Step |
Hyp |
Ref |
Expression |
0 |
|
cqg |
⊢ ~QG |
1 |
|
vr |
⊢ 𝑟 |
2 |
|
cvv |
⊢ V |
3 |
|
vi |
⊢ 𝑖 |
4 |
|
vx |
⊢ 𝑥 |
5 |
|
vy |
⊢ 𝑦 |
6 |
4
|
cv |
⊢ 𝑥 |
7 |
5
|
cv |
⊢ 𝑦 |
8 |
6 7
|
cpr |
⊢ { 𝑥 , 𝑦 } |
9 |
|
cbs |
⊢ Base |
10 |
1
|
cv |
⊢ 𝑟 |
11 |
10 9
|
cfv |
⊢ ( Base ‘ 𝑟 ) |
12 |
8 11
|
wss |
⊢ { 𝑥 , 𝑦 } ⊆ ( Base ‘ 𝑟 ) |
13 |
|
cminusg |
⊢ invg |
14 |
10 13
|
cfv |
⊢ ( invg ‘ 𝑟 ) |
15 |
6 14
|
cfv |
⊢ ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) |
16 |
|
cplusg |
⊢ +g |
17 |
10 16
|
cfv |
⊢ ( +g ‘ 𝑟 ) |
18 |
15 7 17
|
co |
⊢ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) |
19 |
3
|
cv |
⊢ 𝑖 |
20 |
18 19
|
wcel |
⊢ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 |
21 |
12 20
|
wa |
⊢ ( { 𝑥 , 𝑦 } ⊆ ( Base ‘ 𝑟 ) ∧ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ) |
22 |
21 4 5
|
copab |
⊢ { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ ( Base ‘ 𝑟 ) ∧ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ) } |
23 |
1 3 2 2 22
|
cmpo |
⊢ ( 𝑟 ∈ V , 𝑖 ∈ V ↦ { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ ( Base ‘ 𝑟 ) ∧ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ) } ) |
24 |
0 23
|
wceq |
⊢ ~QG = ( 𝑟 ∈ V , 𝑖 ∈ V ↦ { 〈 𝑥 , 𝑦 〉 ∣ ( { 𝑥 , 𝑦 } ⊆ ( Base ‘ 𝑟 ) ∧ ( ( ( invg ‘ 𝑟 ) ‘ 𝑥 ) ( +g ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ) } ) |