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Theorem ltsopr 7597
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4299). 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 7596 . 2 <P Po P
2 ltdfpr 7507 . . . . 5 ((𝑥P𝑦P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
323adant3 1017 . . . 4 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
4 prop 7476 . . . . . . . . . . . 12 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
5 prnminu 7490 . . . . . . . . . . . 12 ((⟨(1st𝑥), (2nd𝑥)⟩ ∈ P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
64, 5sylan 283 . . . . . . . . . . 11 ((𝑥P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
7 prop 7476 . . . . . . . . . . . 12 (𝑦P → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ P)
8 prnmaxl 7489 . . . . . . . . . . . 12 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑞 ∈ (1st𝑦)) → ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠)
97, 8sylan 283 . . . . . . . . . . 11 ((𝑦P𝑞 ∈ (1st𝑦)) → ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠)
106, 9anim12i 338 . . . . . . . . . 10 (((𝑥P𝑞 ∈ (2nd𝑥)) ∧ (𝑦P𝑞 ∈ (1st𝑦))) → (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
1110an4s 588 . . . . . . . . 9 (((𝑥P𝑦P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
12 reeanv 2647 . . . . . . . . 9 (∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠) ↔ (∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞 ∧ ∃𝑠 ∈ (1st𝑦)𝑞 <Q 𝑠))
1311, 12sylibr 134 . . . . . . . 8 (((𝑥P𝑦P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → ∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠))
14133adantl3 1155 . . . . . . 7 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → ∃𝑟 ∈ (2nd𝑥)∃𝑠 ∈ (1st𝑦)(𝑟 <Q 𝑞𝑞 <Q 𝑠))
15 ltsonq 7399 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 7366 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 5026 . . . . . . . . . . . 12 ((𝑟 <Q 𝑞𝑞 <Q 𝑠) → 𝑟 <Q 𝑠)
1817adantl 277 . . . . . . . . . . 11 (((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) ∧ (𝑟 <Q 𝑞𝑞 <Q 𝑠)) → 𝑟 <Q 𝑠)
19 prop 7476 . . . . . . . . . . . . . . . 16 (𝑧P → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ P)
20 prloc 7492 . . . . . . . . . . . . . . . 16 ((⟨(1st𝑧), (2nd𝑧)⟩ ∈ P𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
2119, 20sylan 283 . . . . . . . . . . . . . . 15 ((𝑧P𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
22213ad2antl3 1161 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑟 <Q 𝑠) → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)))
2322ex 115 . . . . . . . . . . . . 13 ((𝑥P𝑦P𝑧P) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧))))
2423adantr 276 . . . . . . . . . . . 12 (((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) → (𝑟 <Q 𝑠 → (𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧))))
2524ad2antrr 488 . . . . . . . . . . 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 7483 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝑥), (2nd𝑥)⟩ ∈ P𝑟 ∈ (2nd𝑥)) → 𝑟Q)
284, 27sylan 283 . . . . . . . . . . . . . . . . . . . 20 ((𝑥P𝑟 ∈ (2nd𝑥)) → 𝑟Q)
29 ax-ia3 108 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ (2nd𝑥) → (𝑟 ∈ (1st𝑧) → (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3029adantl 277 . . . . . . . . . . . . . . . . . . . 20 ((𝑥P𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
31 19.8a 1590 . . . . . . . . . . . . . . . . . . . 20 ((𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3228, 30, 31syl6an 1434 . . . . . . . . . . . . . . . . . . 19 ((𝑥P𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))))
33323ad2antl1 1159 . . . . . . . . . . . . . . . . . 18 (((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))))
3433imp 124 . . . . . . . . . . . . . . . . 17 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
35 df-rex 2461 . . . . . . . . . . . . . . . . 17 (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) ↔ ∃𝑟(𝑟Q ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3634, 35sylibr 134 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) ∧ 𝑟 ∈ (1st𝑧)) → ∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)))
37 ltdfpr 7507 . . . . . . . . . . . . . . . . . . 19 ((𝑥P𝑧P) → (𝑥<P 𝑧 ↔ ∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧))))
3837biimprd 158 . . . . . . . . . . . . . . . . . 18 ((𝑥P𝑧P) → (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧))
39383adant2 1016 . . . . . . . . . . . . . . . . 17 ((𝑥P𝑦P𝑧P) → (∃𝑟Q (𝑟 ∈ (2nd𝑥) ∧ 𝑟 ∈ (1st𝑧)) → 𝑥<P 𝑧))
4039ad2antrr 488 . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑟 ∈ (2nd𝑥)) → (𝑟 ∈ (1st𝑧) → 𝑥<P 𝑧))
4342adantrr 479 . . . . . . . . . . . . 13 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → (𝑟 ∈ (1st𝑧) → 𝑥<P 𝑧))
44 elprnql 7482 . . . . . . . . . . . . . . . . . . . . 21 ((⟨(1st𝑦), (2nd𝑦)⟩ ∈ P𝑠 ∈ (1st𝑦)) → 𝑠Q)
457, 44sylan 283 . . . . . . . . . . . . . . . . . . . 20 ((𝑦P𝑠 ∈ (1st𝑦)) → 𝑠Q)
46 pm3.21 264 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (1st𝑦) → (𝑠 ∈ (2nd𝑧) → (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
4746adantl 277 . . . . . . . . . . . . . . . . . . . 20 ((𝑦P𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
48 19.8a 1590 . . . . . . . . . . . . . . . . . . . 20 ((𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
4945, 47, 48syl6an 1434 . . . . . . . . . . . . . . . . . . 19 ((𝑦P𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))))
50493ad2antl2 1160 . . . . . . . . . . . . . . . . . 18 (((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))))
5150imp 124 . . . . . . . . . . . . . . . . 17 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
52 df-rex 2461 . . . . . . . . . . . . . . . . 17 (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) ↔ ∃𝑠(𝑠Q ∧ (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
5351, 52sylibr 134 . . . . . . . . . . . . . . . 16 ((((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) ∧ 𝑠 ∈ (2nd𝑧)) → ∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)))
54 ltdfpr 7507 . . . . . . . . . . . . . . . . . . . 20 ((𝑧P𝑦P) → (𝑧<P 𝑦 ↔ ∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦))))
5554biimprd 158 . . . . . . . . . . . . . . . . . . 19 ((𝑧P𝑦P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
5655ancoms 268 . . . . . . . . . . . . . . . . . 18 ((𝑦P𝑧P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
57563adant1 1015 . . . . . . . . . . . . . . . . 17 ((𝑥P𝑦P𝑧P) → (∃𝑠Q (𝑠 ∈ (2nd𝑧) ∧ 𝑠 ∈ (1st𝑦)) → 𝑧<P 𝑦))
5857ad2antrr 488 . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . 14 (((𝑥P𝑦P𝑧P) ∧ 𝑠 ∈ (1st𝑦)) → (𝑠 ∈ (2nd𝑧) → 𝑧<P 𝑦))
6160adantrl 478 . . . . . . . . . . . . 13 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → (𝑠 ∈ (2nd𝑧) → 𝑧<P 𝑦))
6243, 61orim12d 786 . . . . . . . . . . . 12 (((𝑥P𝑦P𝑧P) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)) → (𝑥<P 𝑧𝑧<P 𝑦)))
6362adantlr 477 . . . . . . . . . . 11 ((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 ∈ (1st𝑧) ∨ 𝑠 ∈ (2nd𝑧)) → (𝑥<P 𝑧𝑧<P 𝑦)))
6463adantr 276 . . . . . . . . . 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 115 . . . . . . . 8 ((((𝑥P𝑦P𝑧P) ∧ (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))) ∧ (𝑟 ∈ (2nd𝑥) ∧ 𝑠 ∈ (1st𝑦))) → ((𝑟 <Q 𝑞𝑞 <Q 𝑠) → (𝑥<P 𝑧𝑧<P 𝑦)))
6766rexlimdvva 2602 . . . . . . 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 115 . . . . 5 ((𝑥P𝑦P𝑧P) → ((𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) → (𝑥<P 𝑧𝑧<P 𝑦)))
7069rexlimdvw 2598 . . . 4 ((𝑥P𝑦P𝑧P) → (∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)) → (𝑥<P 𝑧𝑧<P 𝑦)))
713, 70sylbid 150 . . 3 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦)))
7271rgen3 2564 . 2 𝑥P𝑦P𝑧P (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦))
73 df-iso 4299 . 2 (<P Or P ↔ (<P Po P ∧ ∀𝑥P𝑦P𝑧P (𝑥<P 𝑦 → (𝑥<P 𝑧𝑧<P 𝑦))))
741, 72, 73mpbir2an 942 1 <P Or P
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3a 978  wex 1492  wcel 2148  wral 2455  wrex 2456  cop 3597   class class class wbr 4005   Po wpo 4296   Or wor 4297  cfv 5218  1st c1st 6141  2nd c2nd 6142  Qcnq 7281   <Q cltq 7286  Pcnp 7292  <P cltp 7296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-enq 7348  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467  df-iltp 7471
This theorem is referenced by:  prplnqu  7621  addextpr  7622  caucvgprprlemk  7684  caucvgprprlemnkltj  7690  caucvgprprlemnkeqj  7691  caucvgprprlemnjltk  7692  caucvgprprlemnbj  7694  caucvgprprlemml  7695  caucvgprprlemlol  7699  caucvgprprlemupu  7701  caucvgprprlemloc  7704  caucvgprprlemaddq  7709  suplocexprlemmu  7719  lttrsr  7763  ltposr  7764  ltsosr  7765  archsr  7783
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