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Definition df-plq0 6282
Description: Define addition on non-negative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)
Assertion
Ref Expression
df-plq0 +Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}
Distinct variable group:   x,y,z,w,v,u,f

Detailed syntax breakdown of Definition df-plq0
StepHypRef Expression
1 cplq0 6147 . 2 class +Q0
2 vx . . . . . . 7 setvar x
32cv 1227 . . . . . 6 class x
4 cnq0 6145 . . . . . 6 class Q0
53, 4wcel 1374 . . . . 5 wff x Q0
6 vy . . . . . . 7 setvar y
76cv 1227 . . . . . 6 class y
87, 4wcel 1374 . . . . 5 wff y Q0
95, 8wa 97 . . . 4 wff (x Q0 y Q0)
10 vw . . . . . . . . . . . . . 14 setvar w
1110cv 1227 . . . . . . . . . . . . 13 class w
12 vv . . . . . . . . . . . . . 14 setvar v
1312cv 1227 . . . . . . . . . . . . 13 class v
1411, 13cop 3353 . . . . . . . . . . . 12 class w, v
15 ceq0 6144 . . . . . . . . . . . 12 class ~Q0
1614, 15cec 6015 . . . . . . . . . . 11 class [⟨w, v⟩] ~Q0
173, 16wceq 1228 . . . . . . . . . 10 wff x = [⟨w, v⟩] ~Q0
18 vu . . . . . . . . . . . . . 14 setvar u
1918cv 1227 . . . . . . . . . . . . 13 class u
20 vf . . . . . . . . . . . . . 14 setvar f
2120cv 1227 . . . . . . . . . . . . 13 class f
2219, 21cop 3353 . . . . . . . . . . . 12 class u, f
2322, 15cec 6015 . . . . . . . . . . 11 class [⟨u, f⟩] ~Q0
247, 23wceq 1228 . . . . . . . . . 10 wff y = [⟨u, f⟩] ~Q0
2517, 24wa 97 . . . . . . . . 9 wff (x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 )
26 vz . . . . . . . . . . 11 setvar z
2726cv 1227 . . . . . . . . . 10 class z
28 comu 5914 . . . . . . . . . . . . . 14 class ·𝑜
2911, 21, 28co 5436 . . . . . . . . . . . . 13 class (w ·𝑜 f)
3013, 19, 28co 5436 . . . . . . . . . . . . 13 class (v ·𝑜 u)
31 coa 5913 . . . . . . . . . . . . 13 class +𝑜
3229, 30, 31co 5436 . . . . . . . . . . . 12 class ((w ·𝑜 f) +𝑜 (v ·𝑜 u))
3313, 21, 28co 5436 . . . . . . . . . . . 12 class (v ·𝑜 f)
3432, 33cop 3353 . . . . . . . . . . 11 class ⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩
3534, 15cec 6015 . . . . . . . . . 10 class [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0
3627, 35wceq 1228 . . . . . . . . 9 wff z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0
3725, 36wa 97 . . . . . . . 8 wff ((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 )
3837, 20wex 1362 . . . . . . 7 wff f((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 )
3938, 18wex 1362 . . . . . 6 wff uf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 )
4039, 12wex 1362 . . . . 5 wff vuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 )
4140, 10wex 1362 . . . 4 wff wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 )
429, 41wa 97 . . 3 wff ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))
4342, 2, 6, 26coprab 5437 . 2 class {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}
441, 43wceq 1228 1 wff +Q0 = {⟨⟨x, y⟩, z⟩ ∣ ((x Q0 y Q0) wvuf((x = [⟨w, v⟩] ~Q0 y = [⟨u, f⟩] ~Q0 ) z = [⟨((w ·𝑜 f) +𝑜 (v ·𝑜 u)), (v ·𝑜 f)⟩] ~Q0 ))}
Colors of variables: wff set class
This definition is referenced by:  dfplq0qs  6285
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