Step |
Hyp |
Ref |
Expression |
1 |
|
dihmeetlem4.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
dihmeetlem4.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dihmeetlem4.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
dihmeetlem4.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dihmeetlem4.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
dihmeetlem4.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
7 |
|
dihmeetlem4.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihmeetlem4.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
9 |
|
eqid |
⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) |
10 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
12 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
14 |
|
eqid |
⊢ ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) = ( ℎ ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ 𝐵 ) ) |
15 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
dihmeetlem4preN |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝐼 ‘ 𝑄 ) ∩ ( 𝐼 ‘ ( 𝑋 ∧ 𝑊 ) ) ) = { 0 } ) |