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Theorem ltsopr 7594
Description: Positive real 'less than' is a weak linear order (in the sense of df-iso 4297). 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 7593 . 2 <P Po P
2 ltdfpr 7504 . . . . 5 ((𝑥P𝑦P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
323adant3 1017 . . . 4 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦))))
4 prop 7473 . . . . . . . . . . . 12 (𝑥P → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ P)
5 prnminu 7487 . . . . . . . . . . . 12 ((⟨(1st𝑥), (2nd𝑥)⟩ ∈ P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
64, 5sylan 283 . . . . . . . . . . 11 ((𝑥P𝑞 ∈ (2nd𝑥)) → ∃𝑟 ∈ (2nd𝑥)𝑟 <Q 𝑞)
7 prop 7473 . . . . . . . . . . . 12 (𝑦P → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ P)
8 prnmaxl 7486 . . . . . . . . . . . 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 2646 . . . . . . . . 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 7396 . . . . . . . . . . . . 13 <Q Or Q
16 ltrelnq 7363 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
1715, 16sotri 5024 . . . . . . . . . . . 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 7473 . . . . . . . . . . . . . . . 16 (𝑧P → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ P)
20 prloc 7489 . . . . . . . . . . . . . . . 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 7480 . . . . . . . . . . . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . . . . . 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 7479 . . . . . . . . . . . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . . . . . . 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 4297 . 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 3595   class class class wbr 4003   Po wpo 4294   Or wor 4295  cfv 5216  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   <Q cltq 7283  Pcnp 7289  <P cltp 7293
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-enq 7345  df-nqqs 7346  df-ltnqqs 7351  df-inp 7464  df-iltp 7468
This theorem is referenced by:  prplnqu  7618  addextpr  7619  caucvgprprlemk  7681  caucvgprprlemnkltj  7687  caucvgprprlemnkeqj  7688  caucvgprprlemnjltk  7689  caucvgprprlemnbj  7691  caucvgprprlemml  7692  caucvgprprlemlol  7696  caucvgprprlemupu  7698  caucvgprprlemloc  7701  caucvgprprlemaddq  7706  suplocexprlemmu  7716  lttrsr  7760  ltposr  7761  ltsosr  7762  archsr  7780
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