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Definition df-ltnqqs 6449
Description: Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 13-Feb-1996.)
Assertion
Ref Expression
df-ltnqqs <Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Detailed syntax breakdown of Definition df-ltnqqs
StepHypRef Expression
1 cltq 6381 . 2 class <Q
2 vx . . . . . . 7 setvar 𝑥
32cv 1242 . . . . . 6 class 𝑥
4 cnq 6376 . . . . . 6 class Q
53, 4wcel 1393 . . . . 5 wff 𝑥Q
6 vy . . . . . . 7 setvar 𝑦
76cv 1242 . . . . . 6 class 𝑦
87, 4wcel 1393 . . . . 5 wff 𝑦Q
95, 8wa 97 . . . 4 wff (𝑥Q𝑦Q)
10 vz . . . . . . . . . . . . . 14 setvar 𝑧
1110cv 1242 . . . . . . . . . . . . 13 class 𝑧
12 vw . . . . . . . . . . . . . 14 setvar 𝑤
1312cv 1242 . . . . . . . . . . . . 13 class 𝑤
1411, 13cop 3378 . . . . . . . . . . . 12 class 𝑧, 𝑤
15 ceq 6375 . . . . . . . . . . . 12 class ~Q
1614, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑧, 𝑤⟩] ~Q
173, 16wceq 1243 . . . . . . . . . 10 wff 𝑥 = [⟨𝑧, 𝑤⟩] ~Q
18 vv . . . . . . . . . . . . . 14 setvar 𝑣
1918cv 1242 . . . . . . . . . . . . 13 class 𝑣
20 vu . . . . . . . . . . . . . 14 setvar 𝑢
2120cv 1242 . . . . . . . . . . . . 13 class 𝑢
2219, 21cop 3378 . . . . . . . . . . . 12 class 𝑣, 𝑢
2322, 15cec 6104 . . . . . . . . . . 11 class [⟨𝑣, 𝑢⟩] ~Q
247, 23wceq 1243 . . . . . . . . . 10 wff 𝑦 = [⟨𝑣, 𝑢⟩] ~Q
2517, 24wa 97 . . . . . . . . 9 wff (𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q )
26 cmi 6370 . . . . . . . . . . 11 class ·N
2711, 21, 26co 5512 . . . . . . . . . 10 class (𝑧 ·N 𝑢)
2813, 19, 26co 5512 . . . . . . . . . 10 class (𝑤 ·N 𝑣)
29 clti 6371 . . . . . . . . . 10 class <N
3027, 28, 29wbr 3764 . . . . . . . . 9 wff (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)
3125, 30wa 97 . . . . . . . 8 wff ((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣))
3231, 20wex 1381 . . . . . . 7 wff 𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣))
3332, 18wex 1381 . . . . . 6 wff 𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣))
3433, 12wex 1381 . . . . 5 wff 𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣))
3534, 10wex 1381 . . . 4 wff 𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣))
369, 35wa 97 . . 3 wff ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))
3736, 2, 6copab 3817 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}
381, 37wceq 1243 1 wff <Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥Q𝑦Q) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~Q𝑦 = [⟨𝑣, 𝑢⟩] ~Q ) ∧ (𝑧 ·N 𝑢) <N (𝑤 ·N 𝑣)))}
Colors of variables: wff set class
This definition is referenced by:  ltrelnq  6461  ordpipqqs  6470
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