Step |
Hyp |
Ref |
Expression |
1 |
|
dihglbc.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihglbc.g |
⊢ 𝐺 = ( glb ‘ 𝐾 ) |
3 |
|
dihglbc.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
dihglbc.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dihglbc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
6 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
10 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
eqid |
⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) |
14 |
1 2 3 4 5 6 7 8 9 10 11 12 13
|
dihglbcpreN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ¬ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) ) |