Theorem ltsopr 7818

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4393). 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 7817	. 2 |- <P Po P.
2		ltdfpr 7728	. . . . 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 1043	. . . 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 7697	. . . . . . . . . . . 12 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
5		prnminu 7711	. . . . . . . . . . . 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 7697	. . . . . . . . . . . 12 |- ( y e. P. -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
8		prnmaxl 7710	. . . . . . . . . . . 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 592	. . . . . . . . 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 2702	. . . . . . . . 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 1181	. . . . . . 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 7620	. . . . . . . . . . . . 13 |- <Q Or Q.
16		ltrelnq 7587	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
17	15, 16	sotri 5131	. . . . . . . . . . . 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 7697	. . . . . . . . . . . . . . . 16 |- ( z e. P. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
20		prloc 7713	. . . . . . . . . . . . . . . 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 1187	. . . . . . . . . . . . . 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 7704	. . . . . . . . . . . . . . . . . . . . 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 1638	. . . . . . . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . . . . 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 7728	. . . . . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . . . 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 7703	. . . . . . . . . . . . . . . . . . . . 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 1638	. . . . . . . . . . . . . . . . . . . 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 1478	. . . . . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . . . . 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 7728	. . . . . . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . 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 2657	. . . . . . 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 2653	. . . 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 2618	. 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 4393	. 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 950	1 |- <P Or P.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   E.wex 1540   e. wcel 2201   A.wral 2509   E.wrex 2510   <.cop 3671   class class class wbr 4087   Po wpo 4390   Or wor 4391   ` cfv 5325   1stc1st 6303   2ndc2nd 6304   Q.cnq 7502   <Q cltq 7507   P.cnp 7513   <P cltp 7517

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4203  ax-sep 4206  ax-nul 4214  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-setind 4634  ax-iinf 4685

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-iun 3971  df-br 4088  df-opab 4150  df-mpt 4151  df-tr 4187  df-eprel 4385  df-id 4389  df-po 4392  df-iso 4393  df-iord 4462  df-on 4464  df-suc 4467  df-iom 4688  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-ima 4737  df-iota 5285  df-fun 5327  df-fn 5328  df-f 5329  df-f1 5330  df-fo 5331  df-f1o 5332  df-fv 5333  df-ov 6023  df-oprab 6024  df-mpo 6025  df-1st 6305  df-2nd 6306  df-recs 6473  df-irdg 6538  df-oadd 6588  df-omul 6589  df-er 6704  df-ec 6706  df-qs 6710  df-ni 7526  df-mi 7528  df-lti 7529  df-enq 7569  df-nqqs 7570  df-ltnqqs 7575  df-inp 7688  df-iltp 7692

This theorem is referenced by:  prplnqu  7842  addextpr  7843  caucvgprprlemk  7905  caucvgprprlemnkltj  7911  caucvgprprlemnkeqj  7912  caucvgprprlemnjltk  7913  caucvgprprlemnbj  7915  caucvgprprlemml  7916  caucvgprprlemlol  7920  caucvgprprlemupu  7922  caucvgprprlemloc  7925  caucvgprprlemaddq  7930  suplocexprlemmu  7940  lttrsr  7984  ltposr  7985  ltsosr  7986  archsr  8004