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Theorem ltpopr 6693
Description: Positive real 'less than' is a partial ordering. Remark ("< is transitive and irreflexive") preceding Proposition 11.2.3 of [HoTT], p. (varies). Lemma for ltsopr 6694. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
ltpopr <P Po P

Proof of Theorem ltpopr
Dummy variables 𝑟 𝑞 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2 prdisj 6590 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
31, 2sylan 267 . . . . . . 7 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
4 ancom 253 . . . . . . 7 ((𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)) ↔ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
53, 4sylnib 601 . . . . . 6 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
65nrexdv 2412 . . . . 5 (𝑠P → ¬ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
7 ltdfpr 6604 . . . . . 6 ((𝑠P𝑠P) → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
87anidms 377 . . . . 5 (𝑠P → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
96, 8mtbird 598 . . . 4 (𝑠P → ¬ 𝑠<P 𝑠)
109adantl 262 . . 3 ((⊤ ∧ 𝑠P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 6604 . . . . . . . . . . 11 ((𝑠P𝑡P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
12113adant3 924 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
13 ltdfpr 6604 . . . . . . . . . . 11 ((𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
14133adant1 922 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1512, 14anbi12d 442 . . . . . . . . 9 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
16 reeanv 2479 . . . . . . . . 9 (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1715, 16syl6bbr 187 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
1817biimpa 280 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
19 simprll 489 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (2nd𝑠))
20 prop 6573 . . . . . . . . . . . . . . . . . 18 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
21 prltlu 6585 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1168 . . . . . . . . . . . . . . . . 17 ((𝑡P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1132 . . . . . . . . . . . . . . . 16 ((𝑡P𝑞 ∈ (1st𝑡) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
24233adant2l 1129 . . . . . . . . . . . . . . 15 ((𝑡P ∧ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
25243expb 1105 . . . . . . . . . . . . . 14 ((𝑡P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1067 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
2726adantlr 446 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
28 prop 6573 . . . . . . . . . . . . . . . . 17 (𝑢P → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ P)
29 prcdnql 6582 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝑢), (2nd𝑢)⟩ ∈ P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3028, 29sylan 267 . . . . . . . . . . . . . . . 16 ((𝑢P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3130adantrl 447 . . . . . . . . . . . . . . 15 ((𝑢P ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3231adantrl 447 . . . . . . . . . . . . . 14 ((𝑢P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
33323ad2antl3 1068 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3433adantlr 446 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3527, 34mpd 13 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (1st𝑢))
3619, 35jca 290 . . . . . . . . . 10 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
3736ex 108 . . . . . . . . 9 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3837rexlimdvw 2436 . . . . . . . 8 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3938reximdv 2420 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4018, 39mpd 13 . . . . . 6 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
41 ltdfpr 6604 . . . . . . . . 9 ((𝑠P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
42413adant2 923 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4342biimprd 147 . . . . . . 7 ((𝑠P𝑡P𝑢P) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4443adantr 261 . . . . . 6 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4540, 44mpd 13 . . . . 5 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → 𝑠<P 𝑢)
4645ex 108 . . . 4 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4746adantl 262 . . 3 ((⊤ ∧ (𝑠P𝑡P𝑢P)) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4810, 47ispod 4041 . 2 (⊤ → <P Po P)
4948trud 1252 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3a 885  wtru 1244  wcel 1393  wrex 2307  cop 3378   class class class wbr 3764   Po wpo 4031  cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6378   <Q cltq 6383  Pcnp 6389  <P cltp 6393
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-enq 6445  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564  df-iltp 6568
This theorem is referenced by:  ltsopr  6694
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