Theorem ltsopr 7806

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4392). 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 7805	. 2 |- <P Po P.
2		ltdfpr 7716	. . . . 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 1041	. . . 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 7685	. . . . . . . . . . . 12 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
5		prnminu 7699	. . . . . . . . . . . 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 7685	. . . . . . . . . . . 12 |- ( y e. P. -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
8		prnmaxl 7698	. . . . . . . . . . . 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 590	. . . . . . . . 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 2701	. . . . . . . . 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 1179	. . . . . . 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 7608	. . . . . . . . . . . . 13 |- <Q Or Q.
16		ltrelnq 7575	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
17	15, 16	sotri 5130	. . . . . . . . . . . 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 7685	. . . . . . . . . . . . . . . 16 |- ( z e. P. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
20		prloc 7701	. . . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . 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 7692	. . . . . . . . . . . . . . . . . . . . 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 1636	. . . . . . . . . . . . . . . . . . . 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 1476	. . . . . . . . . . . . . . . . . . 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 1183	. . . . . . . . . . . . . . . . . 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 2514	. . . . . . . . . . . . . . . . 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 7716	. . . . . . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . . . . . 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 7691	. . . . . . . . . . . . . . . . . . . . 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 1636	. . . . . . . . . . . . . . . . . . . 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 1476	. . . . . . . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . . . . . . 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 2514	. . . . . . . . . . . . . . . . 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 7716	. . . . . . . . . . . . . . . . . . . 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 1039	. . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . 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 2656	. . . . . . 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 2652	. . . 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 2617	. 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 4392	. 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 948	1 |- <P Or P.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   <.cop 3670   class class class wbr 4086   Po wpo 4389   Or wor 4390   ` cfv 5324   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   <Q cltq 7495   P.cnp 7501   <P cltp 7505

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-enq 7557  df-nqqs 7558  df-ltnqqs 7563  df-inp 7676  df-iltp 7680

This theorem is referenced by:  prplnqu  7830  addextpr  7831  caucvgprprlemk  7893  caucvgprprlemnkltj  7899  caucvgprprlemnkeqj  7900  caucvgprprlemnjltk  7901  caucvgprprlemnbj  7903  caucvgprprlemml  7904  caucvgprprlemlol  7908  caucvgprprlemupu  7910  caucvgprprlemloc  7913  caucvgprprlemaddq  7918  suplocexprlemmu  7928  lttrsr  7972  ltposr  7973  ltsosr  7974  archsr  7992