Detailed syntax breakdown of Definition df-add
Step | Hyp | Ref
| Expression |
1 | | caddc 10872 |
. 2
class
+ |
2 | | vx |
. . . . . . 7
setvar 𝑥 |
3 | 2 | cv 1538 |
. . . . . 6
class 𝑥 |
4 | | cc 10867 |
. . . . . 6
class
ℂ |
5 | 3, 4 | wcel 2106 |
. . . . 5
wff 𝑥 ∈ ℂ |
6 | | vy |
. . . . . . 7
setvar 𝑦 |
7 | 6 | cv 1538 |
. . . . . 6
class 𝑦 |
8 | 7, 4 | wcel 2106 |
. . . . 5
wff 𝑦 ∈ ℂ |
9 | 5, 8 | wa 396 |
. . . 4
wff (𝑥 ∈ ℂ ∧ 𝑦 ∈
ℂ) |
10 | | vw |
. . . . . . . . . . . . 13
setvar 𝑤 |
11 | 10 | cv 1538 |
. . . . . . . . . . . 12
class 𝑤 |
12 | | vv |
. . . . . . . . . . . . 13
setvar 𝑣 |
13 | 12 | cv 1538 |
. . . . . . . . . . . 12
class 𝑣 |
14 | 11, 13 | cop 4569 |
. . . . . . . . . . 11
class
〈𝑤, 𝑣〉 |
15 | 3, 14 | wceq 1539 |
. . . . . . . . . 10
wff 𝑥 = 〈𝑤, 𝑣〉 |
16 | | vu |
. . . . . . . . . . . . 13
setvar 𝑢 |
17 | 16 | cv 1538 |
. . . . . . . . . . . 12
class 𝑢 |
18 | | vf |
. . . . . . . . . . . . 13
setvar 𝑓 |
19 | 18 | cv 1538 |
. . . . . . . . . . . 12
class 𝑓 |
20 | 17, 19 | cop 4569 |
. . . . . . . . . . 11
class
〈𝑢, 𝑓〉 |
21 | 7, 20 | wceq 1539 |
. . . . . . . . . 10
wff 𝑦 = 〈𝑢, 𝑓〉 |
22 | 15, 21 | wa 396 |
. . . . . . . . 9
wff (𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) |
23 | | vz |
. . . . . . . . . . 11
setvar 𝑧 |
24 | 23 | cv 1538 |
. . . . . . . . . 10
class 𝑧 |
25 | | cplr 10623 |
. . . . . . . . . . . 12
class
+R |
26 | 11, 17, 25 | co 7277 |
. . . . . . . . . . 11
class (𝑤 +R
𝑢) |
27 | 13, 19, 25 | co 7277 |
. . . . . . . . . . 11
class (𝑣 +R
𝑓) |
28 | 26, 27 | cop 4569 |
. . . . . . . . . 10
class
〈(𝑤
+R 𝑢), (𝑣 +R 𝑓)〉 |
29 | 24, 28 | wceq 1539 |
. . . . . . . . 9
wff 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉 |
30 | 22, 29 | wa 396 |
. . . . . . . 8
wff ((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉) |
31 | 30, 18 | wex 1782 |
. . . . . . 7
wff
∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉) |
32 | 31, 16 | wex 1782 |
. . . . . 6
wff
∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉) |
33 | 32, 12 | wex 1782 |
. . . . 5
wff
∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉) |
34 | 33, 10 | wex 1782 |
. . . 4
wff
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉) |
35 | 9, 34 | wa 396 |
. . 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 7278 |
. 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 1539 |
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)〉))} |