Definition df-plpq 10899

Description: Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 11112, and is intended to be used only by the construction. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +_Q (df-plq 10905) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~_Q (df-enq 10902). (Analogous remarks apply to the other "pre-" operations in the complex number construction that follows.) From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 28-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
df-plpq	⊢ +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-plpq

Step	Hyp	Ref	Expression
1		cplpq 10839	. 2 class +pQ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnpi 10835	. . . 4 class N
5	4, 4	cxp 5673	. . 3 class (N × N)
6	2	cv 1541	. . . . . . 7 class 𝑥
7		c1st 7968	. . . . . . 7 class 1st
8	6, 7	cfv 6540	. . . . . 6 class (1st ‘𝑥)
9	3	cv 1541	. . . . . . 7 class 𝑦
10		c2nd 7969	. . . . . . 7 class 2nd
11	9, 10	cfv 6540	. . . . . 6 class (2nd ‘𝑦)
12		cmi 10837	. . . . . 6 class ·N
13	8, 11, 12	co 7404	. . . . 5 class ((1st ‘𝑥) ·N (2nd ‘𝑦))
14	9, 7	cfv 6540	. . . . . 6 class (1st ‘𝑦)
15	6, 10	cfv 6540	. . . . . 6 class (2nd ‘𝑥)
16	14, 15, 12	co 7404	. . . . 5 class ((1st ‘𝑦) ·N (2nd ‘𝑥))
17		cpli 10836	. . . . 5 class +N
18	13, 16, 17	co 7404	. . . 4 class (((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥)))
19	15, 11, 12	co 7404	. . . 4 class ((2nd ‘𝑥) ·N (2nd ‘𝑦))
20	18, 19	cop 4633	. . 3 class ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩
21	2, 3, 5, 5, 20	cmpo 7406	. 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
22	1, 21	wceq 1542	1 wff +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st ‘𝑥) ·N (2nd ‘𝑦)) +N ((1st ‘𝑦) ·N (2nd ‘𝑥))), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)

This definition is referenced by:  addpipq2  10927  addpqnq  10929  addpqf  10935