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

Definition df-plqqs 6333
Description: Define 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. From Proposition 9-2.3 of [Gleason] p. 117. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
df-plqqs +Q = {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q ))}
Distinct variable group:   x,y,z,w,v,u,f

Detailed syntax breakdown of Definition df-plqqs
StepHypRef Expression
1 cplq 6266 . 2 class +Q
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cnq 6264 . . . . . 6 class Q
53, 4wcel 1390 . . . . 5 wff x Q
6 vy . . . . . . 7 setvar y
76cv 1241 . . . . . 6 class y
87, 4wcel 1390 . . . . 5 wff y Q
95, 8wa 97 . . . 4 wff (x Q y Q)
10 vw . . . . . . . . . . . . . 14 setvar w
1110cv 1241 . . . . . . . . . . . . 13 class w
12 vv . . . . . . . . . . . . . 14 setvar v
1312cv 1241 . . . . . . . . . . . . 13 class v
1411, 13cop 3370 . . . . . . . . . . . 12 class w, v
15 ceq 6263 . . . . . . . . . . . 12 class ~Q
1614, 15cec 6040 . . . . . . . . . . 11 class [⟨w, v⟩] ~Q
173, 16wceq 1242 . . . . . . . . . 10 wff x = [⟨w, v⟩] ~Q
18 vu . . . . . . . . . . . . . 14 setvar u
1918cv 1241 . . . . . . . . . . . . 13 class u
20 vf . . . . . . . . . . . . . 14 setvar f
2120cv 1241 . . . . . . . . . . . . 13 class f
2219, 21cop 3370 . . . . . . . . . . . 12 class u, f
2322, 15cec 6040 . . . . . . . . . . 11 class [⟨u, f⟩] ~Q
247, 23wceq 1242 . . . . . . . . . 10 wff y = [⟨u, f⟩] ~Q
2517, 24wa 97 . . . . . . . . 9 wff (x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q )
26 vz . . . . . . . . . . 11 setvar z
2726cv 1241 . . . . . . . . . 10 class z
28 cplpq 6260 . . . . . . . . . . . 12 class +pQ
2914, 22, 28co 5455 . . . . . . . . . . 11 class (⟨w, v⟩ +pQu, f⟩)
3029, 15cec 6040 . . . . . . . . . 10 class [(⟨w, v⟩ +pQu, f⟩)] ~Q
3127, 30wceq 1242 . . . . . . . . 9 wff z = [(⟨w, v⟩ +pQu, f⟩)] ~Q
3225, 31wa 97 . . . . . . . 8 wff ((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q )
3332, 20wex 1378 . . . . . . 7 wff f((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q )
3433, 18wex 1378 . . . . . 6 wff uf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q )
3534, 12wex 1378 . . . . 5 wff vuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q )
3635, 10wex 1378 . . . 4 wff wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q )
379, 36wa 97 . . 3 wff ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q ))
3837, 2, 6, 26coprab 5456 . 2 class {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q ))}
391, 38wceq 1242 1 wff +Q = {⟨⟨x, y⟩, z⟩ ∣ ((x Q y Q) wvuf((x = [⟨w, v⟩] ~Q y = [⟨u, f⟩] ~Q ) z = [(⟨w, v⟩ +pQu, f⟩)] ~Q ))}
Colors of variables: wff set class
This definition is referenced by:  addpipqqs  6354  dmaddpq  6363
  Copyright terms: Public domain W3C validator