Step |
Hyp |
Ref |
Expression |
1 |
|
pclss.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
pclss.c |
⊢ 𝑈 = ( PCl ‘ 𝐾 ) |
3 |
|
sstr2 |
⊢ ( 𝑋 ⊆ 𝑌 → ( 𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦 ) ) |
4 |
3
|
3ad2ant2 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦 ) ) |
5 |
4
|
adantr |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ) → ( 𝑌 ⊆ 𝑦 → 𝑋 ⊆ 𝑦 ) ) |
6 |
5
|
ss2rabdv |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ⊆ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } ) |
7 |
|
intss |
⊢ ( { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ⊆ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } → ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } ⊆ ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ) |
8 |
6 7
|
syl |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } ⊆ ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ) |
9 |
|
simp1 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → 𝐾 ∈ 𝑉 ) |
10 |
|
sstr |
⊢ ( ( 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → 𝑋 ⊆ 𝐴 ) |
11 |
10
|
3adant1 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → 𝑋 ⊆ 𝐴 ) |
12 |
|
eqid |
⊢ ( PSubSp ‘ 𝐾 ) = ( PSubSp ‘ 𝐾 ) |
13 |
1 12 2
|
pclvalN |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑋 ) = ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } ) |
14 |
9 11 13
|
syl2anc |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑋 ) = ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑋 ⊆ 𝑦 } ) |
15 |
1 12 2
|
pclvalN |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑌 ) = ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ) |
16 |
15
|
3adant2 |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑌 ) = ∩ { 𝑦 ∈ ( PSubSp ‘ 𝐾 ) ∣ 𝑌 ⊆ 𝑦 } ) |
17 |
8 14 16
|
3sstr4d |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑈 ‘ 𝑋 ) ⊆ ( 𝑈 ‘ 𝑌 ) ) |