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 Description: Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-add + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 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 9977 . 2 class +
2 vx . . . . . . 7 setvar 𝑥
32cv 1522 . . . . . 6 class 𝑥
4 cc 9972 . . . . . 6 class
53, 4wcel 2030 . . . . 5 wff 𝑥 ∈ ℂ
6 vy . . . . . . 7 setvar 𝑦
76cv 1522 . . . . . 6 class 𝑦
87, 4wcel 2030 . . . . 5 wff 𝑦 ∈ ℂ
95, 8wa 383 . . . 4 wff (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)
10 vw . . . . . . . . . . . . 13 setvar 𝑤
1110cv 1522 . . . . . . . . . . . 12 class 𝑤
12 vv . . . . . . . . . . . . 13 setvar 𝑣
1312cv 1522 . . . . . . . . . . . 12 class 𝑣
1411, 13cop 4216 . . . . . . . . . . 11 class 𝑤, 𝑣
153, 14wceq 1523 . . . . . . . . . 10 wff 𝑥 = ⟨𝑤, 𝑣
16 vu . . . . . . . . . . . . 13 setvar 𝑢
1716cv 1522 . . . . . . . . . . . 12 class 𝑢
18 vf . . . . . . . . . . . . 13 setvar 𝑓
1918cv 1522 . . . . . . . . . . . 12 class 𝑓
2017, 19cop 4216 . . . . . . . . . . 11 class 𝑢, 𝑓
217, 20wceq 1523 . . . . . . . . . 10 wff 𝑦 = ⟨𝑢, 𝑓
2215, 21wa 383 . . . . . . . . 9 wff (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)
23 vz . . . . . . . . . . 11 setvar 𝑧
2423cv 1522 . . . . . . . . . 10 class 𝑧
25 cplr 9729 . . . . . . . . . . . 12 class +R
2611, 17, 25co 6690 . . . . . . . . . . 11 class (𝑤 +R 𝑢)
2713, 19, 25co 6690 . . . . . . . . . . 11 class (𝑣 +R 𝑓)
2826, 27cop 4216 . . . . . . . . . 10 class ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
2924, 28wceq 1523 . . . . . . . . 9 wff 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
3022, 29wa 383 . . . . . . . 8 wff ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3130, 18wex 1744 . . . . . . 7 wff 𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3231, 16wex 1744 . . . . . 6 wff 𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3332, 12wex 1744 . . . . 5 wff 𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
3433, 10wex 1744 . . . 4 wff 𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
359, 34wa 383 . . 3 wff ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
3635, 2, 6, 23coprab 6691 . 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
371, 36wceq 1523 1 wff + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
 Colors of variables: wff setvar class This definition is referenced by:  addcnsr  9994  axaddf  10004
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