Step |
Hyp |
Ref |
Expression |
0 |
|
c0g |
⊢ 0g |
1 |
|
vg |
⊢ 𝑔 |
2 |
|
cvv |
⊢ V |
3 |
|
ve |
⊢ 𝑒 |
4 |
3
|
cv |
⊢ 𝑒 |
5 |
|
cbs |
⊢ Base |
6 |
1
|
cv |
⊢ 𝑔 |
7 |
6 5
|
cfv |
⊢ ( Base ‘ 𝑔 ) |
8 |
4 7
|
wcel |
⊢ 𝑒 ∈ ( Base ‘ 𝑔 ) |
9 |
|
vx |
⊢ 𝑥 |
10 |
|
cplusg |
⊢ +g |
11 |
6 10
|
cfv |
⊢ ( +g ‘ 𝑔 ) |
12 |
9
|
cv |
⊢ 𝑥 |
13 |
4 12 11
|
co |
⊢ ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) |
14 |
13 12
|
wceq |
⊢ ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 |
15 |
12 4 11
|
co |
⊢ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) |
16 |
15 12
|
wceq |
⊢ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 |
17 |
14 16
|
wa |
⊢ ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) |
18 |
17 9 7
|
wral |
⊢ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) |
19 |
8 18
|
wa |
⊢ ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) |
20 |
19 3
|
cio |
⊢ ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ) |
21 |
1 2 20
|
cmpt |
⊢ ( 𝑔 ∈ V ↦ ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ) ) |
22 |
0 21
|
wceq |
⊢ 0g = ( 𝑔 ∈ V ↦ ( ℩ 𝑒 ( 𝑒 ∈ ( Base ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ( ( 𝑒 ( +g ‘ 𝑔 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝑔 ) 𝑒 ) = 𝑥 ) ) ) ) |