Theorem ltsopr 7953

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4437). 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 7952	. 2 |- <P Po P.
2		ltdfpr 7863	. . . . 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 1048	. . . 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 7832	. . . . . . . . . . . 12 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
5		prnminu 7846	. . . . . . . . . . . 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 7832	. . . . . . . . . . . 12 |- ( y e. P. -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. P. )
8		prnmaxl 7845	. . . . . . . . . . . 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 596	. . . . . . . . 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 2721	. . . . . . . . 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 1186	. . . . . . 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 7755	. . . . . . . . . . . . 13 |- <Q Or Q.
16		ltrelnq 7722	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q. )
17	15, 16	sotri 5178	. . . . . . . . . . . 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 7832	. . . . . . . . . . . . . . . 16 |- ( z e. P. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
20		prloc 7848	. . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . 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 492	. . . . . . . . . . 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 7839	. . . . . . . . . . . . . . . . . . . . 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 1643	. . . . . . . . . . . . . . . . . . . 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 1483	. . . . . . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 7863	. . . . . . . . . . . . . . . . . . 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 1047	. . . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 7838	. . . . . . . . . . . . . . . . . . . . 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 1643	. . . . . . . . . . . . . . . . . . . 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 1483	. . . . . . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 7863	. . . . . . . . . . . . . . . . . . . 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 1046	. . . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 798	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 2676	. . . . . . 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 2672	. . . 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 2637	. 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 4437	. 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 955	1 |- <P Or P.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   <.cop 3708   class class class wbr 4125   Po wpo 4434   Or wor 4435   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   <Q cltq 7642   P.cnp 7648   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-lti 7664  df-enq 7704  df-nqqs 7705  df-ltnqqs 7710  df-inp 7823  df-iltp 7827

This theorem is referenced by:  prplnqu  7977  addextpr  7978  caucvgprprlemk  8040  caucvgprprlemnkltj  8046  caucvgprprlemnkeqj  8047  caucvgprprlemnjltk  8048  caucvgprprlemnbj  8050  caucvgprprlemml  8051  caucvgprprlemlol  8055  caucvgprprlemupu  8057  caucvgprprlemloc  8060  caucvgprprlemaddq  8065  suplocexprlemmu  8075  lttrsr  8119  ltposr  8120  ltsosr  8121  archsr  8139