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Definition df-ltpq 10027
Description: Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10237, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-ltpq <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ltpq
StepHypRef Expression
1 cltpq 9967 . 2 class <pQ
2 vx . . . . . . 7 setvar 𝑥
32cv 1636 . . . . . 6 class 𝑥
4 cnpi 9961 . . . . . . 7 class N
54, 4cxp 5322 . . . . . 6 class (N × N)
63, 5wcel 2157 . . . . 5 wff 𝑥 ∈ (N × N)
7 vy . . . . . . 7 setvar 𝑦
87cv 1636 . . . . . 6 class 𝑦
98, 5wcel 2157 . . . . 5 wff 𝑦 ∈ (N × N)
106, 9wa 384 . . . 4 wff (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))
11 c1st 7406 . . . . . . 7 class 1st
123, 11cfv 6111 . . . . . 6 class (1st𝑥)
13 c2nd 7407 . . . . . . 7 class 2nd
148, 13cfv 6111 . . . . . 6 class (2nd𝑦)
15 cmi 9963 . . . . . 6 class ·N
1612, 14, 15co 6884 . . . . 5 class ((1st𝑥) ·N (2nd𝑦))
178, 11cfv 6111 . . . . . 6 class (1st𝑦)
183, 13cfv 6111 . . . . . 6 class (2nd𝑥)
1917, 18, 15co 6884 . . . . 5 class ((1st𝑦) ·N (2nd𝑥))
20 clti 9964 . . . . 5 class <N
2116, 19, 20wbr 4855 . . . 4 wff ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))
2210, 21wa 384 . . 3 wff ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))
2322, 2, 7copab 4917 . 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
241, 23wceq 1637 1 wff <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
Colors of variables: wff setvar class
This definition is referenced by:  ordpipq  10059  lterpq  10087
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