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Theorem ltpopr 6750
 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 6751. (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 6630 . . . . . . . 8 (𝑠P → ⟨(1st𝑠), (2nd𝑠)⟩ ∈ P)
2 prdisj 6647 . . . . . . . 8 ((⟨(1st𝑠), (2nd𝑠)⟩ ∈ P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
31, 2sylan 271 . . . . . . 7 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)))
4 ancom 257 . . . . . . 7 ((𝑞 ∈ (1st𝑠) ∧ 𝑞 ∈ (2nd𝑠)) ↔ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
53, 4sylnib 611 . . . . . 6 ((𝑠P𝑞Q) → ¬ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
65nrexdv 2429 . . . . 5 (𝑠P → ¬ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠)))
7 ltdfpr 6661 . . . . . 6 ((𝑠P𝑠P) → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
87anidms 383 . . . . 5 (𝑠P → (𝑠<P 𝑠 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑠))))
96, 8mtbird 608 . . . 4 (𝑠P → ¬ 𝑠<P 𝑠)
109adantl 266 . . 3 ((⊤ ∧ 𝑠P) → ¬ 𝑠<P 𝑠)
11 ltdfpr 6661 . . . . . . . . . . 11 ((𝑠P𝑡P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
12113adant3 935 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑡 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡))))
13 ltdfpr 6661 . . . . . . . . . . 11 ((𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
14133adant1 933 . . . . . . . . . 10 ((𝑠P𝑡P𝑢P) → (𝑡<P 𝑢 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1512, 14anbi12d 450 . . . . . . . . 9 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
16 reeanv 2496 . . . . . . . . 9 (∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) ↔ (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ ∃𝑟Q (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
1715, 16syl6bbr 191 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) ↔ ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))))
1817biimpa 284 . . . . . . 7 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → ∃𝑞Q𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))))
19 simprll 497 . . . . . . . . . . 11 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 ∈ (2nd𝑠))
20 prop 6630 . . . . . . . . . . . . . . . . . 18 (𝑡P → ⟨(1st𝑡), (2nd𝑡)⟩ ∈ P)
21 prltlu 6642 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝑡), (2nd𝑡)⟩ ∈ P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
2220, 21syl3an1 1179 . . . . . . . . . . . . . . . . 17 ((𝑡P𝑞 ∈ (1st𝑡) ∧ 𝑟 ∈ (2nd𝑡)) → 𝑞 <Q 𝑟)
23223adant3r 1143 . . . . . . . . . . . . . . . 16 ((𝑡P𝑞 ∈ (1st𝑡) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
24233adant2l 1140 . . . . . . . . . . . . . . 15 ((𝑡P ∧ (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → 𝑞 <Q 𝑟)
25243expb 1116 . . . . . . . . . . . . . 14 ((𝑡P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
26253ad2antl2 1078 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
2726adantlr 454 . . . . . . . . . . . 12 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → 𝑞 <Q 𝑟)
28 prop 6630 . . . . . . . . . . . . . . . . 17 (𝑢P → ⟨(1st𝑢), (2nd𝑢)⟩ ∈ P)
29 prcdnql 6639 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝑢), (2nd𝑢)⟩ ∈ P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3028, 29sylan 271 . . . . . . . . . . . . . . . 16 ((𝑢P𝑟 ∈ (1st𝑢)) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3130adantrl 455 . . . . . . . . . . . . . . 15 ((𝑢P ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3231adantrl 455 . . . . . . . . . . . . . 14 ((𝑢P ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
33323ad2antl3 1079 . . . . . . . . . . . . 13 (((𝑠P𝑡P𝑢P) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 <Q 𝑟𝑞 ∈ (1st𝑢)))
3433adantlr 454 . . . . . . . . . . . 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 294 . . . . . . . . . 10 ((((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) ∧ ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢)))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)))
3736ex 112 . . . . . . . . 9 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3837rexlimdvw 2453 . . . . . . . 8 (((𝑠P𝑡P𝑢P) ∧ (𝑠<P 𝑡𝑡<P 𝑢)) → (∃𝑟Q ((𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑡)) ∧ (𝑟 ∈ (2nd𝑡) ∧ 𝑟 ∈ (1st𝑢))) → (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
3938reximdv 2437 . . . . . . 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 6661 . . . . . . . . 9 ((𝑠P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
42413adant2 934 . . . . . . . 8 ((𝑠P𝑡P𝑢P) → (𝑠<P 𝑢 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢))))
4342biimprd 151 . . . . . . 7 ((𝑠P𝑡P𝑢P) → (∃𝑞Q (𝑞 ∈ (2nd𝑠) ∧ 𝑞 ∈ (1st𝑢)) → 𝑠<P 𝑢))
4443adantr 265 . . . . . 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 112 . . . 4 ((𝑠P𝑡P𝑢P) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4746adantl 266 . . 3 ((⊤ ∧ (𝑠P𝑡P𝑢P)) → ((𝑠<P 𝑡𝑡<P 𝑢) → 𝑠<P 𝑢))
4810, 47ispod 4068 . 2 (⊤ → <P Po P)
4948trud 1268 1 <P Po P
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∧ w3a 896  ⊤wtru 1260   ∈ wcel 1409  ∃wrex 2324  ⟨cop 3405   class class class wbr 3791   Po wpo 4058  ‘cfv 4929  1st c1st 5792  2nd c2nd 5793  Qcnq 6435
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