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Theorem ltpopr 6426
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 6427. (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 6323 . . . . . . . 8 (𝑠 P → ⟨(1st𝑠), (2nd𝑠)⟩ P)
2 prdisj 6340 . . . . . . . 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 588 . . . . . 6 ((𝑠 P 𝑞 Q) → ¬ (𝑞 (2nd𝑠) 𝑞 (1st𝑠)))
65nrexdv 2386 . . . . 5 (𝑠 P → ¬ 𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠)))
7 ltdfpr 6354 . . . . . 6 ((𝑠 P 𝑠 P) → (𝑠<P 𝑠𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠))))
87anidms 379 . . . . 5 (𝑠 P → (𝑠<P 𝑠𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑠))))
96, 8mtbird 585 . . . 4 (𝑠 P → ¬ 𝑠<P 𝑠)
109adantl 262 . . 3 (( ⊤ 𝑠 P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 6354 . . . . . . . . . . 11 ((𝑠 P 𝑡 P) → (𝑠<P 𝑡𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡))))
12113adant3 910 . . . . . . . . . 10 ((𝑠 P 𝑡 P u P) → (𝑠<P 𝑡𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1st𝑡))))
13 ltdfpr 6354 . . . . . . . . . . 11 ((𝑡 P u P) → (𝑡<P u𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu))))
14133adant1 908 . . . . . . . . . 10 ((𝑠 P 𝑡 P u P) → (𝑡<P u𝑟 Q (𝑟 (2nd𝑡) 𝑟 (1stu))))
1512, 14anbi12d 445 . . . . . . . . 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 2453 . . . . . . . . 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 477 . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . . 18 (𝑡 P → ⟨(1st𝑡), (2nd𝑡)⟩ P)
21 prltlu 6335 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ P 𝑞 (1st𝑡) 𝑟 (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1152 . . . . . . . . . . . . . . . . 17 ((𝑡 P 𝑞 (1st𝑡) 𝑟 (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1116 . . . . . . . . . . . . . . . 16 ((𝑡 P 𝑞 (1st𝑡) (𝑟 (2nd𝑡) 𝑟 (1stu))) → 𝑞 <Q 𝑟)
24233adant2l 1113 . . . . . . . . . . . . . . 15 ((𝑡 P (𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu))) → 𝑞 <Q 𝑟)
25243expb 1089 . . . . . . . . . . . . . 14 ((𝑡 P ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1053 . . . . . . . . . . . . 13 (((𝑠 P 𝑡 P u P) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → 𝑞 <Q 𝑟)
2726adantlr 449 . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . . . 17 (u P → ⟨(1stu), (2ndu)⟩ P)
29 prcdnql 6332 . . . . . . . . . . . . . . . . 17 ((⟨(1stu), (2ndu)⟩ P 𝑟 (1stu)) → (𝑞 <Q 𝑟𝑞 (1stu)))
3028, 29sylan 267 . . . . . . . . . . . . . . . 16 ((u P 𝑟 (1stu)) → (𝑞 <Q 𝑟𝑞 (1stu)))
3130adantrl 450 . . . . . . . . . . . . . . 15 ((u P (𝑟 (2nd𝑡) 𝑟 (1stu))) → (𝑞 <Q 𝑟𝑞 (1stu)))
3231adantrl 450 . . . . . . . . . . . . . 14 ((u P ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 <Q 𝑟𝑞 (1stu)))
33323ad2antl3 1054 . . . . . . . . . . . . 13 (((𝑠 P 𝑡 P u P) ((𝑞 (2nd𝑠) 𝑞 (1st𝑡)) (𝑟 (2nd𝑡) 𝑟 (1stu)))) → (𝑞 <Q 𝑟𝑞 (1stu)))
3433adantlr 449 . . . . . . . . . . . 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 2410 . . . . . . . 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 2394 . . . . . . 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 6354 . . . . . . . . 9 ((𝑠 P u P) → (𝑠<P u𝑞 Q (𝑞 (2nd𝑠) 𝑞 (1stu))))
42413adant2 909 . . . . . . . 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 4011 . 2 ( ⊤ → <P Po P)
4948trud 1235 1 <P Po P
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3a 871  wtru 1227   wcel 1370  wrex 2281  cop 3349   class class class wbr 3734   Po wpo 4001  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:  ltsopr  6427
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