ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-plpq Structured version   GIF version

Definition df-plpq 6321
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 6326) works with the equivalence classes of these ordered pairs determined by the equivalence relation ~Q (df-enq 6324). (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 6253 . 2 class +pQ
2 vx . . 3 setvar x
3 vy . . 3 setvar y
4 cnpi 6249 . . . 4 class N
54, 4cxp 4285 . . 3 class (N × N)
62cv 1241 . . . . . . 7 class x
7 c1st 5704 . . . . . . 7 class 1st
86, 7cfv 4844 . . . . . 6 class (1stx)
93cv 1241 . . . . . . 7 class y
10 c2nd 5705 . . . . . . 7 class 2nd
119, 10cfv 4844 . . . . . 6 class (2ndy)
12 cmi 6251 . . . . . 6 class ·N
138, 11, 12co 5452 . . . . 5 class ((1stx) ·N (2ndy))
149, 7cfv 4844 . . . . . 6 class (1sty)
156, 10cfv 4844 . . . . . 6 class (2ndx)
1614, 15, 12co 5452 . . . . 5 class ((1sty) ·N (2ndx))
17 cpli 6250 . . . . 5 class +N
1813, 16, 17co 5452 . . . 4 class (((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx)))
1915, 11, 12co 5452 . . . 4 class ((2ndx) ·N (2ndy))
2018, 19cop 3369 . . 3 class ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩
212, 3, 5, 5, 20cmpt2 5454 . 2 class (x (N × N), y (N × N) ↦ ⟨(((1stx) ·N (2ndy)) +N ((1sty) ·N (2ndx))), ((2ndx) ·N (2ndy))⟩)
221, 21wceq 1242 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  6331
  Copyright terms: Public domain W3C validator