Step |
Hyp |
Ref |
Expression |
1 |
|
lspval.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspval.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lspval.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
1 2 3
|
lspfval |
⊢ ( 𝑊 ∈ LMod → 𝑁 = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ) |
5 |
4
|
fveq1d |
⊢ ( 𝑊 ∈ LMod → ( 𝑁 ‘ 𝑈 ) = ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) = ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) ) |
7 |
|
eqid |
⊢ ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) = ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) |
8 |
|
sseq1 |
⊢ ( 𝑠 = 𝑈 → ( 𝑠 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑡 ) ) |
9 |
8
|
rabbidv |
⊢ ( 𝑠 = 𝑈 → { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } = { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ) |
10 |
9
|
inteqd |
⊢ ( 𝑠 = 𝑈 → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ) |
11 |
|
simpr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → 𝑈 ⊆ 𝑉 ) |
12 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
13 |
12
|
elpw2 |
⊢ ( 𝑈 ∈ 𝒫 𝑉 ↔ 𝑈 ⊆ 𝑉 ) |
14 |
11 13
|
sylibr |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → 𝑈 ∈ 𝒫 𝑉 ) |
15 |
1 2
|
lss1 |
⊢ ( 𝑊 ∈ LMod → 𝑉 ∈ 𝑆 ) |
16 |
|
sseq2 |
⊢ ( 𝑡 = 𝑉 → ( 𝑈 ⊆ 𝑡 ↔ 𝑈 ⊆ 𝑉 ) ) |
17 |
16
|
rspcev |
⊢ ( ( 𝑉 ∈ 𝑆 ∧ 𝑈 ⊆ 𝑉 ) → ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ) |
18 |
15 17
|
sylan |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ) |
19 |
|
intexrab |
⊢ ( ∃ 𝑡 ∈ 𝑆 𝑈 ⊆ 𝑡 ↔ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ∈ V ) |
20 |
18 19
|
sylib |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ∈ V ) |
21 |
7 10 14 20
|
fvmptd3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( ( 𝑠 ∈ 𝒫 𝑉 ↦ ∩ { 𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡 } ) ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ) |
22 |
6 21
|
eqtrd |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ) → ( 𝑁 ‘ 𝑈 ) = ∩ { 𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡 } ) |