Definition df-add 10866

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) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((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

Step	Hyp	Ref	Expression
1		caddc 10858	. 2 class +
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1540	. . . . . 6 class 𝑥
4		cc 10853	. . . . . 6 class ℂ
5	3, 4	wcel 2109	. . . . 5 wff 𝑥 ∈ ℂ
6		vy	. . . . . . 7 setvar 𝑦
7	6	cv 1540	. . . . . 6 class 𝑦
8	7, 4	wcel 2109	. . . . 5 wff 𝑦 ∈ ℂ
9	5, 8	wa 395	. . . 4 wff (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)
10		vw	. . . . . . . . . . . . 13 setvar 𝑤
11	10	cv 1540	. . . . . . . . . . . 12 class 𝑤
12		vv	. . . . . . . . . . . . 13 setvar 𝑣
13	12	cv 1540	. . . . . . . . . . . 12 class 𝑣
14	11, 13	cop 4572	. . . . . . . . . . 11 class ⟨𝑤, 𝑣⟩
15	3, 14	wceq 1541	. . . . . . . . . 10 wff 𝑥 = ⟨𝑤, 𝑣⟩
16		vu	. . . . . . . . . . . . 13 setvar 𝑢
17	16	cv 1540	. . . . . . . . . . . 12 class 𝑢
18		vf	. . . . . . . . . . . . 13 setvar 𝑓
19	18	cv 1540	. . . . . . . . . . . 12 class 𝑓
20	17, 19	cop 4572	. . . . . . . . . . 11 class ⟨𝑢, 𝑓⟩
21	7, 20	wceq 1541	. . . . . . . . . 10 wff 𝑦 = ⟨𝑢, 𝑓⟩
22	15, 21	wa 395	. . . . . . . . 9 wff (𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩)
23		vz	. . . . . . . . . . 11 setvar 𝑧
24	23	cv 1540	. . . . . . . . . 10 class 𝑧
25		cplr 10609	. . . . . . . . . . . 12 class +R
26	11, 17, 25	co 7268	. . . . . . . . . . 11 class (𝑤 +R 𝑢)
27	13, 19, 25	co 7268	. . . . . . . . . . 11 class (𝑣 +R 𝑓)
28	26, 27	cop 4572	. . . . . . . . . 10 class ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
29	24, 28	wceq 1541	. . . . . . . . 9 wff 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩
30	22, 29	wa 395	. . . . . . . 8 wff ((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
31	30, 18	wex 1785	. . . . . . 7 wff ∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
32	31, 16	wex 1785	. . . . . 6 wff ∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
33	32, 12	wex 1785	. . . . 5 wff ∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
34	33, 10	wex 1785	. . . 4 wff ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩)
35	9, 34	wa 395	. . 3 wff ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))
36	35, 2, 6, 23	coprab 7269	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}
37	1, 36	wceq 1541	1 wff + = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}

This definition is referenced by:  addcnsr  10875  axaddf  10885