Definition df-mpq 7412

Description: Define pre-multiplication 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. From Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 28-Aug-1995.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-mpq

Step	Hyp	Ref	Expression
1		cmpq 7344	. 2 class ·pQ
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cnpi 7339	. . . 4 class N
5	4, 4	cxp 4661	. . 3 class (N × N)
6	2	cv 1363	. . . . . 6 class 𝑥
7		c1st 6196	. . . . . 6 class 1st
8	6, 7	cfv 5258	. . . . 5 class (1st ‘𝑥)
9	3	cv 1363	. . . . . 6 class 𝑦
10	9, 7	cfv 5258	. . . . 5 class (1st ‘𝑦)
11		cmi 7341	. . . . 5 class ·N
12	8, 10, 11	co 5922	. . . 4 class ((1st ‘𝑥) ·N (1st ‘𝑦))
13		c2nd 6197	. . . . . 6 class 2nd
14	6, 13	cfv 5258	. . . . 5 class (2nd ‘𝑥)
15	9, 13	cfv 5258	. . . . 5 class (2nd ‘𝑦)
16	14, 15, 11	co 5922	. . . 4 class ((2nd ‘𝑥) ·N (2nd ‘𝑦))
17	12, 16	cop 3625	. . 3 class ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩
18	2, 3, 5, 5, 17	cmpo 5924	. 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)
19	1, 18	wceq 1364	1 wff ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st ‘𝑥) ·N (1st ‘𝑦)), ((2nd ‘𝑥) ·N (2nd ‘𝑦))⟩)

This definition is referenced by:  dfmpq2  7422  mulpipq2  7438