Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.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 |
|
dalem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalem.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalem.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem.ps |
⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
6 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
7 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐾 ∈ HL ) |
8 |
5
|
dalemccea |
⊢ ( 𝜓 → 𝑐 ∈ 𝐴 ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 ) |
10 |
5
|
dalemddea |
⊢ ( 𝜓 → 𝑑 ∈ 𝐴 ) |
11 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 ) |
12 |
5
|
dalemccnedd |
⊢ ( 𝜓 → 𝑐 ≠ 𝑑 ) |
13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑐 ≠ 𝑑 ) |
14 |
|
eqid |
⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 ) |
15 |
3 4 14
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ 𝑐 ≠ 𝑑 ) → ( 𝑐 ∨ 𝑑 ) ∈ ( LLines ‘ 𝐾 ) ) |
16 |
7 9 11 13 15
|
syl31anc |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑑 ) ∈ ( LLines ‘ 𝐾 ) ) |