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Theorem ltpopr 6559
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 6560. (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 6450 . . . . . . . 8 (𝑠 P → ⟨(1st𝑠), (2nd𝑠)⟩ P)
2 prdisj 6467 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ P 𝑞 Q) → ¬ (𝑞 (1st𝑠) 𝑞 (2nd𝑠)))
31, 2sylan 267 . . . . . . 7 ((𝑠 P 𝑞 Q) → ¬ (𝑞 (1st𝑠) 𝑞 (2nd𝑠)))
4 ancom 253 . . . . . . 7 ((𝑞 (1st𝑠) 𝑞 (2nd𝑠)) ↔ (𝑞 (2nd𝑠) 𝑞 (1st𝑠)))
53, 4sylnib 600 . . . . . 6 ((𝑠 P 𝑞 Q) → ¬ (𝑞 (2nd𝑠) 𝑞 (1st𝑠)))
65nrexdv 2406 . . . . 5 (𝑠 P → ¬ 𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠)))
7 ltdfpr 6481 . . . . . 6 ((𝑠 P 𝑠 P) → (𝑠<P 𝑠𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠))))
87anidms 377 . . . . 5 (𝑠 P → (𝑠<P 𝑠𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠))))
96, 8mtbird 597 . . . 4 (𝑠 P → ¬ 𝑠<P 𝑠)
109adantl 262 . . 3 (( ⊤ 𝑠 P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 6481 . . . . . . . . . . 11 ((𝑠 P 𝑡 P) → (𝑠<P 𝑡𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡))))
12113adant3 923 . . . . . . . . . 10 ((𝑠 P 𝑡 P u P) → (𝑠<P 𝑡𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡))))
13 ltdfpr 6481 . . . . . . . . . . 11 ((𝑡 P u P) → (𝑡<P u𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu))))
14133adant1 921 . . . . . . . . . 10 ((𝑠 P 𝑡 P u P) → (𝑡<P u𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu))))
1512, 14anbi12d 442 . . . . . . . . 9 ((𝑠 P 𝑡 P u P) → ((𝑠<P 𝑡 𝑡<P u) ↔ (𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡)) 𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu)))))
16 reeanv 2473 . . . . . . . . 9 (𝑞 Q 𝑟 Q ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) ↔ (𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡)) 𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu))))
1715, 16syl6bbr 187 . . . . . . . 8 ((𝑠 P 𝑡 P u P) → ((𝑠<P 𝑡 𝑡<P u) ↔ 𝑞 Q 𝑟 Q ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))))
1817biimpa 280 . . . . . . 7 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → 𝑞 Q 𝑟 Q ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))))
19 simprll 489 . . . . . . . . . . 11 ((((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 (2nd𝑠))
20 prop 6450 . . . . . . . . . . . . . . . . . 18 (𝑡 P → ⟨(1st𝑡), (2nd𝑡)⟩ P)
21 prltlu 6462 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ P 𝑞 (1st𝑡) 𝑟 (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1167 . . . . . . . . . . . . . . . . 17 ((𝑡 P 𝑞 (1st𝑡) 𝑟 (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1131 . . . . . . . . . . . . . . . 16 ((𝑡 P 𝑞 (1st𝑡) (𝑟 (2nd𝑡) 𝑟 (1stu))) → 𝑞 <Q 𝑟)
24233adant2l 1128 . . . . . . . . . . . . . . 15 ((𝑡 P (𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) → 𝑞 <Q 𝑟)
25243expb 1104 . . . . . . . . . . . . . 14 ((𝑡 P ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1066 . . . . . . . . . . . . 13 (((𝑠 P 𝑡 P u P) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 <Q 𝑟)
2726adantlr 446 . . . . . . . . . . . 12 ((((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 <Q 𝑟)
28 prop 6450 . . . . . . . . . . . . . . . . 17 (u P → ⟨(1stu), (2ndu)⟩ P)
29 prcdnql 6459 . . . . . . . . . . . . . . . . 17 ((⟨(1stu), (2ndu)⟩ P 𝑟 (1stu)) → (𝑞 <Q 𝑟𝑞 (1stu)))
3028, 29sylan 267 . . . . . . . . . . . . . . . 16 ((u P 𝑟 (1stu)) → (𝑞 <Q 𝑟𝑞 (1stu)))
3130adantrl 447 . . . . . . . . . . . . . . 15 ((u P (𝑟 (2nd𝑡) 𝑟 (1stu))) → (𝑞 <Q 𝑟𝑞 (1stu)))
3231adantrl 447 . . . . . . . . . . . . . 14 ((u P ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 <Q 𝑟𝑞 (1stu)))
33323ad2antl3 1067 . . . . . . . . . . . . 13 (((𝑠 P 𝑡 P u P) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 <Q 𝑟𝑞 (1stu)))
3433adantlr 446 . . . . . . . . . . . 12 ((((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 <Q 𝑟𝑞 (1stu)))
3527, 34mpd 13 . . . . . . . . . . 11 ((((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 (1stu))
3619, 35jca 290 . . . . . . . . . 10 ((((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 (2nd𝑠) 𝑞 (1stu)))
3736ex 108 . . . . . . . . 9 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → (((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) → (𝑞 (2nd𝑠) 𝑞 (1stu))))
3837rexlimdvw 2430 . . . . . . . 8 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → (𝑟 Q ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) → (𝑞 (2nd𝑠) 𝑞 (1stu))))
3938reximdv 2414 . . . . . . 7 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → (𝑞 Q 𝑟 Q ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) → 𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu))))
4018, 39mpd 13 . . . . . 6 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → 𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu)))
41 ltdfpr 6481 . . . . . . . . 9 ((𝑠 P u P) → (𝑠<P u𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu))))
42413adant2 922 . . . . . . . 8 ((𝑠 P 𝑡 P u P) → (𝑠<P u𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu))))
4342biimprd 147 . . . . . . 7 ((𝑠 P 𝑡 P u P) → (𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu)) → 𝑠<P u))
4443adantr 261 . . . . . 6 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → (𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu)) → 𝑠<P u))
4540, 44mpd 13 . . . . 5 (((𝑠 P 𝑡 P u P) (𝑠<P 𝑡 𝑡<P u)) → 𝑠<P u)
4645ex 108 . . . 4 ((𝑠 P 𝑡 P u P) → ((𝑠<P 𝑡 𝑡<P u) → 𝑠<P u))
4746adantl 262 . . 3 (( ⊤ (𝑠 P 𝑡 P u P)) → ((𝑠<P 𝑡 𝑡<P u) → 𝑠<P u))
4810, 47ispod 4031 . 2 ( ⊤ → <P Po P)
4948trud 1251 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 884  wtru 1243   wcel 1390  wrex 2301  cop 3369   class class class wbr 3754   Po wpo 4021  cfv 4844  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   <Q cltq 6262  Pcnp 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:  ltsopr  6560
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