Detailed syntax breakdown of Definition df-mqqs
Step | Hyp | Ref
| Expression |
1 | | cmq 7232 |
. 2
class
·Q |
2 | | vx |
. . . . . . 7
setvar 𝑥 |
3 | 2 | cv 1347 |
. . . . . 6
class 𝑥 |
4 | | cnq 7229 |
. . . . . 6
class
Q |
5 | 3, 4 | wcel 2141 |
. . . . 5
wff 𝑥 ∈
Q |
6 | | vy |
. . . . . . 7
setvar 𝑦 |
7 | 6 | cv 1347 |
. . . . . 6
class 𝑦 |
8 | 7, 4 | wcel 2141 |
. . . . 5
wff 𝑦 ∈
Q |
9 | 5, 8 | wa 103 |
. . . 4
wff (𝑥 ∈ Q ∧
𝑦 ∈
Q) |
10 | | vw |
. . . . . . . . . . . . . 14
setvar 𝑤 |
11 | 10 | cv 1347 |
. . . . . . . . . . . . 13
class 𝑤 |
12 | | vv |
. . . . . . . . . . . . . 14
setvar 𝑣 |
13 | 12 | cv 1347 |
. . . . . . . . . . . . 13
class 𝑣 |
14 | 11, 13 | cop 3584 |
. . . . . . . . . . . 12
class
〈𝑤, 𝑣〉 |
15 | | ceq 7228 |
. . . . . . . . . . . 12
class
~Q |
16 | 14, 15 | cec 6507 |
. . . . . . . . . . 11
class
[〈𝑤, 𝑣〉]
~Q |
17 | 3, 16 | wceq 1348 |
. . . . . . . . . 10
wff 𝑥 = [〈𝑤, 𝑣〉]
~Q |
18 | | vu |
. . . . . . . . . . . . . 14
setvar 𝑢 |
19 | 18 | cv 1347 |
. . . . . . . . . . . . 13
class 𝑢 |
20 | | vf |
. . . . . . . . . . . . . 14
setvar 𝑓 |
21 | 20 | cv 1347 |
. . . . . . . . . . . . 13
class 𝑓 |
22 | 19, 21 | cop 3584 |
. . . . . . . . . . . 12
class
〈𝑢, 𝑓〉 |
23 | 22, 15 | cec 6507 |
. . . . . . . . . . 11
class
[〈𝑢, 𝑓〉]
~Q |
24 | 7, 23 | wceq 1348 |
. . . . . . . . . 10
wff 𝑦 = [〈𝑢, 𝑓〉]
~Q |
25 | 17, 24 | wa 103 |
. . . . . . . . 9
wff (𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q
) |
26 | | vz |
. . . . . . . . . . 11
setvar 𝑧 |
27 | 26 | cv 1347 |
. . . . . . . . . 10
class 𝑧 |
28 | | cmpq 7226 |
. . . . . . . . . . . 12
class
·pQ |
29 | 14, 22, 28 | co 5850 |
. . . . . . . . . . 11
class
(〈𝑤, 𝑣〉
·pQ 〈𝑢, 𝑓〉) |
30 | 29, 15 | cec 6507 |
. . . . . . . . . 10
class
[(〈𝑤, 𝑣〉
·pQ 〈𝑢, 𝑓〉)]
~Q |
31 | 27, 30 | wceq 1348 |
. . . . . . . . 9
wff 𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)]
~Q |
32 | 25, 31 | wa 103 |
. . . . . . . 8
wff ((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
) |
33 | 32, 20 | wex 1485 |
. . . . . . 7
wff
∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
) |
34 | 33, 18 | wex 1485 |
. . . . . 6
wff
∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
) |
35 | 34, 12 | wex 1485 |
. . . . 5
wff
∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
) |
36 | 35, 10 | wex 1485 |
. . . 4
wff
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
) |
37 | 9, 36 | wa 103 |
. . 3
wff ((𝑥 ∈ Q ∧
𝑦 ∈ Q)
∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
)) |
38 | 37, 2, 6, 26 | coprab 5851 |
. 2
class
{〈〈𝑥,
𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
))} |
39 | 1, 38 | wceq 1348 |
1
wff
·Q = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) ∧
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~Q ∧
𝑦 = [〈𝑢, 𝑓〉] ~Q ) ∧
𝑧 = [(〈𝑤, 𝑣〉 ·pQ
〈𝑢, 𝑓〉)] ~Q
))} |