| 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 |
|
dalem13.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
| 6 |
|
dalem13.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
| 7 |
|
dalem13.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
| 8 |
|
dalem13.w |
⊢ 𝑊 = ( 𝑌 ∨ 𝐶 ) |
| 9 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
| 10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐾 ∈ HL ) |
| 11 |
1
|
dalemyeo |
⊢ ( 𝜑 → 𝑌 ∈ 𝑂 ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ∈ 𝑂 ) |
| 13 |
1
|
dalemzeo |
⊢ ( 𝜑 → 𝑍 ∈ 𝑂 ) |
| 14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑍 ∈ 𝑂 ) |
| 15 |
|
eqid |
⊢ ( LVols ‘ 𝐾 ) = ( LVols ‘ 𝐾 ) |
| 16 |
1 2 3 4 5 15 6 7 8
|
dalem9 |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑊 ∈ ( LVols ‘ 𝐾 ) ) |
| 17 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
| 18 |
1 5
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
| 19 |
1 4
|
dalemceb |
⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) ) |
| 20 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
| 21 |
20 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) → 𝑌 ≤ ( 𝑌 ∨ 𝐶 ) ) |
| 22 |
17 18 19 21
|
syl3anc |
⊢ ( 𝜑 → 𝑌 ≤ ( 𝑌 ∨ 𝐶 ) ) |
| 23 |
22 8
|
breqtrrdi |
⊢ ( 𝜑 → 𝑌 ≤ 𝑊 ) |
| 24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ≤ 𝑊 ) |
| 25 |
1 2 3 4 5 6 7 8
|
dalem8 |
⊢ ( 𝜑 → 𝑍 ≤ 𝑊 ) |
| 26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑍 ≤ 𝑊 ) |
| 27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ≠ 𝑍 ) |
| 28 |
2 3 5 15
|
2lplnj |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ∧ 𝑊 ∈ ( LVols ‘ 𝐾 ) ) ∧ ( 𝑌 ≤ 𝑊 ∧ 𝑍 ≤ 𝑊 ∧ 𝑌 ≠ 𝑍 ) ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 ) |
| 29 |
10 12 14 16 24 26 27 28
|
syl133anc |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 ) |