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Definition df-mpq 6441
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) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-mpq
StepHypRef Expression
1 cmpq 6373 . 2 class ·pQ
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cnpi 6368 . . . 4 class N
54, 4cxp 4343 . . 3 class (N × N)
62cv 1242 . . . . . 6 class 𝑥
7 c1st 5765 . . . . . 6 class 1st
86, 7cfv 4902 . . . . 5 class (1st𝑥)
93cv 1242 . . . . . 6 class 𝑦
109, 7cfv 4902 . . . . 5 class (1st𝑦)
11 cmi 6370 . . . . 5 class ·N
128, 10, 11co 5512 . . . 4 class ((1st𝑥) ·N (1st𝑦))
13 c2nd 5766 . . . . . 6 class 2nd
146, 13cfv 4902 . . . . 5 class (2nd𝑥)
159, 13cfv 4902 . . . . 5 class (2nd𝑦)
1614, 15, 11co 5512 . . . 4 class ((2nd𝑥) ·N (2nd𝑦))
1712, 16cop 3378 . . 3 class ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩
182, 3, 5, 5, 17cmpt2 5514 . 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
191, 18wceq 1243 1 wff ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
Colors of variables: wff set class
This definition is referenced by:  dfmpq2  6451  mulpipq2  6467
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