Step |
Hyp |
Ref |
Expression |
0 |
|
clspn |
⊢ LSpan |
1 |
|
vw |
⊢ 𝑤 |
2 |
|
cvv |
⊢ V |
3 |
|
vs |
⊢ 𝑠 |
4 |
|
cbs |
⊢ Base |
5 |
1
|
cv |
⊢ 𝑤 |
6 |
5 4
|
cfv |
⊢ ( Base ‘ 𝑤 ) |
7 |
6
|
cpw |
⊢ 𝒫 ( Base ‘ 𝑤 ) |
8 |
|
vt |
⊢ 𝑡 |
9 |
|
clss |
⊢ LSubSp |
10 |
5 9
|
cfv |
⊢ ( LSubSp ‘ 𝑤 ) |
11 |
3
|
cv |
⊢ 𝑠 |
12 |
8
|
cv |
⊢ 𝑡 |
13 |
11 12
|
wss |
⊢ 𝑠 ⊆ 𝑡 |
14 |
13 8 10
|
crab |
⊢ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } |
15 |
14
|
cint |
⊢ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } |
16 |
3 7 15
|
cmpt |
⊢ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } ) |
17 |
1 2 16
|
cmpt |
⊢ ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } ) ) |
18 |
0 17
|
wceq |
⊢ LSpan = ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } ) ) |