Step |
Hyp |
Ref |
Expression |
1 |
|
dalema.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) ) |
2 |
|
dalemb.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
dalemb.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
5 |
1
|
dalemtea |
⊢ ( 𝜑 → 𝑇 ∈ 𝐴 ) |
6 |
1
|
dalemuea |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
8 |
7 2 3
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) → ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) |
9 |
4 5 6 8
|
syl3anc |
⊢ ( 𝜑 → ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) |