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Theorem ltsopr 6569
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4025). 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.
StepHypRef Expression
1 ltpopr 6568 . 2 <P Po P
2 ltdfpr 6488 . . . . 5 ((x P y P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
323adant3 923 . . . 4 ((x P y P z P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
4 prop 6457 . . . . . . . . . . . 12 (x P → ⟨(1stx), (2ndx)⟩ P)
5 prnminu 6471 . . . . . . . . . . . 12 ((⟨(1stx), (2ndx)⟩ P 𝑞 (2ndx)) → 𝑟 (2ndx)𝑟 <Q 𝑞)
64, 5sylan 267 . . . . . . . . . . 11 ((x P 𝑞 (2ndx)) → 𝑟 (2ndx)𝑟 <Q 𝑞)
7 prop 6457 . . . . . . . . . . . 12 (y P → ⟨(1sty), (2ndy)⟩ P)
8 prnmaxl 6470 . . . . . . . . . . . 12 ((⟨(1sty), (2ndy)⟩ P 𝑞 (1sty)) → 𝑠 (1sty)𝑞 <Q 𝑠)
97, 8sylan 267 . . . . . . . . . . 11 ((y P 𝑞 (1sty)) → 𝑠 (1sty)𝑞 <Q 𝑠)
106, 9anim12i 321 . . . . . . . . . 10 (((x P 𝑞 (2ndx)) (y P 𝑞 (1sty))) → (𝑟 (2ndx)𝑟 <Q 𝑞 𝑠 (1sty)𝑞 <Q 𝑠))
1110an4s 522 . . . . . . . . 9 (((x P y P) (𝑞 (2ndx) 𝑞 (1sty))) → (𝑟 (2ndx)𝑟 <Q 𝑞 𝑠 (1sty)𝑞 <Q 𝑠))
12 reeanv 2473 . . . . . . . . 9 (𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠) ↔ (𝑟 (2ndx)𝑟 <Q 𝑞 𝑠 (1sty)𝑞 <Q 𝑠))
1311, 12sylibr 137 . . . . . . . 8 (((x P y P) (𝑞 (2ndx) 𝑞 (1sty))) → 𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠))
14133adantl3 1061 . . . . . . 7 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → 𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠))
15 ltsonq 6382 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 6349 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 4663 . . . . . . . . . . . 12 ((𝑟 <Q 𝑞 𝑞 <Q 𝑠) → 𝑟 <Q 𝑠)
1817adantl 262 . . . . . . . . . . 11 (((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) (𝑟 <Q 𝑞 𝑞 <Q 𝑠)) → 𝑟 <Q 𝑠)
19 prop 6457 . . . . . . . . . . . . . . . 16 (z P → ⟨(1stz), (2ndz)⟩ P)
20 prloc 6473 . . . . . . . . . . . . . . . 16 ((⟨(1stz), (2ndz)⟩ P 𝑟 <Q 𝑠) → (𝑟 (1stz) 𝑠 (2ndz)))
2119, 20sylan 267 . . . . . . . . . . . . . . 15 ((z P 𝑟 <Q 𝑠) → (𝑟 (1stz) 𝑠 (2ndz)))
22213ad2antl3 1067 . . . . . . . . . . . . . 14 (((x P y P z P) 𝑟 <Q 𝑠) → (𝑟 (1stz) 𝑠 (2ndz)))
2322ex 108 . . . . . . . . . . . . 13 ((x P y P z P) → (𝑟 <Q 𝑠 → (𝑟 (1stz) 𝑠 (2ndz))))
2423adantr 261 . . . . . . . . . . . 12 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → (𝑟 <Q 𝑠 → (𝑟 (1stz) 𝑠 (2ndz))))
2524ad2antrr 457 . . . . . . . . . . 11 (((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) (𝑟 <Q 𝑞 𝑞 <Q 𝑠)) → (𝑟 <Q 𝑠 → (𝑟 (1stz) 𝑠 (2ndz))))
2618, 25mpd 13 . . . . . . . . . 10 (((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) (𝑟 <Q 𝑞 𝑞 <Q 𝑠)) → (𝑟 (1stz) 𝑠 (2ndz)))
27 elprnqu 6464 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1stx), (2ndx)⟩ P 𝑟 (2ndx)) → 𝑟 Q)
284, 27sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((x P 𝑟 (2ndx)) → 𝑟 Q)
29 ax-ia3 101 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 (2ndx) → (𝑟 (1stz) → (𝑟 (2ndx) 𝑟 (1stz))))
3029adantl 262 . . . . . . . . . . . . . . . . . . . 20 ((x P 𝑟 (2ndx)) → (𝑟 (1stz) → (𝑟 (2ndx) 𝑟 (1stz))))
31 19.8a 1479 . . . . . . . . . . . . . . . . . . . 20 ((𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))))
3228, 30, 31syl6an 1320 . . . . . . . . . . . . . . . . . . 19 ((x P 𝑟 (2ndx)) → (𝑟 (1stz) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)))))
33323ad2antl1 1065 . . . . . . . . . . . . . . . . . 18 (((x P y P z P) 𝑟 (2ndx)) → (𝑟 (1stz) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)))))
3433imp 115 . . . . . . . . . . . . . . . . 17 ((((x P y P z P) 𝑟 (2ndx)) 𝑟 (1stz)) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))))
35 df-rex 2306 . . . . . . . . . . . . . . . . 17 (𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)) ↔ 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))))
3634, 35sylibr 137 . . . . . . . . . . . . . . . 16 ((((x P y P z P) 𝑟 (2ndx)) 𝑟 (1stz)) → 𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)))
37 ltdfpr 6488 . . . . . . . . . . . . . . . . . . 19 ((x P z P) → (x<P z𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))))
3837biimprd 147 . . . . . . . . . . . . . . . . . 18 ((x P z P) → (𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)) → x<P z))
39383adant2 922 . . . . . . . . . . . . . . . . 17 ((x P y P z P) → (𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)) → x<P z))
4039ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((x P y P z P) 𝑟 (2ndx)) 𝑟 (1stz)) → (𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)) → x<P z))
4136, 40mpd 13 . . . . . . . . . . . . . . 15 ((((x P y P z P) 𝑟 (2ndx)) 𝑟 (1stz)) → x<P z)
4241ex 108 . . . . . . . . . . . . . 14 (((x P y P z P) 𝑟 (2ndx)) → (𝑟 (1stz) → x<P z))
4342adantrr 448 . . . . . . . . . . . . 13 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → (𝑟 (1stz) → x<P z))
44 elprnql 6463 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1sty), (2ndy)⟩ P 𝑠 (1sty)) → 𝑠 Q)
457, 44sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((y P 𝑠 (1sty)) → 𝑠 Q)
46 pm3.21 251 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 (1sty) → (𝑠 (2ndz) → (𝑠 (2ndz) 𝑠 (1sty))))
4746adantl 262 . . . . . . . . . . . . . . . . . . . 20 ((y P 𝑠 (1sty)) → (𝑠 (2ndz) → (𝑠 (2ndz) 𝑠 (1sty))))
48 19.8a 1479 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))))
4945, 47, 48syl6an 1320 . . . . . . . . . . . . . . . . . . 19 ((y P 𝑠 (1sty)) → (𝑠 (2ndz) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)))))
50493ad2antl2 1066 . . . . . . . . . . . . . . . . . 18 (((x P y P z P) 𝑠 (1sty)) → (𝑠 (2ndz) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)))))
5150imp 115 . . . . . . . . . . . . . . . . 17 ((((x P y P z P) 𝑠 (1sty)) 𝑠 (2ndz)) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))))
52 df-rex 2306 . . . . . . . . . . . . . . . . 17 (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) ↔ 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))))
5351, 52sylibr 137 . . . . . . . . . . . . . . . 16 ((((x P y P z P) 𝑠 (1sty)) 𝑠 (2ndz)) → 𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)))
54 ltdfpr 6488 . . . . . . . . . . . . . . . . . . . 20 ((z P y P) → (z<P y𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))))
5554biimprd 147 . . . . . . . . . . . . . . . . . . 19 ((z P y P) → (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) → z<P y))
5655ancoms 255 . . . . . . . . . . . . . . . . . 18 ((y P z P) → (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) → z<P y))
57563adant1 921 . . . . . . . . . . . . . . . . 17 ((x P y P z P) → (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) → z<P y))
5857ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((x P y P z P) 𝑠 (1sty)) 𝑠 (2ndz)) → (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) → z<P y))
5953, 58mpd 13 . . . . . . . . . . . . . . 15 ((((x P y P z P) 𝑠 (1sty)) 𝑠 (2ndz)) → z<P y)
6059ex 108 . . . . . . . . . . . . . 14 (((x P y P z P) 𝑠 (1sty)) → (𝑠 (2ndz) → z<P y))
6160adantrl 447 . . . . . . . . . . . . 13 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → (𝑠 (2ndz) → z<P y))
6243, 61orim12d 699 . . . . . . . . . . . 12 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → ((𝑟 (1stz) 𝑠 (2ndz)) → (x<P z z<P y)))
6362adantlr 446 . . . . . . . . . . 11 ((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) → ((𝑟 (1stz) 𝑠 (2ndz)) → (x<P z z<P y)))
6463adantr 261 . . . . . . . . . 10 (((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) (𝑟 <Q 𝑞 𝑞 <Q 𝑠)) → ((𝑟 (1stz) 𝑠 (2ndz)) → (x<P z z<P y)))
6526, 64mpd 13 . . . . . . . . 9 (((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) (𝑟 <Q 𝑞 𝑞 <Q 𝑠)) → (x<P z z<P y))
6665ex 108 . . . . . . . 8 ((((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) (𝑟 (2ndx) 𝑠 (1sty))) → ((𝑟 <Q 𝑞 𝑞 <Q 𝑠) → (x<P z z<P y)))
6766rexlimdvva 2434 . . . . . . 7 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → (𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠) → (x<P z z<P y)))
6814, 67mpd 13 . . . . . 6 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → (x<P z z<P y))
6968ex 108 . . . . 5 ((x P y P z P) → ((𝑞 (2ndx) 𝑞 (1sty)) → (x<P z z<P y)))
7069rexlimdvw 2430 . . . 4 ((x P y P z P) → (𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)) → (x<P z z<P y)))
713, 70sylbid 139 . . 3 ((x P y P z P) → (x<P y → (x<P z z<P y)))
7271rgen3 2400 . 2 x P y P z P (x<P y → (x<P z z<P y))
73 df-iso 4025 . 2 (<P Or P ↔ (<P Po P x P y P z P (x<P y → (x<P z z<P y))))
741, 72, 73mpbir2an 848 1 <P Or P
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   w3a 884  wex 1378   wcel 1390  wral 2300  wrex 2301  cop 3370   class class class wbr 3755   Po wpo 4022   Or wor 4023  cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   <Q cltq 6269  Pcnp 6275  <P cltp 6279
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 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-enq 6331  df-nqqs 6332  df-ltnqqs 6337  df-inp 6448  df-iltp 6452
This theorem is referenced by:  addextpr  6592  lttrsr  6670  ltposr  6671  ltsosr  6672  archsr  6688
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