   <.cop 3370   class class class wbr 3755   Po wpo 4022   Or wor 4023  (class class class)co 5455  [cec 6040   P.cnp 6275   +P. cpp 6277   <P cltp 6279   ~R cer 6280   R.cnr 6281   <R cltr 6287

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658

This theorem is referenced by:  1ne0sr  6694  addgt0sr  6703  axpre-ltirr  6766  axpre-ltwlin  6767  axpre-lttrn  6768