Step |
Hyp |
Ref |
Expression |
0 |
|
cocv |
⊢ ocv |
1 |
|
vh |
⊢ ℎ |
2 |
|
cvv |
⊢ V |
3 |
|
vs |
⊢ 𝑠 |
4 |
|
cbs |
⊢ Base |
5 |
1
|
cv |
⊢ ℎ |
6 |
5 4
|
cfv |
⊢ ( Base ‘ ℎ ) |
7 |
6
|
cpw |
⊢ 𝒫 ( Base ‘ ℎ ) |
8 |
|
vx |
⊢ 𝑥 |
9 |
|
vy |
⊢ 𝑦 |
10 |
3
|
cv |
⊢ 𝑠 |
11 |
8
|
cv |
⊢ 𝑥 |
12 |
|
cip |
⊢ ·𝑖 |
13 |
5 12
|
cfv |
⊢ ( ·𝑖 ‘ ℎ ) |
14 |
9
|
cv |
⊢ 𝑦 |
15 |
11 14 13
|
co |
⊢ ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) |
16 |
|
c0g |
⊢ 0g |
17 |
|
csca |
⊢ Scalar |
18 |
5 17
|
cfv |
⊢ ( Scalar ‘ ℎ ) |
19 |
18 16
|
cfv |
⊢ ( 0g ‘ ( Scalar ‘ ℎ ) ) |
20 |
15 19
|
wceq |
⊢ ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) |
21 |
20 9 10
|
wral |
⊢ ∀ 𝑦 ∈ 𝑠 ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) |
22 |
21 8 6
|
crab |
⊢ { 𝑥 ∈ ( Base ‘ ℎ ) ∣ ∀ 𝑦 ∈ 𝑠 ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) } |
23 |
3 7 22
|
cmpt |
⊢ ( 𝑠 ∈ 𝒫 ( Base ‘ ℎ ) ↦ { 𝑥 ∈ ( Base ‘ ℎ ) ∣ ∀ 𝑦 ∈ 𝑠 ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) } ) |
24 |
1 2 23
|
cmpt |
⊢ ( ℎ ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ ℎ ) ↦ { 𝑥 ∈ ( Base ‘ ℎ ) ∣ ∀ 𝑦 ∈ 𝑠 ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) } ) ) |
25 |
0 24
|
wceq |
⊢ ocv = ( ℎ ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ ℎ ) ↦ { 𝑥 ∈ ( Base ‘ ℎ ) ∣ ∀ 𝑦 ∈ 𝑠 ( 𝑥 ( ·𝑖 ‘ ℎ ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ ℎ ) ) } ) ) |