Theorem ltsopr 7609

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4309). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)

Assertion

Ref	Expression
ltsopr	|- <P Or P.

Proof of Theorem ltsopr

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

Step	Hyp	Ref	Expression
1		ltpopr 7608	. 2 |- <P Po P.
2		ltdfpr 7519	. . . . 5 |- ( ( x e. P. /\ y e. P. ) -> ( x <P y <-> E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) )
3	2	3adant3 1018	. . . 4 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( x <P y <-> E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) )
4		prop 7488	. . . . . . . . . . . 12 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
5		prnminu 7502	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. /\ q e. ( 2nd ` x ) ) -> E. r e. ( 2nd ` x ) r <Q q )
6	4, 5	sylan 283	. . . . . . . . . . 11 |- ( ( x e. P. /\ q e. ( 2nd ` x ) ) -> E. r e. ( 2nd ` x ) r <Q q )
7		prop 7488	. . . . . . . . . . . 12 |- ( y e. P. -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
8		prnmaxl 7501	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ q e. ( 1st ` y ) ) -> E. s e. ( 1st ` y ) q <Q s )
9	7, 8	sylan 283	. . . . . . . . . . 11 |- ( ( y e. P. /\ q e. ( 1st ` y ) ) -> E. s e. ( 1st ` y ) q <Q s )
10	6, 9	anim12i 338	. . . . . . . . . 10 |- ( ( ( x e. P. /\ q e. ( 2nd ` x ) ) /\ ( y e. P. /\ q e. ( 1st ` y ) ) ) -> ( E. r e. ( 2nd ` x ) r <Q q /\ E. s e. ( 1st ` y ) q <Q s ) )
11	10	an4s 588	. . . . . . . . 9 |- ( ( ( x e. P. /\ y e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> ( E. r e. ( 2nd ` x ) r <Q q /\ E. s e. ( 1st ` y ) q <Q s ) )
12		reeanv 2657	. . . . . . . . 9 |- ( E. r e. ( 2nd ` x ) E. s e. ( 1st ` y ) ( r <Q q /\ q <Q s ) <-> ( E. r e. ( 2nd ` x ) r <Q q /\ E. s e. ( 1st ` y ) q <Q s ) )
13	11, 12	sylibr 134	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> E. r e. ( 2nd ` x ) E. s e. ( 1st ` y ) ( r <Q q /\ q <Q s ) )
14	13	3adantl3 1156	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> E. r e. ( 2nd ` x ) E. s e. ( 1st ` y ) ( r <Q q /\ q <Q s ) )
15		ltsonq 7411	. . . . . . . . . . . . 13 |- <Q Or Q.
16		ltrelnq 7378	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
17	15, 16	sotri 5036	. . . . . . . . . . . 12 |- ( ( r <Q q /\ q <Q s ) -> r <Q s )
18	17	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) /\ ( r <Q q /\ q <Q s ) ) -> r <Q s )
19		prop 7488	. . . . . . . . . . . . . . . 16 |- ( z e. P. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
20		prloc 7504	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. /\ r <Q s ) -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) )
21	19, 20	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( z e. P. /\ r <Q s ) -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) )
22	21	3ad2antl3 1162	. . . . . . . . . . . . . 14 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r <Q s ) -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) )
23	22	ex 115	. . . . . . . . . . . . 13 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( r <Q s -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) ) )
24	23	adantr 276	. . . . . . . . . . . 12 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> ( r <Q s -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) ) )
25	24	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) /\ ( r <Q q /\ q <Q s ) ) -> ( r <Q s -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) ) )
26	18, 25	mpd 13	. . . . . . . . . 10 |- ( ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) /\ ( r <Q q /\ q <Q s ) ) -> ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) )
27		elprnqu 7495	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. /\ r e. ( 2nd ` x ) ) -> r e. Q. )
28	4, 27	sylan 283	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. P. /\ r e. ( 2nd ` x ) ) -> r e. Q. )
29		ax-ia3 108	. . . . . . . . . . . . . . . . . . . . 21 |- ( r e. ( 2nd ` x ) -> ( r e. ( 1st ` z ) -> ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
30	29	adantl 277	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. P. /\ r e. ( 2nd ` x ) ) -> ( r e. ( 1st ` z ) -> ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
31		19.8a 1600	. . . . . . . . . . . . . . . . . . . 20 |- ( ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) -> E. r ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
32	28, 30, 31	syl6an 1444	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. P. /\ r e. ( 2nd ` x ) ) -> ( r e. ( 1st ` z ) -> E. r ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) ) )
33	32	3ad2antl1 1160	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) -> ( r e. ( 1st ` z ) -> E. r ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) ) )
34	33	imp 124	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) /\ r e. ( 1st ` z ) ) -> E. r ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
35		df-rex 2471	. . . . . . . . . . . . . . . . 17 |- ( E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) <-> E. r ( r e. Q. /\ ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
36	34, 35	sylibr 134	. . . . . . . . . . . . . . . 16 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) /\ r e. ( 1st ` z ) ) -> E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) )
37		ltdfpr 7519	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. P. /\ z e. P. ) -> ( x <P z <-> E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) ) )
38	37	biimprd 158	. . . . . . . . . . . . . . . . . 18 |- ( ( x e. P. /\ z e. P. ) -> ( E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) -> x <P z ) )
39	38	3adant2 1017	. . . . . . . . . . . . . . . . 17 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) -> x <P z ) )
40	39	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) /\ r e. ( 1st ` z ) ) -> ( E. r e. Q. ( r e. ( 2nd ` x ) /\ r e. ( 1st ` z ) ) -> x <P z ) )
41	36, 40	mpd 13	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) /\ r e. ( 1st ` z ) ) -> x <P z )
42	41	ex 115	. . . . . . . . . . . . . 14 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ r e. ( 2nd ` x ) ) -> ( r e. ( 1st ` z ) -> x <P z ) )
43	42	adantrr 479	. . . . . . . . . . . . 13 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) -> ( r e. ( 1st ` z ) -> x <P z ) )
44		elprnql 7494	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. /\ s e. ( 1st ` y ) ) -> s e. Q. )
45	7, 44	sylan 283	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. P. /\ s e. ( 1st ` y ) ) -> s e. Q. )
46		pm3.21 264	. . . . . . . . . . . . . . . . . . . . 21 |- ( s e. ( 1st ` y ) -> ( s e. ( 2nd ` z ) -> ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
47	46	adantl 277	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. P. /\ s e. ( 1st ` y ) ) -> ( s e. ( 2nd ` z ) -> ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
48		19.8a 1600	. . . . . . . . . . . . . . . . . . . 20 |- ( ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) -> E. s ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
49	45, 47, 48	syl6an 1444	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. P. /\ s e. ( 1st ` y ) ) -> ( s e. ( 2nd ` z ) -> E. s ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) ) )
50	49	3ad2antl2 1161	. . . . . . . . . . . . . . . . . 18 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) -> ( s e. ( 2nd ` z ) -> E. s ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) ) )
51	50	imp 124	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) /\ s e. ( 2nd ` z ) ) -> E. s ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
52		df-rex 2471	. . . . . . . . . . . . . . . . 17 |- ( E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) <-> E. s ( s e. Q. /\ ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
53	51, 52	sylibr 134	. . . . . . . . . . . . . . . 16 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) /\ s e. ( 2nd ` z ) ) -> E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) )
54		ltdfpr 7519	. . . . . . . . . . . . . . . . . . . 20 |- ( ( z e. P. /\ y e. P. ) -> ( z <P y <-> E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) ) )
55	54	biimprd 158	. . . . . . . . . . . . . . . . . . 19 |- ( ( z e. P. /\ y e. P. ) -> ( E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) -> z <P y ) )
56	55	ancoms 268	. . . . . . . . . . . . . . . . . 18 |- ( ( y e. P. /\ z e. P. ) -> ( E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) -> z <P y ) )
57	56	3adant1 1016	. . . . . . . . . . . . . . . . 17 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) -> z <P y ) )
58	57	ad2antrr 488	. . . . . . . . . . . . . . . 16 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) /\ s e. ( 2nd ` z ) ) -> ( E. s e. Q. ( s e. ( 2nd ` z ) /\ s e. ( 1st ` y ) ) -> z <P y ) )
59	53, 58	mpd 13	. . . . . . . . . . . . . . 15 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) /\ s e. ( 2nd ` z ) ) -> z <P y )
60	59	ex 115	. . . . . . . . . . . . . 14 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ s e. ( 1st ` y ) ) -> ( s e. ( 2nd ` z ) -> z <P y ) )
61	60	adantrl 478	. . . . . . . . . . . . 13 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) -> ( s e. ( 2nd ` z ) -> z <P y ) )
62	43, 61	orim12d 787	. . . . . . . . . . . 12 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) -> ( ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) -> ( x <P z \/ z <P y ) ) )
63	62	adantlr 477	. . . . . . . . . . 11 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) -> ( ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) -> ( x <P z \/ z <P y ) ) )
64	63	adantr 276	. . . . . . . . . 10 |- ( ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) /\ ( r <Q q /\ q <Q s ) ) -> ( ( r e. ( 1st ` z ) \/ s e. ( 2nd ` z ) ) -> ( x <P z \/ z <P y ) ) )
65	26, 64	mpd 13	. . . . . . . . 9 |- ( ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) /\ ( r <Q q /\ q <Q s ) ) -> ( x <P z \/ z <P y ) )
66	65	ex 115	. . . . . . . 8 |- ( ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) /\ ( r e. ( 2nd ` x ) /\ s e. ( 1st ` y ) ) ) -> ( ( r <Q q /\ q <Q s ) -> ( x <P z \/ z <P y ) ) )
67	66	rexlimdvva 2612	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> ( E. r e. ( 2nd ` x ) E. s e. ( 1st ` y ) ( r <Q q /\ q <Q s ) -> ( x <P z \/ z <P y ) ) )
68	14, 67	mpd 13	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. /\ z e. P. ) /\ ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) -> ( x <P z \/ z <P y ) )
69	68	ex 115	. . . . 5 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) -> ( x <P z \/ z <P y ) ) )
70	69	rexlimdvw 2608	. . . 4 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) -> ( x <P z \/ z <P y ) ) )
71	3, 70	sylbid 150	. . 3 |- ( ( x e. P. /\ y e. P. /\ z e. P. ) -> ( x <P y -> ( x <P z \/ z <P y ) ) )
72	71	rgen3 2574	. 2 |- A. x e. P. A. y e. P. A. z e. P. ( x <P y -> ( x <P z \/ z <P y ) )
73		df-iso 4309	. 2 |- ( <P Or P. <-> ( <P Po P. /\ A. x e. P. A. y e. P. A. z e. P. ( x <P y -> ( x <P z \/ z <P y ) ) ) )
74	1, 72, 73	mpbir2an 943	1 |- <P Or P.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 979   E.wex 1502   e. wcel 2158   A.wral 2465   E.wrex 2466   <.cop 3607   class class class wbr 4015   Po wpo 4306   Or wor 4307   ` cfv 5228   1stc1st 6153   2ndc2nd 6154   Q.cnq 7293   <Q cltq 7298   P.cnp 7304   <P cltp 7308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-mi 7319  df-lti 7320  df-enq 7360  df-nqqs 7361  df-ltnqqs 7366  df-inp 7479  df-iltp 7483

This theorem is referenced by:  prplnqu  7633  addextpr  7634  caucvgprprlemk  7696  caucvgprprlemnkltj  7702  caucvgprprlemnkeqj  7703  caucvgprprlemnjltk  7704  caucvgprprlemnbj  7706  caucvgprprlemml  7707  caucvgprprlemlol  7711  caucvgprprlemupu  7713  caucvgprprlemloc  7716  caucvgprprlemaddq  7721  suplocexprlemmu  7731  lttrsr  7775  ltposr  7776  ltsosr  7777  archsr  7795