Definition df-ltpq 10326

Description: Define pre-ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10537, 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)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-ltpq

Step	Hyp	Ref	Expression
1		cltpq 10266	. 2 class <pQ
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1532	. . . . . 6 class 𝑥
4		cnpi 10260	. . . . . . 7 class N
5	4, 4	cxp 5547	. . . . . 6 class (N × N)
6	3, 5	wcel 2110	. . . . 5 wff 𝑥 ∈ (N × N)
7		vy	. . . . . . 7 setvar 𝑦
8	7	cv 1532	. . . . . 6 class 𝑦
9	8, 5	wcel 2110	. . . . 5 wff 𝑦 ∈ (N × N)
10	6, 9	wa 398	. . . 4 wff (𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N))
11		c1st 7681	. . . . . . 7 class 1st
12	3, 11	cfv 6349	. . . . . 6 class (1st ‘𝑥)
13		c2nd 7682	. . . . . . 7 class 2nd
14	8, 13	cfv 6349	. . . . . 6 class (2nd ‘𝑦)
15		cmi 10262	. . . . . 6 class ·N
16	12, 14, 15	co 7150	. . . . 5 class ((1st ‘𝑥) ·N (2nd ‘𝑦))
17	8, 11	cfv 6349	. . . . . 6 class (1st ‘𝑦)
18	3, 13	cfv 6349	. . . . . 6 class (2nd ‘𝑥)
19	17, 18, 15	co 7150	. . . . 5 class ((1st ‘𝑦) ·N (2nd ‘𝑥))
20		clti 10263	. . . . 5 class <N
21	16, 19, 20	wbr 5058	. . . 4 wff ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥))
22	10, 21	wa 398	. . 3 wff ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
23	22, 2, 7	copab 5120	. 2 class {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}
24	1, 23	wceq 1533	1 wff <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st ‘𝑥) ·N (2nd ‘𝑦)) <N ((1st ‘𝑦) ·N (2nd ‘𝑥)))}

This definition is referenced by:  ordpipq  10358  lterpq  10386