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Definition df-add 6722
Description: Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
Assertion
Ref Expression
df-add + = {⟨⟨x, y⟩, z⟩ ∣ ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩))}
Distinct variable group:   x,y,z,w,v,u,f

Detailed syntax breakdown of Definition df-add
StepHypRef Expression
1 caddc 6714 . 2 class +
2 vx . . . . . . 7 setvar x
32cv 1241 . . . . . 6 class x
4 cc 6709 . . . . . 6 class
53, 4wcel 1390 . . . . 5 wff x
6 vy . . . . . . 7 setvar y
76cv 1241 . . . . . 6 class y
87, 4wcel 1390 . . . . 5 wff y
95, 8wa 97 . . . 4 wff (x y ℂ)
10 vw . . . . . . . . . . . . 13 setvar w
1110cv 1241 . . . . . . . . . . . 12 class w
12 vv . . . . . . . . . . . . 13 setvar v
1312cv 1241 . . . . . . . . . . . 12 class v
1411, 13cop 3370 . . . . . . . . . . 11 class w, v
153, 14wceq 1242 . . . . . . . . . 10 wff x = ⟨w, v
16 vu . . . . . . . . . . . . 13 setvar u
1716cv 1241 . . . . . . . . . . . 12 class u
18 vf . . . . . . . . . . . . 13 setvar f
1918cv 1241 . . . . . . . . . . . 12 class f
2017, 19cop 3370 . . . . . . . . . . 11 class u, f
217, 20wceq 1242 . . . . . . . . . 10 wff y = ⟨u, f
2215, 21wa 97 . . . . . . . . 9 wff (x = ⟨w, v y = ⟨u, f⟩)
23 vz . . . . . . . . . . 11 setvar z
2423cv 1241 . . . . . . . . . 10 class z
25 cplr 6285 . . . . . . . . . . . 12 class +R
2611, 17, 25co 5455 . . . . . . . . . . 11 class (w +R u)
2713, 19, 25co 5455 . . . . . . . . . . 11 class (v +R f)
2826, 27cop 3370 . . . . . . . . . 10 class ⟨(w +R u), (v +R f)⟩
2924, 28wceq 1242 . . . . . . . . 9 wff z = ⟨(w +R u), (v +R f)⟩
3022, 29wa 97 . . . . . . . 8 wff ((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩)
3130, 18wex 1378 . . . . . . 7 wff f((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩)
3231, 16wex 1378 . . . . . 6 wff uf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩)
3332, 12wex 1378 . . . . 5 wff vuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩)
3433, 10wex 1378 . . . 4 wff wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩)
359, 34wa 97 . . 3 wff ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩))
3635, 2, 6, 23coprab 5456 . 2 class {⟨⟨x, y⟩, z⟩ ∣ ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩))}
371, 36wceq 1242 1 wff + = {⟨⟨x, y⟩, z⟩ ∣ ((x y ℂ) wvuf((x = ⟨w, v y = ⟨u, f⟩) z = ⟨(w +R u), (v +R f)⟩))}
Colors of variables: wff set class
This definition is referenced by:  addcnsr  6731
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