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Definition df-plpq 6440
Description: Define pre-addition 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. This "pre-addition" operation works directly with ordered pairs of integers. The actual positive fraction addition +Q (df-plqqs 6445) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6443). (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.)
Assertion
Ref Expression
df-plpq +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-plpq
StepHypRef Expression
1 cplpq 6372 . 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 . . . . . . 7 class 𝑥
7 c1st 5765 . . . . . . 7 class 1st
86, 7cfv 4902 . . . . . 6 class (1st𝑥)
93cv 1242 . . . . . . 7 class 𝑦
10 c2nd 5766 . . . . . . 7 class 2nd
119, 10cfv 4902 . . . . . 6 class (2nd𝑦)
12 cmi 6370 . . . . . 6 class ·N
138, 11, 12co 5512 . . . . 5 class ((1st𝑥) ·N (2nd𝑦))
149, 7cfv 4902 . . . . . 6 class (1st𝑦)
156, 10cfv 4902 . . . . . 6 class (2nd𝑥)
1614, 15, 12co 5512 . . . . 5 class ((1st𝑦) ·N (2nd𝑥))
17 cpli 6369 . . . . 5 class +N
1813, 16, 17co 5512 . . . 4 class (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥)))
1915, 11, 12co 5512 . . . 4 class ((2nd𝑥) ·N (2nd𝑦))
2018, 19cop 3378 . . 3 class ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩
212, 3, 5, 5, 20cmpt2 5514 . 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
221, 21wceq 1243 1 wff +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
Colors of variables: wff set class
This definition is referenced by:  dfplpq2  6450
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