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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
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