Step |
Hyp |
Ref |
Expression |
1 |
|
lspval.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspval.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lspval.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
elex |
⊢ ( 𝑊 ∈ 𝑋 → 𝑊 ∈ V ) |
5 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ 𝑤 ) = ( Base ‘ 𝑊 ) ) |
6 |
5 1
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( Base ‘ 𝑤 ) = 𝑉 ) |
7 |
6
|
pweqd |
⊢ ( 𝑤 = 𝑊 → 𝒫 ( Base ‘ 𝑤 ) = 𝒫 𝑉 ) |
8 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( LSubSp ‘ 𝑤 ) = ( LSubSp ‘ 𝑊 ) ) |
9 |
8 2
|
eqtr4di |
⊢ ( 𝑤 = 𝑊 → ( LSubSp ‘ 𝑤 ) = 𝑆 ) |
10 |
9
|
rabeqdv |
⊢ ( 𝑤 = 𝑊 → { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } = { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) |
11 |
10
|
inteqd |
⊢ ( 𝑤 = 𝑊 → ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) |
12 |
7 11
|
mpteq12dv |
⊢ ( 𝑤 = 𝑊 → ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } ) = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ) |
13 |
|
df-lsp |
⊢ LSpan = ( 𝑤 ∈ V ↦ ( 𝑠 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ∩ { 𝑡 ∈ ( LSubSp ‘ 𝑤 ) ∣ 𝑠 ⊆ 𝑡 } ) ) |
14 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
15 |
14
|
pwex |
⊢ 𝒫 𝑉 ∈ V |
16 |
15
|
mptex |
⊢ ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ∈ V |
17 |
12 13 16
|
fvmpt |
⊢ ( 𝑊 ∈ V → ( LSpan ‘ 𝑊 ) = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ) |
18 |
4 17
|
syl |
⊢ ( 𝑊 ∈ 𝑋 → ( LSpan ‘ 𝑊 ) = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ) |
19 |
3 18
|
syl5eq |
⊢ ( 𝑊 ∈ 𝑋 → 𝑁 = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ) |