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Theorem ltsopr 6427
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4004). 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 6426 . 2 <P Po P
2 ltdfpr 6354 . . . . 5 ((x P y P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
323adant3 910 . . . 4 ((x P y P z P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
4 prop 6323 . . . . . . . . . . . 12 (x P → ⟨(1stx), (2ndx)⟩ P)
5 prnminu 6337 . . . . . . . . . . . 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 6323 . . . . . . . . . . . 12 (y P → ⟨(1sty), (2ndy)⟩ P)
8 prnmaxl 6336 . . . . . . . . . . . 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 509 . . . . . . . . 9 (((x P y P) (𝑞 (2ndx) 𝑞 (1sty))) → (𝑟 (2ndx)𝑟 <Q 𝑞 𝑠 (1sty)𝑞 <Q 𝑠))
12 reeanv 2453 . . . . . . . . 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 1048 . . . . . . 7 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → 𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠))
15 ltsonq 6251 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 6218 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 4643 . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . 16 (z P → ⟨(1stz), (2ndz)⟩ P)
20 prloc 6339 . . . . . . . . . . . . . . . 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 1054 . . . . . . . . . . . . . 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 460 . . . . . . . . . . 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 6330 . . . . . . . . . . . . . . . . . . . . 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 1460 . . . . . . . . . . . . . . . . . . . 20 ((𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz))))
3228, 30, 31syl6an 1299 . . . . . . . . . . . . . . . . . . 19 ((x P 𝑟 (2ndx)) → (𝑟 (1stz) → 𝑟(𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)))))
33323ad2antl1 1052 . . . . . . . . . . . . . . . . . 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 2286 . . . . . . . . . . . . . . . . 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 6354 . . . . . . . . . . . . . . . . . . 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 909 . . . . . . . . . . . . . . . . 17 ((x P y P z P) → (𝑟 Q (𝑟 (2ndx) 𝑟 (1stz)) → x<P z))
4039ad2antrr 460 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 13 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → (𝑟 (1stz) → x<P z))
44 elprnql 6329 . . . . . . . . . . . . . . . . . . . . 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 1460 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty))))
4945, 47, 48syl6an 1299 . . . . . . . . . . . . . . . . . . 19 ((y P 𝑠 (1sty)) → (𝑠 (2ndz) → 𝑠(𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)))))
50493ad2antl2 1053 . . . . . . . . . . . . . . . . . 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 2286 . . . . . . . . . . . . . . . . 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 6354 . . . . . . . . . . . . . . . . . . . 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 908 . . . . . . . . . . . . . . . . 17 ((x P y P z P) → (𝑠 Q (𝑠 (2ndz) 𝑠 (1sty)) → z<P y))
5857ad2antrr 460 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . 13 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → (𝑠 (2ndz) → z<P y))
6243, 61orim12d 687 . . . . . . . . . . . 12 (((x P y P z P) (𝑟 (2ndx) 𝑠 (1sty))) → ((𝑟 (1stz) 𝑠 (2ndz)) → (x<P z z<P y)))
6362adantlr 449 . . . . . . . . . . 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 2414 . . . . . . 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 2410 . . . 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 2380 . 2 x P y P z P (x<P y → (x<P z z<P y))
73 df-iso 4004 . 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 835 1 <P Or P
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 616   w3a 871  wex 1358   wcel 1370  wral 2280  wrex 2281  cop 3349   class class class wbr 3734   Po wpo 4001   Or wor 4002  cfv 4825  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   <Q cltq 6139  Pcnp 6145  <P cltp 6149
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-mi 6160  df-lti 6161  df-enq 6200  df-nqqs 6201  df-ltnqqs 6206  df-inp 6314  df-iltp 6318
This theorem is referenced by:  lttrsr  6506  ltposr  6507  ltsosr  6508
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