Theorem ltsopr 6560

Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4024). 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 6559	. 2 |- <P Po P.
2		ltdfpr 6481	. . . . 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 923	. . . 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 6450	. . . . . . . . . . . 12 |- (x e. P. -> <.( 1st ` x) , ( 2nd ` x) >. e. P.)
5		prnminu 6464	. . . . . . . . . . . 12 |- (( <.( 1st ` x) , ( 2nd ` x) >. e. P. /\ q e. ( 2nd ` x)) -> E. r e. ( 2nd ` x) r <Q q)
6	4, 5	sylan 267	. . . . . . . . . . 11 |- ((x e. P. /\ q e. ( 2nd ` x)) -> E. r e. ( 2nd ` x) r <Q q)
7		prop 6450	. . . . . . . . . . . 12 |- (y e. P. -> <.( 1st ` y) , ( 2nd ` y) >. e. P.)
8		prnmaxl 6463	. . . . . . . . . . . 12 |- (( <.( 1st ` y) , ( 2nd ` y) >. e. P. /\ q e. ( 1st ` y)) -> E. s e. ( 1st ` y) q <Q s)
9	7, 8	sylan 267	. . . . . . . . . . 11 |- ((y e. P. /\ q e. ( 1st ` y)) -> E. s e. ( 1st ` y) q <Q s)
10	6, 9	anim12i 321	. . . . . . . . . 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 522	. . . . . . . . 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 2473	. . . . . . . . 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 137	. . . . . . . 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 1061	. . . . . . 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 6375	. . . . . . . . . . . . 13 |- <Q Or Q.
16		ltrelnq 6342	. . . . . . . . . . . . 13 |- <Q C_ ( Q. X. Q.)
17	15, 16	sotri 4662	. . . . . . . . . . . 12 |- (( r <Q q /\ q <Q s) -> r <Q s)
18	17	adantl 262	. . . . . . . . . . 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 6450	. . . . . . . . . . . . . . . 16 |- (z e. P. -> <.( 1st ` z) , ( 2nd ` z) >. e. P.)
20		prloc 6466	. . . . . . . . . . . . . . . 16 |- (( <.( 1st ` z) , ( 2nd ` z) >. e. P. /\ r <Q s) -> ( r e. ( 1st ` z) \/ s e. ( 2nd ` z)))
21	19, 20	sylan 267	. . . . . . . . . . . . . . 15 |- ((z e. P. /\ r <Q s) -> ( r e. ( 1st ` z) \/ s e. ( 2nd ` z)))
22	21	3ad2antl3 1067	. . . . . . . . . . . . . 14 |- (((x e. P. /\ y e. P. /\ z e. P.) /\ r <Q s) -> ( r e. ( 1st ` z) \/ s e. ( 2nd ` z)))
23	22	ex 108	. . . . . . . . . . . . 13 |- ((x e. P. /\ y e. P. /\ z e. P.) -> ( r <Q s -> ( r e. ( 1st ` z) \/ s e. ( 2nd ` z))))
24	23	adantr 261	. . . . . . . . . . . 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 457	. . . . . . . . . . 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 6457	. . . . . . . . . . . . . . . . . . . . 21 |- (( <.( 1st ` x) , ( 2nd ` x) >. e. P. /\ r e. ( 2nd ` x)) -> r e. Q.)
28	4, 27	sylan 267	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. P. /\ r e. ( 2nd ` x)) -> r e. Q.)
29		ax-ia3 101	. . . . . . . . . . . . . . . . . . . . 21 |- ( r e. ( 2nd ` x) -> ( r e. ( 1st ` z) -> ( r e. ( 2nd ` x) /\ r e. ( 1st ` z))))
30	29	adantl 262	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. P. /\ r e. ( 2nd ` x)) -> ( r e. ( 1st ` z) -> ( r e. ( 2nd ` x) /\ r e. ( 1st ` z))))
31		19.8a 1479	. . . . . . . . . . . . . . . . . . . 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 1320	. . . . . . . . . . . . . . . . . . 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 1065	. . . . . . . . . . . . . . . . . 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 115	. . . . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . . . . 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 137	. . . . . . . . . . . . . . . 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 6481	. . . . . . . . . . . . . . . . . . 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 147	. . . . . . . . . . . . . . . . . 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 922	. . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . 14 |- (((x e. P. /\ y e. P. /\ z e. P.) /\ r e. ( 2nd ` x)) -> ( r e. ( 1st ` z) -> x <P z))
43	42	adantrr 448	. . . . . . . . . . . . 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 6456	. . . . . . . . . . . . . . . . . . . . 21 |- (( <.( 1st ` y) , ( 2nd ` y) >. e. P. /\ s e. ( 1st ` y)) -> s e. Q.)
45	7, 44	sylan 267	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. P. /\ s e. ( 1st ` y)) -> s e. Q.)
46		pm3.21 251	. . . . . . . . . . . . . . . . . . . . 21 |- ( s e. ( 1st ` y) -> ( s e. ( 2nd ` z) -> ( s e. ( 2nd ` z) /\ s e. ( 1st ` y))))
47	46	adantl 262	. . . . . . . . . . . . . . . . . . . 20 |- ((y e. P. /\ s e. ( 1st ` y)) -> ( s e. ( 2nd ` z) -> ( s e. ( 2nd ` z) /\ s e. ( 1st ` y))))
48		19.8a 1479	. . . . . . . . . . . . . . . . . . . 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 1320	. . . . . . . . . . . . . . . . . . 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 1066	. . . . . . . . . . . . . . . . . 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 115	. . . . . . . . . . . . . . . . 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 2306	. . . . . . . . . . . . . . . . 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 137	. . . . . . . . . . . . . . . 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 6481	. . . . . . . . . . . . . . . . . . . 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 147	. . . . . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . . . . 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 921	. . . . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . 14 |- (((x e. P. /\ y e. P. /\ z e. P.) /\ s e. ( 1st ` y)) -> ( s e. ( 2nd ` z) -> z <P y))
61	60	adantrl 447	. . . . . . . . . . . . 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 699	. . . . . . . . . . . 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 446	. . . . . . . . . . 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 261	. . . . . . . . . 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 108	. . . . . . . 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 2434	. . . . . . 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 108	. . . . 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 2430	. . . 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 139	. . 3 |- ((x e. P. /\ y e. P. /\ z e. P.) -> (x <P y -> (x <P z \/ z <P y)))
72	71	rgen3 2400	. 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 4024	. 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 848	1 |- <P Or P.

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628   /\ w3a 884  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301   <.cop 3369   class class class wbr 3754   Po wpo 4021   Or wor 4022   ` cfv 4844   1stc1st 5704   2ndc2nd 5705   Q.cnq 6257   <Q cltq 6262   P.cnp 6268   <P cltp 6272

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-bnd 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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253

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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-mi 6283  df-lti 6284  df-enq 6324  df-nqqs 6325  df-ltnqqs 6330  df-inp 6441  df-iltp 6445

This theorem is referenced by:  addextpr  6583  lttrsr  6642  ltposr  6643  ltsosr  6644  archsr  6660