 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-plpq Structured version   Visualization version   GIF version

Definition df-plpq 10176
 Description: Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10389, 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 10182) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 10179). (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) ↦ ⟨(((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 10116 . 2 class +pQ
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cnpi 10112 . . . 4 class N
54, 4cxp 5441 . . 3 class (N × N)
62cv 1521 . . . . . . 7 class 𝑥
7 c1st 7543 . . . . . . 7 class 1st
86, 7cfv 6225 . . . . . 6 class (1st𝑥)
93cv 1521 . . . . . . 7 class 𝑦
10 c2nd 7544 . . . . . . 7 class 2nd
119, 10cfv 6225 . . . . . 6 class (2nd𝑦)
12 cmi 10114 . . . . . 6 class ·N
138, 11, 12co 7016 . . . . 5 class ((1st𝑥) ·N (2nd𝑦))
149, 7cfv 6225 . . . . . 6 class (1st𝑦)
156, 10cfv 6225 . . . . . 6 class (2nd𝑥)
1614, 15, 12co 7016 . . . . 5 class ((1st𝑦) ·N (2nd𝑥))
17 cpli 10113 . . . . 5 class +N
1813, 16, 17co 7016 . . . 4 class (((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥)))
1915, 11, 12co 7016 . . . 4 class ((2nd𝑥) ·N (2nd𝑦))
2018, 19cop 4478 . . 3 class ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩
212, 3, 5, 5, 20cmpo 7018 . 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
221, 21wceq 1522 1 wff +pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨(((1st𝑥) ·N (2nd𝑦)) +N ((1st𝑦) ·N (2nd𝑥))), ((2nd𝑥) ·N (2nd𝑦))⟩)
 Colors of variables: wff setvar class This definition is referenced by:  addpipq2  10204  addpqnq  10206  addpqf  10212
 Copyright terms: Public domain W3C validator