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

Definition df-mpq 10324
 Description: Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 10536, 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.) (New usage is discouraged.)
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 10264 . 2 class ·pQ
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cnpi 10259 . . . 4 class N
54, 4cxp 5521 . . 3 class (N × N)
62cv 1537 . . . . . 6 class 𝑥
7 c1st 7673 . . . . . 6 class 1st
86, 7cfv 6328 . . . . 5 class (1st𝑥)
93cv 1537 . . . . . 6 class 𝑦
109, 7cfv 6328 . . . . 5 class (1st𝑦)
11 cmi 10261 . . . . 5 class ·N
128, 10, 11co 7139 . . . 4 class ((1st𝑥) ·N (1st𝑦))
13 c2nd 7674 . . . . . 6 class 2nd
146, 13cfv 6328 . . . . 5 class (2nd𝑥)
159, 13cfv 6328 . . . . 5 class (2nd𝑦)
1614, 15, 11co 7139 . . . 4 class ((2nd𝑥) ·N (2nd𝑦))
1712, 16cop 4534 . . 3 class ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩
182, 3, 5, 5, 17cmpo 7141 . 2 class (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
191, 18wceq 1538 1 wff ·pQ = (𝑥 ∈ (N × N), 𝑦 ∈ (N × N) ↦ ⟨((1st𝑥) ·N (1st𝑦)), ((2nd𝑥) ·N (2nd𝑦))⟩)
 Colors of variables: wff setvar class This definition is referenced by:  mulpipq2  10354  mulpqnq  10356  mulpqf  10361
 Copyright terms: Public domain W3C validator