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 |
|
dalemc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalemc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalemc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalempnes.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
6 |
|
dalempnes.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
7 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
8 |
1 4
|
dalempeb |
⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) ) |
9 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
10 |
1
|
dalemqea |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
11 |
1
|
dalemrea |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
12 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
13 |
12 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
14 |
9 10 11 13
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
15 |
12 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) ) |
16 |
7 8 14 15
|
syl3anc |
⊢ ( 𝜑 → 𝑃 ≤ ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) ) |
17 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
18 |
3 4
|
hlatjass |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) ) |
19 |
9 17 10 11 18
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) ) |
20 |
16 19
|
breqtrrd |
⊢ ( 𝜑 → 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
21 |
20 6
|
breqtrrdi |
⊢ ( 𝜑 → 𝑃 ≤ 𝑌 ) |