Step |
Hyp |
Ref |
Expression |
1 |
|
ltrniotaval.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
ltrniotaval.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
ltrniotaval.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
ltrniotaval.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
ltrniotaval.f |
⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
7 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) |
10 |
|
eqid |
⊢ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
11 |
|
eqid |
⊢ ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) |
12 |
|
eqid |
⊢ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ) → 𝑦 = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ) → 𝑦 = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) ) |
13 |
6 1 7 8 2 3 9 10 11 12 4 5
|
cdlemg1fvawlemN |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑅 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑅 ) ≤ 𝑊 ) ) |