Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatcc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
dihjatcc.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
3 |
|
dihjatcc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dihjatcc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dihjatcc.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dihjatcc.s |
⊢ ⊕ = ( LSSum ‘ 𝑈 ) |
7 |
|
dihjatcc.i |
⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
dihjatcc.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
9 |
|
dihjatcc.p |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
10 |
|
dihjatcc.q |
⊢ ( 𝜑 → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
11 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
12 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
13 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) |
14 |
|
eqid |
⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) |
15 |
|
eqid |
⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
16 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
17 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
18 |
|
eqid |
⊢ ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑃 ) = ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑃 ) |
19 |
|
eqid |
⊢ ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) = ( ℩ 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑑 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑄 ) |
20 |
|
eqid |
⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑑 ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ◡ ( 𝑎 ‘ 𝑑 ) ) ) |
21 |
|
eqid |
⊢ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) = ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( I ↾ ( Base ‘ 𝐾 ) ) ) |
22 |
|
eqid |
⊢ ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑑 ) ∘ ( 𝑏 ‘ 𝑑 ) ) ) ) = ( 𝑎 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) , 𝑏 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( 𝑑 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ↦ ( ( 𝑎 ‘ 𝑑 ) ∘ ( 𝑏 ‘ 𝑑 ) ) ) ) |
23 |
11 1 2 3 12 4 5 6 7 13 8 9 10 14 15 16 17 18 19 20 21 22
|
dihjatcclem4 |
⊢ ( 𝜑 → ( 𝐼 ‘ ( ( 𝑃 ∨ 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ⊆ ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ) |
24 |
11 1 2 3 12 4 5 6 7 13 8 9 10 23
|
dihjatcclem2 |
⊢ ( 𝜑 → ( 𝐼 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐼 ‘ 𝑃 ) ⊕ ( 𝐼 ‘ 𝑄 ) ) ) |