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Theorem ltsopr 6694
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4034). Proposition 11.2.3 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Dec-2019.)
Assertion
Ref Expression
ltsopr <P Or P

Proof of Theorem ltsopr
Dummy variables 𝑟 𝑞 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltpopr 6693 . 2 <P Po P
2 ltdfpr 6604 . . . . 5 ((𝑥P𝑦P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
323adant3 924 . . . 4 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
4 prop 6573 . . . . . . . . . . . 12 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
5 prnminu 6587 . . . . . . . . . . . 12 ((⟨(1st𝑥), (2nd𝑥)⟩ ∈ P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
64, 5sylan 267 . . . . . . . . . . 11 ((𝑥P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
7 prop 6573 . . . . . . . . . . . 12 (𝑦P → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ P)
8 prnmaxl 6586 . . . . . . . . . . . 12 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑞 ∈ (1st𝑦)) → ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠)
97, 8sylan 267 . . . . . . . . . . 11 ((𝑦P𝑞 ∈ (1st𝑦)) → ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠)
106, 9anim12i 321 . . . . . . . . . 10 (((𝑥P𝑞 ∈ (2nd𝑥)) ∧ (𝑦P𝑞 ∈ (1st𝑦))) → (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
1110an4s 522 . . . . . . . . 9 (((𝑥P𝑦P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
12 reeanv 2479 . . . . . . . . 9 (∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠) ↔ (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
1311, 12sylibr 137 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → ∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠))
14133adantl3 1062 . . . . . . 7 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → ∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠))
15 ltsonq 6496 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 6463 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 4720 . . . . . . . . . . . 12 ((𝑟 <Q 𝑞𝑞 <Q 𝑠) → 𝑟 <Q 𝑠)
1817adantl 262 . . . . . . . . . . 11 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → 𝑟 <Q 𝑠)
19 prop 6573 . . . . . . . . . . . . . . . 16 (𝑧P → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ P)
20 prloc 6589 . . . . . . . . . . . . . . . 16 ((⟨(1st𝑧), (2nd𝑧)⟩ ∈ P𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
2119, 20sylan 267 . . . . . . . . . . . . . . 15 ((𝑧P𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
22213ad2antl3 1068 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
2322ex 108 . . . . . . . . . . . . 13 ((𝑥P𝑦P𝑧P) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧))))
2423adantr 261 . . . . . . . . . . . 12 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧))))
2524ad2antrr 457 . . . . . . . . . . 11 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧))))
2618, 25mpd 13 . . . . . . . . . 10 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
27 elprnqu 6580 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝑥), (2nd𝑥)⟩ ∈ P𝑟 ∈ (2nd𝑥)) → 𝑟Q)
284, 27sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((𝑥P𝑟 ∈ (2nd𝑥)) → 𝑟Q)
29 ax-ia3 101 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ (2nd𝑥) → (𝑟 ∈ (1st𝑧) → (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3029adantl 262 . . . . . . . . . . . . . . . . . . . 20 ((𝑥P𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
31 19.8a 1482 . . . . . . . . . . . . . . . . . . . 20 ((𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3228, 30, 31syl6an 1323 . . . . . . . . . . . . . . . . . . 19 ((𝑥P𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))))
33323ad2antl1 1066 . . . . . . . . . . . . . . . . . 18 (((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))))
3433imp 115 . . . . . . . . . . . . . . . . 17 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
35 df-rex 2312 . . . . . . . . . . . . . . . . 17 (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) ↔ ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3634, 35sylibr 137 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → ∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))
37 ltdfpr 6604 . . . . . . . . . . . . . . . . . . 19 ((𝑥P𝑧P) → (𝑥<P 𝑧 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3837biimprd 147 . . . . . . . . . . . . . . . . . 18 ((𝑥P𝑧P) → (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧))
39383adant2 923 . . . . . . . . . . . . . . . . 17 ((𝑥P𝑦P𝑧P) → (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧))
4039ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧))
4136, 40mpd 13 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧)
4241ex 108 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → 𝑥<P 𝑧))
4342adantrr 448 . . . . . . . . . . . . 13 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → (𝑟 ∈ (1st𝑧) → 𝑥<P 𝑧))
44 elprnql 6579 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑠 ∈ (1st𝑦)) → 𝑠Q)
457, 44sylan 267 . . . . . . . . . . . . . . . . . . . 20 ((𝑦P𝑠 ∈ (1st𝑦)) → 𝑠Q)
46 pm3.21 251 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (1st𝑦) → (𝑠 ∈ (2nd𝑧) → (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
4746adantl 262 . . . . . . . . . . . . . . . . . . . 20 ((𝑦P𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
48 19.8a 1482 . . . . . . . . . . . . . . . . . . . 20 ((𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
4945, 47, 48syl6an 1323 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))))
50493ad2antl2 1067 . . . . . . . . . . . . . . . . . 18 (((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))))
5150imp 115 . . . . . . . . . . . . . . . . 17 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
52 df-rex 2312 . . . . . . . . . . . . . . . . 17 (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) ↔ ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
5351, 52sylibr 137 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → ∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))
54 ltdfpr 6604 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑦P) → (𝑧<P 𝑦 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
5554biimprd 147 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑦P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
5655ancoms 255 . . . . . . . . . . . . . . . . . 18 ((𝑦P𝑧P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
57563adant1 922 . . . . . . . . . . . . . . . . 17 ((𝑥P𝑦P𝑧P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
5857ad2antrr 457 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
5953, 58mpd 13 . . . . . . . . . . . . . . 15 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → 𝑧<P 𝑦)
6059ex 108 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → 𝑧<P 𝑦))
6160adantrl 447 . . . . . . . . . . . . 13 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → (𝑠 ∈ (2nd𝑧) → 𝑧<P 𝑦))
6243, 61orim12d 700 . . . . . . . . . . . 12 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)) → (𝑥<P 𝑧𝑧<P 𝑦)))
6362adantlr 446 . . . . . . . . . . 11 ((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)) → (𝑥<P 𝑧𝑧<P 𝑦)))
6463adantr 261 . . . . . . . . . 10 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → ((𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)) → (𝑥<P 𝑧𝑧<P 𝑦)))
6526, 64mpd 13 . . . . . . . . 9 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → (𝑥<P 𝑧𝑧<P 𝑦))
6665ex 108 . . . . . . . 8 ((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 <Q 𝑞𝑞 <Q 𝑠) → (𝑥<P 𝑧𝑧<P 𝑦)))
6766rexlimdvva 2440 . . . . . . 7 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠) → (𝑥<P 𝑧𝑧<P 𝑦)))
6814, 67mpd 13 . . . . . 6 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (𝑥<P 𝑧𝑧<P 𝑦))
6968ex 108 . . . . 5 ((𝑥P𝑦P𝑧P) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) → (𝑥<P 𝑧𝑧<P 𝑦)))
7069rexlimdvw 2436 . . . 4 ((𝑥P𝑦P𝑧P) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) → (𝑥<P 𝑧𝑧<P 𝑦)))
713, 70sylbid 139 . . 3 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦)))
7271rgen3 2406 . 2 𝑥P𝑦P𝑧P (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦))
73 df-iso 4034 . 2 (<P Or P ↔ (<P Po P ∧ ∀𝑥P𝑦P𝑧P (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦))))
741, 72, 73mpbir2an 849 1 <P Or P
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629  w3a 885  wex 1381  wcel 1393  wral 2306  wrex 2307  cop 3378   class class class wbr 3764   Po wpo 4031   Or wor 4032  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:  prplnqu  6718  addextpr  6719  caucvgprprlemk  6781  caucvgprprlemnkltj  6787  caucvgprprlemnkeqj  6788  caucvgprprlemnjltk  6789  caucvgprprlemnbj  6791  caucvgprprlemml  6792  caucvgprprlemlol  6796  caucvgprprlemupu  6798  caucvgprprlemloc  6801  caucvgprprlemaddq  6806  lttrsr  6847  ltposr  6848  ltsosr  6849  archsr  6866
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