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Theorem ltsopr 6422
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4000). 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 6421 . 2 <P Po P
2 ltdfpr 6349 . . . . 5 ((x P y P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
323adant3 908 . . . 4 ((x P y P z P) → (x<P y𝑞 Q (𝑞 (2ndx) 𝑞 (1sty))))
4 prop 6318 . . . . . . . . . . . 12 (x P → ⟨(1stx), (2ndx)⟩ P)
5 prnminu 6332 . . . . . . . . . . . 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 6318 . . . . . . . . . . . 12 (y P → ⟨(1sty), (2ndy)⟩ P)
8 prnmaxl 6331 . . . . . . . . . . . 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 507 . . . . . . . . 9 (((x P y P) (𝑞 (2ndx) 𝑞 (1sty))) → (𝑟 (2ndx)𝑟 <Q 𝑞 𝑠 (1sty)𝑞 <Q 𝑠))
12 reeanv 2451 . . . . . . . . 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 1046 . . . . . . 7 (((x P y P z P) (𝑞 (2ndx) 𝑞 (1sty))) → 𝑟 (2ndx)𝑠 (1sty)(𝑟 <Q 𝑞 𝑞 <Q 𝑠))
15 ltsonq 6246 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 6213 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 4638 . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . 16 (z P → ⟨(1stz), (2ndz)⟩ P)
20 prloc 6334 . . . . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . . 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 6325 . . . . . . . . . . . . . . . . . . . . 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 1458 . . . . . . . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . . . . . . 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 2284 . . . . . . . . . . . . . . . . 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 6349 . . . . . . . . . . . . . . . . . . 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 907 . . . . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . . . . . . . . . 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 1458 . . . . . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . . . . 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 2284 . . . . . . . . . . . . . . . . 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 6349 . . . . . . . . . . . . . . . . . . . 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 906 . . . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . 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 2412 . . . . . . 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 2408 . . . 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 2378 . 2 x P y P z P (x<P y → (x<P z z<P y))
73 df-iso 4000 . 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 833 1 <P Or P
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 613   w3a 869  wex 1357   wcel 1369  wral 2278  wrex 2279  cop 3345   class class class wbr 3730   Po wpo 3997   Or wor 3998  cfv 4820  1st c1st 5679  2nd c2nd 5680  Qcnq 6129   <Q cltq 6134  Pcnp 6140  <P cltp 6144
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 529  ax-in2 530  ax-io 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-13 1380  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-coll 3838  ax-sep 3841  ax-nul 3849  ax-pow 3893  ax-pr 3910  ax-un 4111  ax-setind 4195  ax-iinf 4229
This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 870  df-3an 871  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ne 2182  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2287  df-v 2531  df-sbc 2736  df-csb 2824  df-dif 2891  df-un 2893  df-in 2895  df-ss 2902  df-nul 3196  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-uni 3547  df-int 3582  df-iun 3625  df-br 3731  df-opab 3785  df-mpt 3786  df-tr 3821  df-eprel 3992  df-id 3996  df-po 3999  df-iso 4000  df-iord 4044  df-on 4046  df-suc 4049  df-iom 4232  df-xp 4269  df-rel 4270  df-cnv 4271  df-co 4272  df-dm 4273  df-rn 4274  df-res 4275  df-ima 4276  df-iota 4785  df-fun 4822  df-fn 4823  df-f 4824  df-f1 4825  df-fo 4826  df-f1o 4827  df-fv 4828  df-ov 5430  df-oprab 5431  df-mpt2 5432  df-1st 5681  df-2nd 5682  df-recs 5833  df-irdg 5869  df-oadd 5911  df-omul 5912  df-er 6008  df-ec 6010  df-qs 6014  df-ni 6153  df-mi 6155  df-lti 6156  df-enq 6195  df-nqqs 6196  df-ltnqqs 6201  df-inp 6309  df-iltp 6313
This theorem is referenced by:  addextpr  6445  lttrsr  6503  ltposr  6504  ltsosr  6505
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